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Contents
Distinguishing numbers of finite 4-valent vertex-transitive graphs
Florian Lehner, Gabriel Verret........................173
Divergence zero quaternionic vector fields and Hamming graphs
Jasna Prezelj, Fabio Vlacci..........................189
A Mobius-type gluing technique for obtaining edge-critical graphs
Simona Bonvicini, Andrea Vietri.......................209
On resolving sets in the point-line incidence graph of PG(n, q)
Daniele Bartoli, Gyorgy Kiss, Stefano Marcugini, Fernanda Pambianco ..231
Building maximal green sequences via component preserving mutations
Eric Bucher, John Machacek, Evan Runburg, Abe Yeck, Ethan Zewde . . . 249
The Cayley isomorphism property for the group Cf x Cp
Grigory Ryabov................................277
On the divisibility of binomial coefficients
Silvia Casacuberta ..............................297
Distance-balanced graphs and travelling salesman problems
Matteo Cavaleri, Alfredo Donno ....................... 311
On generalized truncations of complete graphs
Xue Wang, Fu-Gang Yin, Jin-Xin Zhou...................325
Properties of double Roman domination on cardinal products of graphs
Antoaneta Klobucar, Ana Klobucar......................337
On the Smith normal form of the Varchenko matrix
Tommy Wuxing Cai, Yue Chen, Lili Mu...................351
Frobenius groups which are the automorphism groups of orientably-regular maps
Hai-Peng Qu, Yan Wang, Kai Yuan......................363
Volume 19, Number 2, Fall/Winter 2020, Pages 173-374
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ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 19 (2020) 173-187
https://doi.org/10.26493/1855-3974.1849.148
(Also available at http://amc-journal.eu)
Distinguishing numbers of finite 4-valent vertex-transitive graphs*
*
Florian Lehner t ©
Institute of Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
Gabriel Verret * ©
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
Received 8 November 2018, accepted 25 February 2020, published online 13 November 2020
The distinguishing number of a graph G is the smallest k such that G admits a k-colouring for which the only colour-preserving automorphism of G is the identity. We determine the distinguishing number of finite 4-valent vertex-transitive graphs. We show that, apart from one infinite family and finitely many examples, they all have distinguishing number 2.
Keywords: Vertex colouring, symmetry breaking in graph, distinguishing number, vertex-transitive graphs.
Math. Subj. Class. (2020): 05C15, 05E18
1 Introduction
All graphs in this paper will be finite. A distinguishing colouring of a graph is a colouring which is not preserved by any non-identity automorphism. The distinguishing number D(G) of a graph G is the least number of colours needed for a distinguishing colouring of the vertices of G. These concepts were first introduced by Albertson and Collins [1] and have since received considerable attention.
*	We would like to thank the anonymous referees for a number of helpful suggestions. t Supported by the Austrian Science Fund (FWF), grant J 3850-N32.
*	Gabriel Verret is grateful to the N.Z. Marsden Fund for its support (via grant UOA1824).
E-mail addresses: mail@florian-lehner.net (Florian Lehner), g.verret@auckland.ac.nz (Gabriel Verret)
Abstract
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
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It is an easy observation that a graph has distinguishing number 1 if and only if its automorphism group is trivial. Hence, by [6] almost all graphs have distinguishing number 1. This obviously is not true for non-trivial vertex-transitive graphs which always have non-trivial automorphisms. However, it seems that the vast majority of vertex-transitive graphs still have the lowest possible distinguishing number, namely 2. Hence let us call a vertex-transitive graph exceptional if its distinguishing number is not equal to 2.
One of the most interesting results concerning distinguishing numbers of vertex-transitive graphs is that, apart from the complete and edgeless graphs, there are only finitely many exceptional vertex-primitive graphs [3, 14]. It is only natural to ask whether something similar holds for vertex-transitive graphs as well. As a first step, Huning et al. recently determined the exceptional 3-valent vertex-transitive graphs and their distinguishing numbers.
Theorem 1.1 ([7, Corollary 2.2]). The exceptional connected 3-valent vertex-transitive graphs are
1.	K4 and K3,3, with distinguishing number 4, and
2.	Q3 = K4 x K2 and the Petersen graph, with distinguishing number 3.
This result shows that there are only finitely many connected 3-valent vertex-transitive exceptional graphs. This is not true for 4-valent graphs, as shown by the following family of graphs. For n > 3, the wreath graph Wn is the lexicographic product Cn [2Ki] of a cycle of length n with an edgeless graph of order 2, see Figure 1.
Figure 1: The wreath graph W10.
It is easy to see that wreath graphs form an infinite family of connected exceptional 4-valent vertex-transitive graphs, thus providing a negative answer to [7, Question 2]. Our main result shows that this is the only such family, that is, apart from the wreath graphs, there are only finitely many connected exceptional 4-valent vertex-transitive graphs.
Theorem 1.2. The exceptional connected 4-valent vertex-transitive graphs are
1.	K5 and K4,4 = W4, with distinguishing number 5, and
2.	K3 □ K3, K4 □ K2, K5 x K2 and Wn for some n > 3, n = 4, with distinguishing number 3.
In particular, there is no example with distinguishing number 4. This leads us to the following question.
F. Lehner and G. Verret: Distinguishing numbers of finite 4-valent vertex-transitive graphs 175
Question 1.3. For A > 5, is there a connected A-valent vertex-transitive graph G with
D(G) = A?
More generally, one could ask about "gaps" in the set of distinguishing numbers of connected A-valent vertex-transitive graphs, as a subset of {2,..., A + 1}.
Using lexicographic products, it is not hard to construct infinite families of connected exceptional vertex-transitive graphs with fixed valency.
Example 1.4. Let Hi be a connected vertex-transitive graph of valency Ai and let H2 be a vertex-transitive graph of valency A2 on n2 vertices. Then the lexicographic product Hi[H2] is connected, has valency Ain2 + A2 and its distinguishing number is at least D(H2) + 1. For an infinite family of examples that are not lexicographic products, note that, for every n > 3 and every d > 2, the graph (Cn[d2Ki]) □ K2 has valency 2d2 + 1 and distinguishing number strictly greater than d.
We hence pose the following (informal) problem.
Problem 1.5. Is there a "natural small family" F of exceptional graphs such that, for every positive integer k, all but finitely many k-valent connected exceptional vertex-transitive graphs are contained in F ?
2 Definitions and auxiliary results
Throughout this paper, all graphs are assumed to be finite and simple. Graph theoretic notions that are not explicitly defined will be taken from [5].
An automorphism of a graph is an adjacency preserving permutation of its vertices. The group of all automorphisms of a graph G is denoted by Aut G. We say that a graph is vertex-transitive if its automorphism group is transitive (that is, for every pair of vertices, there exists an automorphism mapping the first to the second).
An arc in a graph G is an ordered pair of adjacent vertices, or equivalently, a walk of length 1 in G. An s-arc is a non-backtracking walk of length s in G, i.e. a sequence of vertices v0,... ,vs where vi is adjacent to vi+i for 0 < i < s - 1, and vi-i = Vi+i for 1 < i < s - 1. The automorphism group Aut G acts on the set of edges, arcs, and s-arcs of G in an obvious way. Call a graph edge-transitive, arc-transitive, or s-arc-transitive, if the action of Aut G on edges, arcs, or s-arcs is transitive, respectively. Analogously define arc-regular and s-arc-regular.
The local group at a vertex v is the permutation group induced by the stabiliser of v acting on its neighbourhood N(v). Note that, for vertex-transitive graphs, this does not depend on the choice of v (up to permutation equivalence). We say that a graph is locally r, if the local group is isomorphic to r.
A graph G is called k-connected if it remains connected after removing any set of at most k - 1 vertices and all incident edges, and k-edge connected if it remains connected after removing any set of k edges. The following result about the connectivity of vertex-transitive graphs is due to Watkins [16].
Lemma 2.1. A vertex-transitive graph with valency r is at least 2r -connected.
If we impose additional properties on the set of vertices to be removed, then we can remove much larger sets without disconnecting the graph. The following lemma follows easily from results in [15].
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Lemma 2.2. If G is a k-valent vertex-transitive graph with k > 4 and girth g > 5, then there is a g-cycle C in G such that G — C is 2-edge connected.
Proof. By [15, Theorem 4.5], there is a g-cycle C such that the edges with one endpoint in C and the other endpoint in H := G — C form a minimum (w.r.t. cardinality) cut separating two cycles in G. Assume that H was not 2-edge connected and let e be a cut-edge of H. Let A and B be the two components of H — e. By [15, Lemma 3.3], the minimum degree of H is 2, so A and B each contain at most one vertex of degree 1, and thus there are cycles in both components. Now either the cut separating A U C from B, or the cut separating B U C from A contains strictly fewer edges than the cut separating C from H, contradicting the minimality.	□
We will also need the notion of distinguishing index D'(G) of a graph G, which is the least number of colours needed for a distinguishing colouring of the edges of G. Here are a few results giving upper bounds on D'(G). The first two are Theorems 2.8 and 3.2 in [11].
Theorem 2.3. Let G be a connected graph that is neither a symmetric nor a bisymmetric tree. If the maximum degree A(G) of G is at least 3, then D'(G) < A(G) — 1 unless G is K4 or K3 3.
Theorem 2.4. If G is a graph of order at least 7 with a Hamiltonian path, then D' (G) < 2.
Lemma 2.5. If G is a connected graph on 5 or more vertices, then AutL(G) is permu-tationally equivalent to Aut G with its natural action on E(G). Furthermore, in this case D'(G) < D(G), unless G is a tree.
Proof. The first part is a variant of Whitney's theorem due to Jung [8], the second part
follows by applying [10, Theorem 1.3] to a distinguishing colouring with D(G) colours.
□
In the remainder of this section, we discuss some known results on distinguishing numbers and determine the distinguishing numbers of several graphs that will occur in the proof of Theorem 1.2. The following lemma gives a general bound on distinguishing numbers and was independently proved in [4] and [9].
Lemma 2.6. If G is a connected graph with maximum degree A, then D(G) < A + 1, with equality if and only if G is either C5, or Kn or Kn,n for some n > 1.
For n > 2, we define a family of graphs Cn,K33 as follows. For 1 < i < n, let Hi be disjoint copies of K3,3 with bipartition V(Hi) = Xi uYj. Let Cn,K33 be the graph obtained from this collection by adding a matching between Xi and Yi+1 for 1 < i < n — 1, and between Xn and Yi, see Figure 2.
Lemma 2.7. The following graphs have distinguishing number at most 2:
(1)	The line graph of every non-exceptional 3-valent graph;
(2)	The line graphs of the following graphs: the Petersen graph, Q3, K3 □ K3, K5 x K2, and Wn for every n > 3;
(3)	The bipartite complement of the Heawood graph;
(4)	The 4-dimensional hypercube Q4;
F. Lehner and G. Verret: Distinguishing numbers of finite 4-valent vertex-transitive graphs 177
(5)	The (4,6)-cage, and
(6)	The graph Cn K3 3 for n > 2.
Figure 2: Distinguishing colouring of C6 K3 3.
Proof. Lemma 2.5 immediately implies (1).
For (2), it suffices to observe that all the base graphs have at least 7 vertices and a Hamiltonian path, and then apply Theorem 2.4 and Lemma 2.5 .
For (3) note that the bipartite complement of the Heawood graph has the same automorphism group as the Heawood graph and thus also the same distinguishing number. By Theorem 1.1, this distinguishing number is 2.
(4) follows from [2], where distinguishing numbers of all hypercubes were determined. For the proof of (5) first note that the (4, 6)-cage is bipartite and any two vertices in each of its parts have exactly one neighbour in common. Let v be any vertex, let Vj for 1 < i < 4 be the neighbours of v, and let Vj 1 < i < 3 be the neighbours of Vj.
Colour v white, for 1 < i < 4 colour Vj black, and colour Vj black if i < j and white otherwise. Finally colour the common neighbours of v22 and v32, and v22 and v33 black and all other vertices at distance 3 from v white, see Figure 3.
v
Figure 3: Colouring of the (4, 6)-cage, all vertices at distance 3 from v not shown in the picture are coloured white.
Let y be a colour preserving automorphism. Then 7 must fix v, since it is the only white vertex with 4 black neighbours. Furthermore 7 must fix all neighbours of v since they have a different number of black neighbours. It must also fix the two black vertices at distance
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3 from v for the same reason. Now it is easy to see that y has to fix all vertices at distance
2	from v and hence it is the identity.
For (6), consider the colouring shown in Figure 2. Note that the automorphism group has two orbits on edges: those that belong to a copy of K3 3, and those that don't, which we call matching edges. There is a unique matching edge both of whose endpoints are coloured white. Every colour preserving automorphism must fix this edge and the matching it is contained in. The colours on the remaining edges in this matching make sure that every colour preserving automorphism must fix this matching pointwise, and thus must fix every matching between two copies of K3 3 setwise. It is now easy to see that a colour preserving automorphism fixes all vertices of C6,K3,3. Finally note that this colouring can be generalised to a colouring of Cn,K3 3 for any number n > 2.	□
3	The proof of Theorem 1.2
In this section, we prove our main result. Determining the distinguishing numbers of the exceptional graphs is straightforward and will be left to the reader.
To show that the remaining graphs have distinguishing number 2, we distinguish cases according to the local group of A := Aut G. Define the type of an edge uv as the size of the orbit of u under the action of the local group at v. By the orbit-stabiliser lemma, this is the index of Auv in Av. Since by vertex transitivity |Av | = |Au|, this also shows that the type is well-defined, i.e. it does not depend on the endpoint of the edge.
Note that since the orbits of the local group at v partition the neighbourhood of v the types of edges incident to v correspond to a partition of 4. Since G is vertex-transitive, this partition is the same for every vertex. Since the only partitions of 4 that do not contain a part of size 1 are (2, 2) and (4), we split up the proof of Theorem 1.2 into the following three cases:
1.	There are edges of type 1. This case is treated in Section 3.1.
2.	All edges have type 2. This is treated in Section 3.2.
4. All edges have type 4, and hence G is arc-transitive. For this case, see Section 3.3. 3.1 Graphs with edges of type 1
Let Gt> 2 be the graph obtained from G by removing all edges of type 1. Note that the components of Gt>2 form a system of imprimitivity for A. We will need the following results.
Lemma 3.1. Assume that every vertex of G is incident to a unique type 1-edge, Gt>2 is not connected, and any two components of Gt>2 are connected by at most one type 1-edge. Then G has a distinguishing 2-colouring.
Proof. Let k be the number of vertices in a component of Gt>2. Consider the graph H obtained from G by contracting every component of Gt>2 to a single vertex. By our assumptions, H is a k-regular graph and it follows from Lemma 2.6 that its distinguishing number is at most k +1. Let c! be a distinguishing colouring of H with colours {0,1,..., k}. We now colour G in the following way: in every component of Gt>2, we colour as many vertices black as the colour of the corresponding vertex of H suggests.
F. Lehner and G. Verret: Distinguishing numbers of finite 4-valent vertex-transitive graphs 179
Since c' is distinguishing, any automorphism which preserves the resulting colouring has to fix all components of Gt>2 setwise. As every type 1 edge is uniquely identified by the components it connects, each type 1 edge and hence also every vertex must be fixed by
Lemma 3.2. Let G be a connected vertex-transitive graph. Assume that Gt>2 is not connected, let H be a component of Gt>2 and let v e H. If H admits a 2-colouring c' such that the only automorphism of H fixing v and preserving c' is the identity, then G has a distinguishing 2-colouring.
Proof. Denote the components of Gt>2 by Hi,..., HR. Note that each Hj is isomorphic to H. Let vi e Hi. Note that the graph obtained from G by contracting the components Hi,..., Hr is connected and vertex-transitive and thus at least 2-connected. Hence G-Hi is connected, and thus (G - Hi) + vi is connected as well.
For i e {2,..., R}, pick some shortest path from Hj to vi in (G - Hi) + vi and let vj and ej be the first vertex and edge of this path, respectively. Without loss of generality we may assume that the number of black vertices in c' is not exactly one—otherwise change the colour of v to obtain a colouring with this property. Let nj: H ^ Hj be an isomorphism which maps v to vj. Such an isomorphism exists because G (and thus also H) is vertex-transitive. Now define a colouring c of G by
Let y be an automorphism of G preserving c. We show that y fixes every vertex and thus c is distinguishing.
First, note that y must fix vi, since vi is the only black vertex in Hi which in turn is the only component with a unique black vertex.
Next we show that, for i = 1, every Hj must be fixed pointwise by y. Assume not. Let Hj be a component such that the distance from Hj to vi is minimal, among the components that are not fixed pointwise. The endpoint uj of ej which does not lie in Hj is either vi, or it lies in some component Hj which is closer to vi. Hence uj is fixed by y. Since ei has type 1, y must also fix vj and thus induce an automorphism of Hj. By hypothesis, this induced automorphism is trivial and thus y fixes Hj pointwise.
Finally, let x e Hi - vi. Then x is incident to an edge of type 1 which connects Hi to a different component Hj. Since the other endpoint of this edge is fixed by y, the same must be true for x.	□
Corollary 3.3. Let G be a connected, vertex-transitive graph and let H be a component of Gt>2. If H has a distinguishing 2-colouring, then so does G.
Proof. If H is the only component of Gt>2, then a distinguishing colouring of H is also distinguishing for G, otherwise apply Lemma 3.2.	□
Theorem 3.4. Let G be a connected 4-valent vertex-transitive graph containing edges of type 1. Then D(G) = 2, unless G is K4 □ K2.
every colour-preserving automorphism.
□
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Proof. If all edges are of type 1, then Av = 1 and thus colouring one vertex black and all other vertices white yields a distinguishing colouring.
Next assume that the local group has two orbits of size 1 and one orbit of size 2. In this case Gt>2 is a union of cycles. If there is only one such cycle, then it must have length 6 or more, and hence G is 2-distinguishable by Corollary 3.3. If there is more than one, then the conditions of Lemma 3.2 are satisfied.
Finally consider the case where the local group has one orbit of size 1 and one orbit of size 3. All components of Gt>2 are isomorphic to some 3-regular vertex-transitive graph G'. Also note that the induced action of A on G' is arc-transitive.
If G' has distinguishing number 2, then we can apply Corollary 3.3 to obtain a distinguishing 2-colouring of G. By Theorem 1.1, the only other possibility is that G' is isomorphic to one of K4, K33, Q3 or the Petersen graph.
If Gt>2 is connected, then G is obtained from G' by adding edges of type 1. Since A is arc-transitive on G', no edge of type 1 can connect two neighbours (in G') of the same vertex. Otherwise any two neighbours of this vertex would have to be connected by an edge, contradicting the fact that each vertex of G is adjacent to only one edge of type 1 . Hence an edge of type 1 can't connect vertices at distance at most 2 in G'. This rules out K4, K3 3 and the Petersen graph as possibilities for G', since they have diameter at most 2. The only way to add edges with respect to this constraint in the cube Q3 yields G = K4 4 which does not contain edges of type 1 .
Thus we can assume that Gt>2 is not connected. Both the Petersen graph and Q3 have colourings satisfying the condition of Lemma 3.2, see Figure 4. Hence if G' is one of them, then G has a distinguishing 2-colouring.
Figure 4: Colourings satisfying the condition of Lemma 3.2, v is the square vertex.
We may thus assume that G' is either K4 or K3 3. By Lemma 3.1 we may assume that there is a pair of components of Gt>2 connected by multiple type 1 edges. Since G is vertex-transitive and each vertex is incident to a unique edge of type 1, the number of type 1 edges between any pair of adjacent components of Gt>2 is independent of the choice of the pair. Furthermore, recall that A acts arc-transitively on G'. Hence if two adjacent vertices in a component H are both adjacent to the same component H' (via type 1 edges), then all vertices of H are adjacent to H'. For G' = K4, this is the only possibility, and the resulting graph is G = K4 □ K2. For G' = K3,3, the above observation tells us that all vertices in the same bipartite class of a component send their type 1 edges to the same component, and hence G = Gn,K3,3 (see Figure 2) for some n > 2, which has distinguishing number 2. □
#
F. Lehner and G. Verret: Distinguishing numbers of finite 4-valent vertex-transitive graphs 181
3.2 Graphs with only edges of type 2
In this section, we assume that all edges of G are of type 2. This implies that A has two orbits on arcs and therefore at most two orbits on edges. We distinguish two subcases according to whether G is edge-transitive or not.
3.2.1	Edge-transitive case
Theorem 3.5. Let G be a connected 4-valent graph that is vertex- and edge-transitive but not arc-transitive. Then D(G) = 2.
Proof. In this case, A has two orbits on arcs and each arc is in a different orbit than its inverse arc. By removing one of the two orbits, G becomes an arc-transitive directed graph in which every vertex has in- and out-degree 2. There is some s > 1 such that A acts regularly on directed s-arcs (see for example [12, Lemma 5.4(v)]).
Let P = (v0,..., vs) be a directed s-arc in G. Suppose for a contradiction that there is an arc from vs to v0. Clearly, in this case s > 2, as G does not contain any 2-cycles. There is an automorphism fixing (v0,..., vs_i) pointwise, but not fixing vs. Therefore, the second out-neighbour vs = vs of vs-1 must also have v0 as an out-neighbour. By directed 2-arc-transitivity we conclude that for any vertex v4 on P, the out-neighbours of vj are exactly the in-neighbours of vi+2, so the digraph is a directed wreath graph and G is arc-transitive, which gives the desired contradiction.
We may thus assume that there is no arc from vs to v0. Colour the vertices of P black and the remaining vertices white. Note that v0 is the unique black vertex with no black in-neighbour. Hence v0 and thus all of P must be fixed by any colour-preserving automorphism. By s-arc-regularity, this implies that the colouring is distinguishing and G has distinguishing number 2.	□
3.2.2	Non-edge-transitive case
If G is not edge-transitive, then there must be 2 orbits on edges each of which forms a disjoint union of cycles. Denote the two subgraphs induced by the edge orbits by G1 and G2. By transitivity, all cycles in G1 have the same length, the same is true for G2.
We will inductively construct a distinguishing colouring from partial colourings of G. Let c be a partial colouring of G with domain V C V(G), that is, c is a function from V to some set C of colours. An extension of c is a colouring c of G such that c and c coincide on Vc .
Lemma 3.6. Let G be a connected 4-valent vertex-transitive but not edge-transitive graph and assume that all edges have type 2. Let G1 and G2 be the subgraphs induced by the two edge orbits. Let V' be a set of vertices of G and let C be a cycle in G1 which is disjoint from V' and contains a neighbour v of some vertex in V'. Then there is a cycle D in G1 which is disjoint from V' (possibly D = C) and a partial 2-colouring c of G with domain C U D such that
•	C and D both contain either 1 or 2 black vertices, and
•	if y G Aut G fixes V' pointwise and fixes any extension of c, then 7 fixes V' U C U D pointwise.
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Proof. Call a vertex u a twin of v if there is an automorphism in the stabiliser of V' that moves u to v. Note that v has at most one twin, since there is an edge in G2 connecting v to some w in V', and w has only one other neighbour in G2.
If v has no twin then every automorphism that fixes V' pointwise must fix v. Set D = C, colour v and one of its neighbours on C black and colour the remaining vertices of C white. Then every automorphism which fixes V' as well as an extension of this colouring must fix v and its black neighbour and thus also fixes C.
Next assume that v has a twin that lies on C. Again let D = C and colour v and one of its neighbours in C black, but make sure that the black neighbour of v is not a twin of v. The same argument as above tells us that this colouring has the desired properties.
Finally assume that v has a twin u that lies outside of C. Let D be the cycle in Gi containing u and observe that D is also disjoint from V'. Colour v and one of its neighbours in C black, colour one of the neighbours of u in D black, and colour the remaining vertices of C U D white. Any automorphism that fixes V' as well as an extension of this colouring must fix u and v and their respective black neighbours, whence we have found the desired colouring.	□
Theorem 3.7. Let G be a connected 4-valent vertex-transitive but not edge-transitive graph and assume that all edges have type 2. Then D(G) = 2.
Proof. Let G1 and G2 be the subgraphs induced by the two edge orbits respectively and without loss of generality assume that cycles in G1 are at least as long as cycles in G2.
If G1 consists of a single cycle then this cycle must have length at least 6. Hence there is a distinguishing 2-colouring of G1 which must also be distinguishing 2-colouring of G. Hence we may assume that G1 consists of more than one cycle.
If cycles in G1 have length at least 4, then let C1 be a cycle in G1 and let v1 be a vertex on this cycle. Now inductively apply Lemma 3.6. For the first step, let V' = {v1}. In each step, pick a cycle C = C1 which contains a G2-neighbour of V', colour it according to the lemma and add the vertices of CUD to V'. The graph obtained from G by contracting every cycle in G1 is connected and vertex-transitive. Hence, by Lemma 2.1 it is 2-connected and remains connected after removing C1. In particular, the above colouring procedure assigns colours to all vertices except those in C1 . Finally colour v1 and its neighbours on C1 black, and colour the rest of C1 white.
We claim that the resulting colouring is distinguishing. Clearly, every colour-preserving automorphism must fix v1 since it is the only black vertex both of whose neighbours in G1 are black (recall that C1 is the only cycle in G1 containing 3 black vertices). Using Lemma 3.6 inductively, we see that every colour-preserving automorphism must fix every cycle pointwise, except possibly C1. Hence the colouring is distinguishing unless the two neighbours of v1 in G1 have the same G2-neighbourhood. In this case, by vertex-transitivity any two vertices at distance 2 in G1 have the same G2-neighbourhood. If cycles in G1 have length 5 or more, this implies that vertices have degree at least 3 in G2 which is a contradiction. If cycles in G1 have length 4, then so do cycles in G2 and G is a graph obtained by identifying antipodal points of 4-cycles, i.e., a wreath graph, which contradicts the assumption that G is not edge transitive.
It remains to deal with the case when both G1 and G2 are disjoint unions of 3-cycles. Let H be the graph with vertices these 3-cycles, with two such 3-cycles being adjacent in H if they share a vertex in G. It is easy to see that H is regular of valency 3 and G = L(H).
F. Lehner and G. Verret: Distinguishing numbers of finite 4-valent vertex-transitive graphs 183
By Theorem 2.3, we have D(G) = D'(H) < 2, unless H is K4 or K3,3. Finally, note that
L(K4) =W3 while L(K3,3) = K3 □ K3.	'	□
3.3 Arc-transitive graphs
We first prove a few lemmas to show that we can restrict ourselves to graphs with girth 4.
Lemma 3.8. Let G be a connected 4-valent arc-transitive graph. If G has girth 3, then G is either K5 or W3, or the line graph of a 3-valent arc-transitive graph.
Proof. Follows from [13, Theorem 5.1(1)]).	□
Lemma 3.9. Let G be a connected graph of minimal valency at least 3 and girth g > 5. If G is s-arc-transitive, then s < g — 3, unless G is a Moore graph of girth 5, or the incidence graph of a projective plane.
Proof. Assume for a contradiction that G is (g — 2)-arc-transitive. Let C = (v0,..., vg_i) be a cycle of length g. Note that (v0,..., vg_2) is a (g — 2)-arc and that its endpoints have a common neighbour. By (g — 2)-arc-transitivity, every (g — 2)-arc has this property. Let vg_2 be a neighbour of vg_3 outside of C. Then (v0,..., vg_3, vg_2) is a (g — 2)-arc, whence vg_2 and v0 have a common neighbour Now the closed walk (v0,vg_i,vg_2,vg_3,v'g_2,v'g_i,v0) shows that g < 6.
If g = 5, then the fact that the endpoints of every 3-arc have a common neighbour implies that G has diameter 2 and is thus a Moore graph.
If g = 6, then an analogous argument as above yields that G has diameter 3. If G was not bipartite, then for v G V(G) there would be an edge connecting two vertices x and y at the same distance from v, and since g = 6 we have d(x, v) = d(y, v) = 3. But then there is a 4-arc from v to x whence by the above argument v and x have a common neighbour, contradicting d(x, v) = 3.
Hence G is bipartite and every vertex at distance 2 from a given vertex v has a unique
common neighbour with v. It follows that G is the incidence graph of a projective plane.
□
Lemma 3.10. Let G be a connected 4-valent arc-transitive graph of girth at least 5, then D(G) = 2.
Proof. Let g be the girth of G and let s be such that G is s-arc-transitive but not (s + 1)-arc-transitive. Note that there is no 4-valent Moore graph, and that there is a unique 4-valent graph that is the incidence graph of a projective plane, namely the (4, 6)-cage. By Lemmas 2.7 and 3.9 we may thus assume that s < g — 3.
By Lemma 2.2, there is a cycle C = (v0,..., vg_1) such that G — C is 2-edge connected. Let P = (vs+1, vs,..., v1) and let X be its pointwise stabiliser. Note that P is an s-arc and thus X is not transitive on N(v1) \ {v2} (otherwise G would be (s + ^-arc-transitive). Let v0 be a neighbour of v1 that is in a different orbit than v0 under X.
Note that the subgraph induced by the vertices {v0, v0, v1,..., vg_2} is a tree since any additional edge between these vertices would give a cycle of length less than g. Denote this tree by T and let H be the subgraph obtained from G by removing all vertices of T. Observe that v'0 has degree at most 3 in G — C. If H is not connected, then there is one component of H that is connected to v'0 by a unique edge. Removing that edge from G — C
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vo
Vg-1
Figure 5: The tree T in the proof of Lemma 3.10.
would disconnect it, contradicting the fact that G - C is 2-edge connected. It follows that H is connected.
Colour all vertices of T black and colour vg-1 white. Inductively colour the vertices of G as follows: Let x be a vertex at minimal distance to vg-1 in H that has not been coloured yet. If x is fixed by the pointwise stabiliser in A of all previously coloured points, then colour it white. Otherwise colour it black.
We claim that this colouring is distinguishing. First note that if an automorphism fixes two neighbours u and w of a vertex v, then it must also fix v, since otherwise the image of v would also be a common neighbour of u and w contradicting g > 5. Note that this implies that all vertices in H with a neighbour outside of H are coloured white. Indeed, at the time such a vertex x is considered for colouring, two of its neighbours are already coloured: its predecessor on a shortest vg-1-x-path in H and its neighbour outside of H. Hence by the previous observation, x is coloured white.
Next we show that v1 is the only black vertex with three black neighbours. By the above observations it is the only such vertex in T. Now let x be a black vertex in H. Then at most one neighbour of x was coloured before x (otherwise we would have coloured x white). Furthermore, if P is a shortest vg-1-x-path in H, then P U C contains an s-arc ending in x. Hence the pointwise stabiliser of x and all vertices coloured before x does not act transitively on the remaining neighbours of x, whence at most one of them will be coloured black.
Let y be a colour preserving automorphism. The above discussion shows that y must fix v1. Furthermore all neighbours of T are white, so y must preserve T setwise. Since there is no automorphism of G that fixes (v1,..., vg-2) and moves v0 to v0, y must fix T pointwise. Finally assume that there is a vertex in H that is not fixed by y and let x be the first such vertex that was coloured in the inductive procedure. Clearly, x is coloured black. Let y be the neighbour of x on a shortest vg-1-x-path P, and let S be an s-arc contained in C U P. Then S is pointwise stabilised by y, and since the orbit of x under the pointwise stabiliser of S is not a singleton, it contains exactly one other element x'. Every automorphism that fixes x and S also fixes x' and vice versa. Hence at most one of x and x' can be coloured black and thus neither of them can be moved by y.	□
Next we give some results for the case when G has girth exactly 4. Note that in this case, there must be vertices at distance 2 from each other with 2 or more common neighbours. The following two lemmas follow from results in [13].
Lemma 3.11. Let G be a connected 4-valent arc-transitive graph. If there are two vertices at distance 2 with 3 or more common neighbours, then G is isomorphic to either K5 x K2 or Wn for some n > 4.
v
F. Lehner and G. Verret: Distinguishing numbers of finite 4-valent vertex-transitive graphs 185
Proof. If there are vertices with 4 common neighbours, then by [13, Lemma 4.3], G is a wreath graph. Otherwise, Subcase II.A of the proof [13, Theorem 3.3] implies that G =
K5 x K2.	□
Lemma 3.12. Let G be a connected 4-valent 2-arc-transitive graph. If G has girth 4 but no two vertices at distance 2 have more than 2 common neighbours, then G is isomorphic to either Q4, or the bipartite complement of the Heawood graph.
Proof. By 2-arc-transitivity, every edge is contained in at least three 4-cycles. Subcase II.B of the proof of [13, Theorem 3.3] then implies that G is isomorphic to one of the two graphs as claimed.	□
The hardest case to deal with is when the graph is locally D4. In this case, we take advantage of the following structural property. Note that D4 in its natural action on 4 points admits a unique system of imprimitivity with 2 blocks of size 2. We say that a 2-arc (v0, vi, v2) is straight, if {v0, v2} is a block with respect to the local group at vi, and crooked otherwise. Note that, of the three 2-arcs starting with a given arc, one is straight and two are crooked. Further note that fixing a crooked 2-arc fixes all neighbours of its midpoint. Finally, note that A acts transitively on crooked 2-arcs of G. Call a cycle in G straight, if all sub-arcs of length 2 are straight.
Theorem 3.13. Let G be a connected 4-valent arc-transitive graph, then D(G) = 2 unless G is K5, K3 □ K3, K5 x K2, or Wn for some n > 3.
Proof. By Lemmas 3.8, 3.10, as well as Lemma 2.7, we can assume that G has girth 4. By Lemma 3.11, we can assume that no two vertices have more than two common neighbours.
Since G is arc-transitive, the local group must be a transitive subgroup of S4. If the local group is 2-transitive, then G is 2-arc-transitive and this case is handled with Lemmas 3.12 and 2.7.
If the local group is C4 or V4, then G is arc-regular. One can then colour one vertex v and three of its neighbours black, and colour the remaining vertices white. Any colour preserving automorphism must fix the arc from v to its unique white neighbour, thus the colouring is distinguishing.
The last remaining case is that G is locally D4. Suppose first that G contains a 4-cycle that is not straight. Let (u, v, w, x) be a 4-cycle of G such that (u, v, w) is a crooked 2-arc.
We claim that any automorphism fixing u and all of its neighbours must be the identity. By arc-transitivity and connectedness it is enough to show that such an automorphism must fix all neighbours of v. Since no pair of vertices has more than two common neighbours, u and w are the only two common neighbours of v and x. In particular, if an automorphism fixes w and all its neighbours, then it must also fix u. Hence it fixes a crooked 2-arc with midpoint v, and thus it fixes v and all of its neighbours, thus proving our claim.
Let y be the unique vertex such that (v, w, y) is a straight 2-arc, and let P = (u, v, w, y). Suppose that y is adjacent to u. Let u' be the unique vertex other than u such that (u', v, w) is crooked. Note that there is an automorphism fixing v and w (and thus y) and mapping u to u', and thus y is adjacent to u', and v and y have at least 3 common neighbours (u, u', and w), contradicting an earlier hypothesis. We conclude that y is not adjacent to u and thus the induced subgraph on P is a path of length 3. Colour P black and colour the remaining vertices white. Since (u, v, w) is crooked, but (v, w, y) is straight, every colour preserving
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automorphism fixes P pointwise, and thus it fixes v and all its neighbours. Hence, by the above claim, this colouring is distinguishing.
From now on, we can assume that all 4-cycles of G are straight. Let C be the set of all 4-cycles. Note that every edge is contained in a unique straight 4-cycle, whence C forms a partition of E(G). Furthermore, any two elements of C intersect in at most one vertex, since otherwise there would be vertices with 3 or more common neighbours.
Now consider the auxiliary graph G' with vertex set C and an edge between two vertices if the 4-cycles have a vertex in common. Note that G' is a 4-valent graph on |C | = |E(4G)| = |V(2G)| vertices.
Note that A has a natural induced action on G', and this is easily seen to be locally D4. Furthermore any distinguishing colouring of L(G') corresponds to a distinguishing colouring of G. By Lemma 2.5 and the above observations D(G') > D(L(G')) > D(G). Hence if D(G') = 2, then D(G) = 2 and we are done. By induction, we may thus assume that G' is one of K5, K3 □ K3, K5 x K2, or Wn for some n > 3. If G' = K5, then by Lemma 2.7(2), we have D(L(G')) = 2 and we are done. Finally note that G' = K5 is not possible, since A induces a transitive, locally D4 action on G', but K5 admits no such action.	□
ORCID iDs
Florian Lehner © https://orcid.org/0000-0002-0599-2390 Gabriel Verret© https://orcid.org/0000-0003-1766-4834
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ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 19 (2020) 189-208
https://doi.org/10.26493/1855-3974.2033.974
(Also available at http://amc-journal.eu)
Divergence zero quaternionic vector fields and Hamming graphs
Jasna Prezelj * ©
Fakulteta za matematiko in fiziko, Jadranska 19,1000 Ljubljana, Slovenija, and UP FAMNIT, Glagoljaska 8, 6000 Koper, Slovenija, and IMFM, Jadranska 19, Ljubljana, Slovenija
Fabio Vlaccif ©
DiSPeS Universita di Trieste Piazzale Europa 1, Trieste, Italy
We give a possible extension of the definition of quaternionic power series, partial derivatives and vector fields in the case of two (and then several) non commutative (quaternionic) variables. In this setting we also investigate the problem of describing zero functions which are not null functions in the formal sense. A connection between an analytic condition and a graph theoretic property of a subgraph of a Hamming graph is shown, namely the condition that polynomial vector field has formal divergence zero is equivalent to connectedness of subgraphs of Hamming graphs H(d, 2). We prove that monomials in variables z and w are always linearly independent as functions only in bidegrees
Keywords: Quaternionic power series, bidegree full functions, Hamming graph, linearly independent quaternionic monomials.
Math. Subj. Class. (2020): 30G35, 15A03, 05C10
*The first author was partially supported by research program P1-0291 and by research projects J1-7256 and J1-9104 at Slovenian Research Agency. Part of the paper was written when the first author was visiting the DiMal at University of Florence and she wishes to thank this institution for its hospitality.
^The second author was partially supported by Progetto MIUR di Rilevante Interesse Nazionale PRIN 201011 Varietal reali e complesse: geometria, topologia e analisi armonica. The research that led to the present paper was partially supported by a grant of the group GNSAGA of Istituto Nazionale di Alta Matematica "F: Severi".
E-mail addresses: jasna.prezelj@fmf.uni-lj.si (Jasna Prezelj), fvlacci@units.it (Fabio Vlacci)
Dedicated to the memory of Marjan Jerman.
Received 3 July 2019, accepted 1 July 2020, published online 13 November 2020
Abstract
(p, 0), (p, 1), (0, q), (1, q) and (2,2).
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
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1	Introduction
Complex holomorphic vector fields with divergence zero represent an important tool for the description of the groups of volume preserving automorphisms of Cn with n > 1 (we refer the reader to [1] and [2] for a thorough description of this topic). In this paper we investigate generalizations of complex holomorphic vector fields in the quaternionic setting, and for this purpose we restrict our research to mappings represented by convergent quaternionic power series.
We introduce an alternative definition of partial derivative, namely as a first order approximation (which is not linear) and using this new notion of partial derivatives we define the corresponding divergence in the quaternionic setting. We show that quaternionic vector fields with divergence zero are bidegree full (see Section 2.2 for definition) and that the divergence zero condition on quaternionic vector fields is equivalent to finding connected subgraphs of Hamming graphs. The paper is structured as follows: Section 2 contains the description of our setting with basic definitions and notions, such as partial derivatives and divergence. Moreover, bidegree full functions are introduced together with some basic facts about Hamming graphs. Section 3 is devoted to vector fields and their properties, in particular it contains the main result, Theorem 3.4, on quaternionic vector fields with divergence zero and explains the connection between divergence zero vector fields and Hamming graphs. In Section 4 we prove the theorem on linear independence of monomials.
2	Preliminaries
2.1 Convergent quaternionic power series
In this section we introduce the basic concepts and notions to deal with generalizations of complex holomorphic power series in the quaternionic setting.
We denote by H the algebra of quaternions, H = {z = x0 + x1i + x2j + x3k, x0,..., x3 G R}, where i, j, k are imaginary units satisfying i2 = j2 = k2 = -1, ij = k, jk = i, ki = j. Denote by S the sphere of imaginary unit quaternions, i.e. the set of quaternions 1 such that 12 = -1; notice that for a quaternion z we have z2 = x2 - x2 - x2 - x3 + 2x0(xii + x2j + x3k), therefore the condition z2 = -1 implies z = xii + x2j + x3k and - xi - x2 - x2 — -1. Given any nonreal quaternion z, there exist (and are uniquely determined) an imaginary unit 1, and two real numbers x, y (with y > 0) such that z = x + 1y. With this notation, the conjugate of z will be z := x - 1y. Each imaginary unit 1 generates (as a real algebra) a copy of a complex plane denoted by C/. We call such a complex plane a slice.
A product of nonzero quaternionic coefficients and the variables z, w of degree d is called a generalized quaternionic monomial of degree d. Let Hd[z, w] denote the set of all finite sums of generalized quaternionic monomials of degree d, which we call generalized quaternionic homogenous polynomials of degree d. For example, the generalized quaternionic polynomial a0za1wa2wa3 + 60z261w63 + c0wc1zc2wc3 belongs to H3 [z, w]. Let
H[z,w] :=0 Hd[z,w]
d>0
be the ring of generalized quaternionic polynomials in the variables z, w over the quaternions. We consider polynomials P G H[z, w] as formal (left and right) linear combinations.
J. Prezelj and F Vlacci: Divergence zero quaternionic vector fields and Hamming graphs 191
It turns out (see Section 4) that there are several polynomials defining the same polynomial function. We therefore identify a given polynomial function P with the equivalence class [P] of all polynomials defining the same function. The set of all polynomial functions coincides with real polynomials in 8 variables with quaternionic coefficients (see [3]).
We consider the right-submodule Hrhs [z, w] of H[z, w] which consists of all generalized quaternionic polynomials whose generalized monomials have coefficients on the right-hand side. To be precise, given the multiindex a = («i,..., ad) G {0,1}d, called a word on letters 0,1, we define the length of a to be |a| :=J2f=1 a^. Then we put
(z, w)a := (zaiw1-ai) • • • (zadw1-ad).
For integers p, q > 0,p + q = d, denote by ap,q a multiindex with |ap'q | = p. There are
(p, q) a
(z,w)c
(p) such multiindices. We call the pair (p, q) a bidegree. The (pure) monomials of degree d can be written in the form
and hence define
Hrhs,(p,q)[z,w] := {Pp,q(z, w) = ^ (z, w)"™cap,q ; Cap,q G H},
ap'q, |ap,q |=p
Hrhs,d[z,w] := {Pp,q(z, w) = ^ (z, w)"™cap,, ; cap,q G H,p + q = d}
ap'q, |ap'q |=p
so that Hrhs [z, w] = ed>0Hrhs,d [z, w].
Our basic assumption on regularity, for the definition of the class of quaternionic series we are interested in, is that any such a series f
f(z,w)= E E fp.q^(z,w)	(2.X)
p,q>0 AeAp,,
converges absolutely on H2. Notice that absolute convergence implies uniform convergence on compact sets of H2. The notation fp,q,A(z, w) G Hd[z, w] stands for generalized monomials containing p copies of z and q copies of w with p + q = d and the sets Ap,q are supposed finite. The set of all such series f will be denoted by H[z, w]. Putting
d
fd(z,w):=E E fP,d-p, A(z,w) p=0 \eAp,d-p
any f G H[z, w] also has a homogenous expansion f (z,w) = ^d>0 fd(z,w). Uniform convergence on compact sets of H2 means that given any e > 0 and a compact set K C H2, there exists a natural number de K such that for any generalized polynomial of the form
de,K	d
P(z,w)=E fd(z,w)+ E E	E	fp,d-p,A(z,w)
d=0	d>de,K P=0 AeAp,d_pCAp,d-p
the uniform estimate |f (z, w) - P (z, w)|K < e holds. Let the norm of the term fp,q,A be |fp,q,A(z,w)| = |z|p|w|qCp,q,A (with Cp,qjA > 0) and define Cp,q := £AeA Cp^a. The
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absolute convergence at the point (z0, w0) in the domain of definition of f means that
53 53 lfp,q,A(zo,wo)| = 53 NPK|qcp,q < to
p,q>0 xeAp,g	p,q>o
and implies uniform convergence on compact sets of B(0, |z0|) x B(0, |w0|).
Any series f G H[z, w] uniquely defines a function of two quaternionic variables, but as in the case of polynomials, there are many series defining the same function. We say that two quaternionic series are equivalent if each of them defines the same quaternionic function. This is an equivalence relation, and so we identify the function f with the corresponding equivalence class [f] of all series in H[z, w] defining the same function. To avoid too many notations, we will say that a given function belongs to H[z, w] if it has a series representative in H[z,w]. By abuse of notation, if f g H[z,w], we also denote by [f] the set of all series which determine the same function. Since uniqueness of the power series for a function f is not granted (see next paragraphs and (2.2)), the absolute convergence of a chosen power series for a given function f g H[z, w] is not a consequence of uniform convergence on compact sets, as in the complex or real case, and has to be additionally required. In the sequel we focus our attention on the right H-module Hrhs [z, w] in H[z, w] of (absolutely convergent) power series with coefficients on the right. We extend all the above definitions also to series of three or more variables. Notice that in HrhSid[z1; z2,..., zn] there are nd different (pure) monomials. If we assume only uniform convergence of series in Hrhs[z1, z2,..., zn], given an uniformly convergent series f (zi, z2,..., z„) = d>0 fd(zi, z2,..., z„), for R > 0, £ > 0 there is a do G N such that for each d > d0 and p = (p1;... ,pn) G Nd with |p| = J21 P1 = d, where p denotes the multiindex, whose j th element pj is the total degree of zj in the corresponding monomial, the estimate |fp,>(z1, z2,..., zn)| < £ holds on the ball Bn(0, R) c Hn. As a consequence, on the ball Bn(0, R/(n + 1)), we have the estimate |fp,A(z1; z2,..., zn) | < £/(n + 1)|p| (with |p| = p1 = d), so that for (z1,z2 ,...,z„) G B n(0,R/(n + 1)) we have
|fp>A(z1,z2,... ,z„)| < £
|p|=d
which implies that the series f (z1; z2,..., zn) = J2d>0 fd(z1, z2,..., zn) is not just uniformly but also absolutely convergent. Once more, we observe that, in general, in H[z1; z2,..., zn] one has to assume absolute convergence for a proper definition of series, since the number of different generalized monomials can grow faster than exponentially; for example, if the polynomial P is as in (2.2), then the sums J2fc=o P(z, ak), with ak G H, are identically 0 for any m G N and they contain 6m different generalized monomials. Let us mention another right-submodule of Hrhs[z, w], namely, the submodule of slice-regular functions in the sense of Ghiloni-Perotti (see [5]), denoted by hgp [z, w]. It is generated by (pure) monomials of the form zkwl, k, l G N0, with this precise order, so any element of HGP[z, w] has a unique power series expansion and uniform convergence on compact sets in H implies absolute convergence. Slice-regular functions in the sense of Ghiloni-Perotti can be also seen as the kernel of a suitable partial differential operator. Notice that hgp[z,w] c Hrhs [z, w] cH[z, w].
Unfortunately, also in Hrhs[z, w] there are several power series which define the same function. In general the monomials of a given bidegree are not (right) linearly independent as functions. As far as we know, very little is known about this question except for linear
n + 1
n
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independence of monomials of bidegrees (p, 1) and (1, q) as proved in [6, Proposition 2.4]. In Section 4 we prove that monomials of bidegrees (p, 0), (p, 1), (1, q), (0, q) and (2, 2) are linearly independent but monomials of bidegree (3,2) (and all other bidegrees) are not necessarily: since the square of the commutator of z and w is real, i.e. [z, w]2 G R, the polynomial of bidegree (3,2),
P(z, w) = —z2wzw + z2w2z + zwz2w — zw2z2 — wz2wz + wzwz2 = [[z, w]2, z] (2.2)
is identically zero as a function but it is not (formally) equal to the null polynomial. Therefore, even here there is no one-to-one correspondence between power series and functions. However, as we will see, this fact does not affect the generality of the problem we are interested in (see also Remark 3.8 and Example 4.4). We realized that there exists a submodule HBF[z, w] in Hrhs[z, w] which gives rise to vector fields with nice analytic properties, but these vector fields could not in general be detected using just analytic tools, due to the fact that we are not able to describe formal properties of the series defining the zero class [0]. Nevertheless, it turns out that these vector fields have representatives in their corresponding classes of power series with specific symmetry properties and for them all the results stated are valid within a given bidegree up to adding a polynomial which defines the identically-zero function. Example 4.4 is a special case where analytic conditions imply the existence of this special type of representatives in the classes of power series and these representatives are unique.
We remark that HGP [z, w] contains, as a particular case, the right submodule of slice-regular functions in one variable denoted by SR as introduced in [4] (see also the monograph [3]): it is the class Hrhs[z] := Hrhs[z, 1]. Vaguely speaking it is defined to be the class of functions f: H ^ H such that the limit
lim h-1(f (z + h) — f (z)) h^0
exists if h and z belong to the same slice. These functions turn out to be quaternionic analytic and their power expansions are unique.
In general, there is no standard way of introducing a notion of (partial) derivative for quaternionic functions (see for instance [4, 5]). For example, for the slice-regular function f (z) = z2a the limit of the differential quotient
lim h-1(f (z + h) — f (z)) = lim(h-1zh + z + h)a
h^0 h^0
does not exist unless h and z belong to the same slice.
We introduce new differential operators dz, dw : H[z, w] ^ H[z, w, h], which can be interpreted as partial derivatives for a convergent power series as in (2.1) with respect to each of the variables z, w in a given direction h.
Definition 2.1. For a function f G H[z, w] and z0, w0, h0 G H we define the quaternion
dzf (z0, w0)[h0] to be the limit
dzf (z0, w0)[h0] := lim ^(f (z0 + th0,w0) — f (z0, w0)), t G R, or equivalently
f (z0 + th0,w0) — f (z0, w0) = tdzf (z0,w0)[h0] + o(|t|);
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similarly
dw f (zo, wo)[ho] := lim 1 (f (zo, wo + tho) - f (zo, wo)), t € R,
defines dw f (zo, wo)[ho]. The function dz f in three variables (z, w, h) is then defined to be
(dzf)(z,w, h) := dzf(z,w)[h],
and similarly
(dwf )(z,w, h) := dwf (z, w)[h].
We use the notation dzf (z, w)[h], dwf (z, w)[h] also to denote the resulting functions of three variables in order to emphasize the special role the variable h plays. Both the operators dz, dw are additive and right-H-linear, namely
dz(f (z, w)a + g(z, w)b)[h] = dzf (z, w)[h]a + dzg(z, w)[h]b, dw(f (z, w)a + g(z, w)b)[h] = dwf (z, w)[h]a + dwg(z, w)[h]b.
The resulting functions are additive and real-homogenous in the variable h, but not linear in h. Furthermore, the Leibniz rule holds. In the language of analysis on manifolds, for a fixed h, the partial derivative dzf (z, w)[h] is the Lie derivative of the function f along the constant vector field X = (h, 0) evaluated at (z, w) and dwf (z, w)[h] is the Lie derivative of the function f along the constant vector field X = (0, h) evaluated at (z, w). In practice, for polynomial function represented by a polynomial, each of the operators dz, dw acts by replacing one occurrence of the prescribed variable at a time in each monomial of fd with h € H as in the following example
dz (zwz2wa)[h] = (hwz2w + zwhzw + zwzhw)a.
If lfp,q,\(z, w)| = |z|p|w|qthen we can estimate
\dz fp,q,X (z,w)[h]| < p|z|p-1|w|q |h|Cp,q,A
and \dwfp,q,A(z, w)[h]\ < q\z\p\w\q—1 \h\cp,qjA, which, in view of the assumed absolute convergence of the power series, implies that the power series can be differentiated term by term. Therefore operators dz, dw are well-defined as mappings from quaternionic analytic functions of two variables to quaternionic analytic functions of three variables. This motivates the following definition of partial derivatives for series:
Definition 2.2. Given a series f € H[z, w],
f(z,w)= E E fp,q,A(z,w) p,q>o AeAP,q
the series dz f is defined as
(dzf)(z,w,h):=£ ^ 4fp,q,A(z,w)[h].	(2.3)
p,q>o A£Ap,q
The operator dw is defined similarly. Note that the operators dz, dw map series in H[z, w] to series in H[z, w, h].
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We also use the notation dzf (z,w)[h] for the series to indicate the special role the variable h plays.
Linearity of the derivation implies that if a function is represented by two different series f and g, then also the series dzf (z, w)[h] and dzg(z, w)[h] represent the same function.
The following result motivates the introduction of the differential operators dz ,dw.
Lemma2.3. Let f G HIhs[z,w] be a series. If dz f (z,w)[h] is the null-series, then f (z,w) is (formally) independent of z and so is also the corresponding function. An analogous result holds for w.
Proof. It suffices to prove the first assertion for polynomials P{pq) of bidegree (p, q) for each (p, q). We proceed by induction on q. For q = 0 and P(Pto) (z,w) = zpcp we have
dz PM(z,w)[h] = (hzp—1 + zhzp-2 + ■■■ + zp— 1 h) cp = 0
formally, so cp = 0. Moreover, by [6, Proposition 2.4] the same holds if dz P(pio) (z, w)[h] = 0 as a function. If q> 0 write
P(p,q)(z,w) = zP(p—l,q)(z,w) + wP(p,q-l) (z,w)
and then the formal identity
dz P(p,q)(z,w)[h] = hP(p-ltq)(z,w) + zdz P(p-1,q)(z,w)[h]+ wdz P(p,q-1)(z,w)[h] =0
implies
P(p-I,q)(z,w) = 0, dzP(p-1,q)(z,w)[h] = 0 and dzP(p,q-1)(z,w)[h] = 0
formally. By induction hypothesis, dzP^pq-i)(z,w)[h] being formally 0 implies
P(p,q-l)(z, w) = wq— 1 Cq— 1, so P(p,q) = wqCq-1.	□
Remark 2.4. In analogy to the one variable case one could also define the (differential) operator
9zf(z,w) := dzf(z,w)[l].
In short, the operator dz replaces one occurrence of the variable z at a time with 1. This operator is a derivation. Using the notation from the above Lemma, the expression dzP(p,q) (z, w) is a polynomial of bidegree (p — 1, q) (similarly for w). Furthermore, this operator coincides with the corresponding (Cullen) derivative, when f is a slice-regular function (see [4]).
However, a result like the one in Lemma 2.3 does not hold when considering dz instead of dz. Indeed,
dz (zw — wz) = w — w = 0
but the neither the series f (z, w) = zw — wz nor the corresponding function do not depend on w only.
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2.2 Bidegree full series
For p, q positive integers, consider the series
Sp,q(z,w) := Y iz,w)aP'" ■
ap'q, |ap>q l=p p+q=d
It is clear that Spq(z, w) = Sq,p(w, z). We also have this important identity
dz Sp+I,q (z,w)[h] = dw Sp,q+1(z,w)[h]■	(2.4)
If z and w commute, then Sp,q(z, w) = (p+q) zpwq. Definition 2.5. We define
HBF [z,w] := l Y Sp,q(z,w)ap,q, ap,q £ H [p+q=d
and
HBF[z,w] :=® HBf [z,w].
d>0
We say that HBF [z, w] is the right module of bidegree full (in short BF) polynomials in the variables z, w. The equivalence class of BF polynomials is called a bidegree polynomial function. Similarly, we define the right module of bidegree full series to consist of all converging power series of the form
f(z,w) = Y fd(z,w),
,w)
d=0
with fd(z, w) e HfF [z, w] and denote it by HBF [z, w]. The equivalence class of a BF series is called a bidegree full function.
The following result shows that bidegree full polynomials form an interesting class of polynomials.
Lemma 2.6. For any real number p and any d e N, the polynomial
d times
(z — pw)d := (z — pw) ■ ■ ■ (z — pw)
is bidegree full. If
i
P(z,w) = ^2 E Sp,q(z,w)aPiq
d=0 p,q>0, p+q=d
is a bidegree full polynomial of degree d, then it also has a decomposition
P(z,w) = Y	— pw)dTp,d(n)\ ap,q, with Vp4(n) e R.
d=0 p+q=d \n=0	)
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Proof. Indeed, from direct calculations, it follows that
(z — ^w)d = (z — ^w) • • • (z — ^w) ^^ Sp,q(z, w)(—^)q.
p,q> 0, p+q=d
The second statement follows from the fact (proved in [2] by induction on d with an argument which applies to our setting) that the polynomials {xd, (x — 1)d,..., (x — d)d} form a basis of real polynomials of order less or equal to d and consequently polynomials
zd, (z — w)d,..., (z — dw)d form a basis of HfF [z, w]	□
The term (z — ^w)d is well-defined also for ^ G H. But the equality
<9w (z — ^w)d = — pdz (z — ^w)d	(2.5)
holds if and only if ^ G R.
Remark 2.7. As a consequence of Lemma 2.6, from any F G SR, in the variable u
F(u) = udad,
d>0
one gets a bidegree full series by replacing u with z — ^w, ^ g R namely
f (z, w) = ^2(z — ^w)dad G Hbf[z, w].
d>0
2.3 Basics on Hamming graphs
Since the monomials we are dealing with are described by words on two letters, the Hamming graphs are natural objects to associate with such monomials.
Definition 2.8. Given d,q G N, the graph (V, E) is a Hamming graph H(d, q) if the set of vertices V consists of all words of length d on q different letters and there is an edge e(v1, v2) G E between two vertices v1,v2 if they differ in precisely one letter.
The Hamming graph H(d, q) is, equivalently, the Cartesian product of d complete graphs Kq. We are interested in Hamming graphs on two letters, 0,1, i.e. on hypercubes. A layer Lp, 0 < p < d, is a set of vertices which contain p copies of 1. It is easy to see that the following result holds:
Lemma 2.9. Any two subsequent layers of the hypercube form a connected subgraph.
Proof. The case d =1 is trivial since it consists of letters 0 and 1 and an edge connecting them. Assume that d > 1 and take p G {0,...,d — 1}. Let Lp+1 and Lp be two subsequent layers, and let a, a G Lp+1 differ for one transposition of indices 0 and 1 on positions l, m. Without loss of generality we assume that l = 1 and m = 2. We may also assume that
a = 01ai and a = 10ai.
Define
P = 00a1.
Since a and P differ in precisely one letter, there is an edge between a and P and of course also an edge between P and a, so there exist a path between any two vertices in Lp+1, since all other multiindices in Lp+1 are permutations of letters of a. By the same reason there exist a path connecting any two vertices in Lp which proves the lemma.	□
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3 Quaternionic vector fields
In this section, using the partial derivatives dz, dw, we define an operator divergence for quaternionic vector fields in two variables. We show that there is a large class of vector fields with good properties of analyticity.
Definition 3.1. Given the series f, g G H[z, w], then X = (f, g) is called a vector field in H2, in short we write X G VH. If f, g G Hrhs[z,w], then we write X G VHrhs. In particular, we say that a vector field X = (f, g) is bidegree full (in short BF) if f, g are bidegree full and we use the notation X g VHbf. A vector field X = (f, g) defines a vector mapping [X] := ([f ], [g]): H2 ^ H2.
We assume from now on that the vector fields under consideration belong to VHrhs. Next we introduce the following
Definition 3.2. Given the vector field X = (f, g) G VHrhs, we define the operator Div by
DivX(z, w)[h] := dzf (z, w)[h] + dwg(z, w)[h],
where the partial differential operators are used in the sense of (2.3). A vector field X(z, w)
has divergence zero if Div X(z, w)[h] is the null series.
Clearly for a vector field, divergence zero implies divergence zero as a function. Example 3.3. The vector field X(z, w) = (zw + wz, —w2) has divergence zero, since
Div(zw + wz, —w2)[h] = hw + wh — (hw + wh) = 0
and the divergence of the vector field Y(z, w) = X(z, w) + (0, [[z, w]2, z]) = (zw +
wz, —w2 + [[z, w]2, z]) is
Div(zw + wz, —w2 + [[z, w]2, z])[h] = hw + wh — (hw + wh)
+ [[z, h][z, w], z] + [[z, w][z, h], z].
This shows that Div Y is not a null-series, but Div Y, considered as a vector mapping, vanishes identically.
The vector field (z2w, —zw2) does not have divergence zero:
Div(z2w, —zw2)[h] = (hz + zh)w — z(hw + wh) = hzw — zwh = 0
and also [hzw — zwh] = 0. By identity (2.5) any vector field (z —	1), ^ G R
has divergence zero. Such vector fields are called shear vector fields and they generate a 1-parameter family of automorphisms of H2, namely
$i(z, w) = (z, w) + t(z —	1)a, a G H,t G R,
called shears. In the complex analytic case by a famous result due to Andersen (see [1]) every volume preserving automorphism of C2 (these are holomorphic automorphisms f: C2 ^ C2 with determinant det Jf (z, w) = 1) is approximable by a finite composition of shears. In search for analogous results in the quaternionic setting, it is then necessary to
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prove that any polynomial divergence zero vector field is generated by a shear vector field. Because of identity (2.4), any vector field
Xptq(z,w) = (Sp+ltq(z,w), -SPtq+1(z,w))	(3.1)
has divergence zero. It can be shown using Lemma 2.6 that every vector field XPA is a sum of shear vector fields. The interested reader can find the details in [6]. The next theorem shows that any divergence zero vector field is generated by such vector fields XPiq.
Theorem 3.4. Let X = (f,g) be a vector field with divergence zero, then f and g are bidegree full.
Remark 3.5. Example 4.4 shows that for any vector field X with components of bidegrees (3,2) and (2,3), the condition DivX(z, w)[h] = 0 as a function of three variables implies that the mapping representing the vector field X has a bidegree full representative.
Corollary 3.6. If X is a vector field with divergence zero, then X is of the form X = £ XPtqaPtq, aPiq G H with XpA as in (3.1).
Before proceeding to the proof, let us show an example with vector fields of the form
X(z, w) = (f(z, w), g(z, w)) = (z2wa\ + zwza2 + wz2a3, —w2zb\ — wzwb2 — zw2b3). We first calculate the partial derivatives separately.
dzf(z, w)[h] = (zhw + hzw)a\ + (hwz + zwh)a2 + (whz + wzh)a3l dwg(z, w)[h] = —(whz + hwz)b\ — (hzw + wzh)b2 — (zwh + zhw)b3.
The sum of the partial derivatives is zero if and only if monomials of the same type cancel out, for example we have conditions zhw (ax — b3) = 0 and hzw (ax — b2) = 0 which imply ai = b3 and a\ = b2 and similarly for other terms. We represent these equalities by means of a bipartite graph on {ai, a2, a3} U {&i, b2, 63} in which there is an edge between ai and bj if and only if they are equal. The graph is given in Figure 1.
Figure 1: Bipartite graph.
Proof of Theorem 3.4. Let f(z,w) = £ fPtq(z,w), g(z,w) = £ gP,q(z, w) be the decompositions of series / and g with respect to the bidegrees. Then X = (/, g) has divergence zero if and only if
9zfp+i,q(z,w)[h] + dwgPtq+1(z,w)[h} = 0.
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Let A = {a G {0, 1}d+1, |a| = p + 1} and B = {3 G {0,1}d+1, |3| = p}. Write
fp+i,q (z,w) = ^(z,w)aAa
aeA
9p,q+ i(z,w) = - (z,w)p Bp.
pes
The monomials in the sum dz fp+i,q(z, w)[h] (and similarly dzgp,q+i(z,w)[h]) are of the following form:
(z,w)ai h(z,w)a2 Aa where a = ai1a2. For any such a there is exactly one 3, namely
3 = ai0a2
such that in the sum dw(z, w)pBp(h) there is the monomial of the same type (but multiplied by a different constant)
-(z,w)ai h(z,w)a2Bp.
Zero divergence implies that Aa = Bp for any such pair a, ¡.
Define a bipartite graph on the vertices V = A U B. There is an edge between a word a G A and a word 3 G B iff the word 3 is obtained from the word a by replacing one of the letters 1 by 0. So by definition, in this particular case, we are considering a subgraph of the Hamming graph H(d +1,2), spanned on edges from the set AUB which represent two subsequent layers in the corresponding hypercube, A = Lp+1 and B = Lp. By Lemma 2.9 this subgraph is connected. This implies that all Aa = A for some constant A and hence the same holds for all Bp so all the coefficients are the same and this means that fp+i,q, gp,q+i are bidegree full.	□
Remark 3.7. We should point out that the analytic condition on a vector field of two quater-nionic variables having divergence zero is equivalent to connectedness of subgraphs of a Hamming graph. We could proceed analogously in higher dimensions. In three variables we would consider H(d, 3), graphs on three letters, where d is the degree, but in this case the divergence zero condition translates into looking for cycles of order 3 of a particular form. Its analysis turns out to be more complicated than the two-dimensional case and is not related to connectedness of subgraphs of Hamming graphs. For example, the divergence zero condition for the vector field X(z, w, u) = (f (z, w, u), g(z, w, u), h(z, w, u)) in the case
f (z, w, u) = zwuai + wzua,2 + wuza3 + zuwa4 + uzwa5 + uwza6, g(z, w, u) = w2ubi + wuwb2 + uw2b3, h(z, w, u) = wu2ci + uwuc2 + u2wc3,
gives equations a1 + b1 + c2 = 0, a2 + b1 + c1 = 0,... and they can be represented as a 2-simplicial complex with 2-cells being triangles with vertices (a1, b2,c2), (a2,b1,c1) and so forth (see Figure 2).
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a 5
Figure 2: 2-simplicial complex describing the divergence 0 condition for 3 variables.
Remark 3.8. Notice that if we can write a vector mapping as a vector field X = (fi,gi) + (/2,32) such that (/1,31) has divergence zero and such that each of /2 and go are not formally 0 but identically equal to 0 as functions, then the flow of [A"] coincides with the flow of [(/1, g\)}. Furthermore, the flow of [(/2,32)] exists and is the identity mapping, so it does not affect the problem of approximating a flow by shears.
4 Linear independence of monomials
In this section we consider the problem of linear independence of monomials in Hrhs (s, iv); in particular we exhibit an algorithm for determining linear independence of monomials in HrhSj(Pj9)[^, w]. We point out that this approach does not involve the computation of independent monomials in 8 real variables of degree p in the first 4 variables and of degree q in the last 4 variables.
We prove the following result.
Theorem 4.1. Given a bidegree (p, q), the set of all distinct monomials in HrhSj(p9) [z, if] is linearly independent if and only if (p,q) equals (p,0), (0 ,</), (p, 1), (1, q) or (2, 2).
The proof below gives an explicit algorithm for calculating the basis of the kernel of the linear mapping
"p+2,2
Ap,2 :	^ HThB[z, w], *4p,2(ci, ..., c„p+2,2) = ^ (-> w)a"ck,
1
where ak G {0, I j'' are all distinct multiindices of length p, |afc| = p. with I.....
"P+2.2 = (pf )•
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Proof. The cases in bidegrees (p, 0), (0, q) are trivial and the cases in bidegrees (p, 1), (1, q) were proved in [6, Proposition 2.4]. Assume that z G C/ \ R and choose any imaginary unit J orthogonal to I. Then we can write w G H in the form z0 + zi J, where z0, zi G C/ are uniquely determined. This choice of coordinates provides us with a frame which determines the identification H = C/ x C/. In other words, if w = z0 + z1 J ~ (z0, z1) G C/ x C/, then, since z J = Jz, we have
w2 = z^ - |zi|2 + (zZjzi + z0zi)J;
similarly
[z,w] = z1(z — z)J and [z,w]2 = -|(z — z)z1|2.
We recall that the polynomial introduced in (2.2) is precisely P(z, w) = [[z, w]2, z].
We begin with polynomial w2 G HrhSj(0j2) [z, w] and develop an algorithm for producing monomials in Hrhs (12) [z, w] which we describe with respect to the above identification and then proceed inductively.
First of all, without loss of generality, we (may and will) assume that z is unitary, so z-i = z. Let A0 = B0 = C0 = {w2}. Define the sets A1 = {zw2},B1 = {wzw},C1 = {w2z}. The monomial in A1 was obtained by adding z to the monomial w2 on the left hand side, the one in Bi by adding one z after the first w of the monomial w2 and the monomial in C1 z was obtained by adding a z on the right hand side of the monomial w2. If w = z0 + z1 J (and then w2 = z2 — |z112 + (z"0z1 + z0z1) J) we have
Ai 3 zw2 = z(z^ — |zi|2 + (z"0zi, z0zi) J)zz
= (z2 — |zi|2 + (z2z"0zi + z2z0zi)J)z = fi(z, z0,zi)z, Bi 3 wzw = (z0 + zi J)z(z0 + zi J)zz
= ((z2 — z2|zi|2 + (zbzi + z2z0zi)J)z = /2(z, z0,zi)z, Ci 3 w2z = (z2 — |z112 + (zbzi, z0zi) J)z = /3(z, z0, zi)z.
We identify w2 with the vector (z2, — |zi|2, z0zi, z0zi) G C| and identify the function /i with the vector ui = (z2, — |zi|2, z2z0zi,z2z0zi) G C|, /2 with the vector u2 = (z2, — z2|zi|2, z"0zi, z2z0zi) G C| and the function /3 with the vector u3 = (z2, — |zi|2, z0zi, z0zi) G C4. We notice that ui is obtained from u3 by multiplying the first two components by 1 and the last two by z2,i.e. u i = u3 *(1, 1,z2,z2), where * denotes the componentwise multiplication in C| defined as follows: if (a, b, c, d) G C| and (x, y, u, v) G C| then
(a, b, c, d) * (x, y, u, v) := (ax, by, cu, dv). Notice that the componentwise multiplication * in C| is commutative, i.e.
(x, y, u, v) * (a, b, c, d) = (a, b, c, d) * (x, y, u, v)
for any (a, b, c, d) G C| and (x, y, u, v) G C| and has no zero divisors.
Similarly u2 = u3 * (1, z2,1, z2) and u3 = u3 * (1,1,1,1). Consider a quaternionic (right-hand side) null linear combination of the monomials which generates Hrhs(i2) [z, w], namely zw2a + wzwb + w2zc = 0 with a, b, c G H.
In terms of the vectors ui, u2, u3, we can write the same null linear combination as
u3 * (1,1, z2, z2)za + u3 * (1, z2,1, z2)zb + u3 * (1,1,1,1)zc =
= u3 * [(1,1, z2, z2)za + (1, z2, 1, z2)zb + (1, 1,1,1)zc] = 0. ^
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If we write a = a0 + ai J, b = bo + &i Jc = c0 + cJ (according to the adopted frame) and look at the first component in (4.1), since u3 = 0, we get the equation
z(ao + ai J) + z(bo + bo J) + z(co + ci J) = z((ao + ai J + bo + bo J + co + ci J)
= z[(ao + bo + co) + (ai + bi + ci) J] = 0
which implies ao + bo + co =0 and ai + bi + ci = 0. From the vanishing of the second component in (4.1) we get the equation
1 • z(ao + ai J) + z2 z(bo + bo J) + z(co + ci J)
= z(ao + ai J) + zz2(bo + bi J) + z(co + ci J) = z[(ao + ai J) + z2(bo + bi J) + (co + ci J)] = z[(ao + z2bo + co) + (ai + biz2 + ci) J] = 0
which implies ao + z2bo + co = 0 and (ai + z2bi + ci) J = 0. From the vanishing of the third component in (4.1) we get the equation
z2 Jz(ao + ai J) + Jz(bo + bo J) + Jz(co + ci J)
= Jz(z2(ao + ai J) + (bo + bi J) + (co + ci J) = Jz[(z2ao + bo + co) + (z2ai + bi + ci) J) = 0
which implies z2ao + bo + co = 0 and z2ai + bi + ci =0. Here we can replace z2 with z2 since we are allowed to plug in any z G C/, in particular we can plug in z and use the fact that z = z. From the vanishing of the last component in (4.1) we get the equation
z2 Jz(ao + ai J) + z2 Jz(bo + bi J) + Jz(co + ci J)
= Jz(z2(ao + ai J) + z2(bo + bi J) + (co + ci J) = Jz[(z2ao + z2bo + co) + (z2ai + z2bi + ci) J] = 0
which implies z2ao + z2bo + co = 0 and z2ai + z2bi + ci = 0. Also in these equations, one can substitute zz with z.1 In the vectorial version, we can write the above-given equations as
(1, 1, z2, z2)ao + (1, z2,1, z2)bo + (1, 1, 1,1)co = 0, (1, 1, z2, z2)ai + (1,z2,1,z2)bi + (1, 1, 1,1)ci = 0.
Therefore, the linear dependence of the monomials zw2,wzw and w2z (generators of Hrhs,(i,2)[z, w]) is equivalent to the linear dependence of the vector functions Xi(z) = (1,1, z2, z2), X2 (z) = (1,z2,1,z2) and X3(z) = (1,1,1,1). In this case, the vector functions Xi (z), X2 (z) and X3 (z) are evidently linearly independent and so are the monomials
zw2, wzw and w2z.
To generalize the formalization of the above ideas we introduce the operator adz and adopt the identification w2 ~ (z2, — |zi |2, zozi, zozi) G C|. For q G H \ {0}, |q| = 1, let
1 In all equations we can plug in the variable z instead of z and we can consider the conjugated equation. Then the linear independence in question is equivalent to (one of) the above equations with real coefficients.
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adq (w) := qwg. This transformation represents a rotation in H which keeps fixed the slice which contains q. Notice that if z G C/ \ R and w = zo + zi J (with zo, zi G C/), then
and
Then
adz (w) = zo + ziz2 J zw2 = (adz (w))2z = adz (w2)z, adz (adz (w)) = adz2 (w).
adz(w2) = zw2z ^ (z^, — |zi |2, z2zozi, z2zozi)
= (z2, —|zi|2, zozi, zozi) * (1, 1, z2, z2), w adz(w) = wzwz ~ (z2, —z2|zi |2, zozi, z2zozi)
= (z^, —|zi|2, zozi, zozi) * (1, z2,1, z2), w2adz(1) = w2 ~ (z2, —|zi|2, zozi, zozi)
= (z2, —|zi|2, zozi, zozi) * (1,1,1,1).
Define the functions , , idz : C| ^ C| by
[(a, b, c, d)] = (a, 6, c, d) * Xi(z), ^z[(a, b, c, d)] = (a, b, c, d) * X2(z), idz[(a, b, c, d)] = (a, b, c, d) * X3(z).
Since , and idz are linear in C| and can be represented by diagonal matrices, we identify the maps with the diagonals of the corresponding matrices: = (1,1, z2, z2) =
Xi(z), ^z = (1, z2, 1,z2) = X2(z), and idz = (1,1,1,1) = X3(z). In this sense the functions , and idz are linearly independent since Xi(z), X2(z) and X3(z) are linearly independent as vectors.
Now one can write the monomial zw2 G Ai as [(1,1,1,1)] =: vi (z), the monomial wzw G Bi as [(1,1,1,1)] =: v2(z) and the monomial w2z G Ci as idz[(1,1,1,1)] =: v3(z). In the sequel the functions like vi, v2, v3 will be called vector functions. With this identification we have Ai = ^>z(Ao), Bi = (Bo) and Ci = idz(Co). Notice that one can also write Ao = Ao U Bo U Co and Bo = Bo U Co. We proceed by inductive construction and define
Ap = ^z(Ap_i U Bp-i U Cp-i), Bp = ^z(Bp-i U Cp-i) and Cp = idz(Cp-i).
The set Ap contains all monomials, obtained by adding a z on the left hand side to all bidegree (p — 1,2) monomials, the set Bp is obtained by adding a z after the first w of the monomials in Bp-i and in Cp-i and the set Cp is obtained by adding a z on the right hand side to the monomials in Cp-i.
Let us describe the sets Ap, Bp, Cp together with the corresponding vector functions and compute the kernels of Ap,2 for p = 2, 3. Notice that with the adopted identifications, it turns out that Cp = {w2zp} for any p > 0 and this implies that the vector function associated with the unique monomial in Cp is the same, namely (1,1,1,1) for any p > 0. Here we list the sets Ap Bp and Cp (together with the description of monomials as
J. Prezelj and F. Vlacci: Divergence zero quaternionic vector fields and Hamming graphs 205
vector functions) for p = 2:
	vector function		monomial
	i vi(z)	= (1, 1,z4,z4)	~ z2w2
A2 =	1 v2 (z)	= (1,z2,z2,z4)	~ zwzw
	[ v3(z)	= (1, 1,z2,z2)	~ zw2z
B2 =	i v4(z) I v5(z)	= (1, Z4, 1,z4) = (1,Z2, 1,z2)	~ wz2w ~ wzwz
C2 =	{ v6 (z)	= (1,1,1,1)	~ w2z2 }
Notice that each of the components of the vector functions vk (z) is generated by {1, z2, z4, z2, z4}. We look for the functional kernel of the linear mapping A2,2(ci,..., c6) = J2k=i vk (z)ck where the vector functions vk (z) are listed above; in other words we are imposing conditions on ck's to have £k=1 vk(z)ck = 0 as a function of z. From the vanishing of the first component we only get one equation, from the vanishing of the second, third and fourth components the (linear) equations are obtained by imposing the vanishing of coefficients in the basis {1, z2, z4 z2, z4}. In this way we obtain a homogeneous linear system whose corresponding matrix is
111111 101001 010010 000100 0 0 0 1 1 1 0 110 0 0 100000 000001 001010 110100
The matrix M2 has trivial kernel and this proves the linear independence of sets of distinct monomials in Hrhs,(2,2) [z, w]. For p = 3, using the same approach, we get
M2
A3 =
B3 =
vector function		monomial
'vi(z) =	(1, 1,z6,z6)	32 ~ z3w2
v2 (z) =	(1, Z2, z4, z6)	2 ~ z2wzw
v3 (z) =	(1, 1,z4,z4)	22 ~ z2w2z
^ ( \ v4 (z) =	(1, Z4, z2, z6)	~ zwz2w
v5(z) =	(1, Z2, z2, z4)	~ zwzwz
. v6(z) =	(1, 1,z2,z2)	22 ~ zw2z2
i vr(z) =	(1,Z6,1,z6)	3 ~ wz3w
{ vs(z) =	(1, Z4,1,z4)	2 ~ wz2wz
I vg(z) =	(1,Z2,1,z2)	2 ~ wzwz2
{ v10(z) =	= (1,1,1,1)	~ w2z3 }
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Notice that each of the sets A3, B3, C3 contains only linearly independent monomials. We look for the functional kernel of the linear mapping A3j2(c^..., c10) = Efc=i vk(z)ck where the vector functions vk (z) are listed above; in other words we are imposing conditions on ck's to have J2k=1 vk(z)ck = 0 as a function of z. We list the equations in the same order as in the previous case. The homogeneous linear system in this case has as corresponding matrix
M3
1	1	1	1	1	1	1	1	1	1
1	0	1	0	0	1	0	0	0	1
0	1	0	0	1	0	0	0	1	0
0	0	0	1	0	0	0	1	0	0
0	0	0	0	0	0	1	0	0	0
0	0	0	0	0	0	1	1	1	1
0	0	0	1	1	1	0	0	0	0
0	1	1	0	0	0	0	0	0	0
1	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0	1
0	0	0	0	0	1	0	0	1	0
0	0	1	0	1	0	0	1	0	0
1	1	0	1	0	0	1	0	0	0
whose kernel is spanned by the vector (0, -1,1,1,0, -1,0, -1,1,0). The generator of the kernel represents precisely the polynomial P(z, w) = [[z, w]2, z] in (2.2).
In the same way one can verify that the kernel of the mapping A4,2 in Hrhs,(4,2) [z, w] is three-dimensional with generators P1(z,w) = zP(z,w) and P2(z,w) = P(z,w)z, obtained from the polynomial P, and the polynomial
Q(z,w) = [[z, w]z[z, w], z].
(4.2)
The latter is a zero function since [z, w]z[z, w] = -|(z — z)z1|2z. The polynomial Q is formally linearly independent from the other two generators since it contains the term wz3wz, which does not appear in P1 or P2. Then the kernel of the mapping A5,2 in Hrhs,(5,2) [z, w] is six-dimensional, generated by z2P (z, w), zP (z, w)z, P (z, w)z2, zQ(z, w), Q(z, w)z and [[z2, w], z]. By a similar argument as in bidegree (4, 2), the first five polynomials are formally linearly independent and the last one contains the term wz4wz, which does not appear in the first five polynomials.
In fact it is easy to see that in general the first component of the vector functions in Ap, Bp and Cp is always 1, whereas in the second component terms containing 1, z2,..., z2(p) will appear; similarly, in the third and the fourth component only terms containing 1, z2,..., z2p will show up.
Let us count the number of equations obtained by imposing the vanishing of coefficients of Ap,2. There is only one equation coming from the first component (which is redundant) and the last three components give (22 — 1)(p + 1) equations, whereas we have (p+2) = (p + 2)(p + 1)/2 formally different monomials, so we see that the dimension of the kernel grows quadratically in the bidegree (p, 2). If p = 2 we have 6 monomials and 9+1 equations and if p =3 there are 10 monomials and 12 + 1 equations. If p = 5 we have for the first time that the number of equations (which is 19) is smaller than the number of monomials (which is 21 ).
J. Prezelj and F. Vlacci: Divergence zero quaternionic vector fields and Hamming graphs 207
By a similar procedure one would expect (23 — 1)(p+1) equations for (p+3) monomials in the submodule Hrhs (p 3)[z, w] and so forth, but it turns out the for q = 3 there are 7 linearly independent monomials of degree 3 in z0, z1, z, ¿1 in the expression of w3, with the first component giving a redundant equation as before, therefore we get less equations.
The same procedure applied to Hrhs (p1) [z, w] is equivalent to looking only at the sets Ap and Cp and their union, since Bp is empty. Moreover in Hrhs(p1) [z, w] the generating monomials have as corresponding vector functions (1,1, z2k, z2k), k = 0,... ,p and they are obviously linearly independent. This is an alternative proof of Proposition 2.4 in [6].
It is clear that if a set of distinct monomials {m>(z, w)}^eA is not linearly independent in the submodule Hrhs (p q) [z, w], so the set {«"m^fz, w)}^eA is not linearly independent in HrhSj(p+"jq) [z, w] for each n G N and because of symmetry the set {mx(w, z)}a£a is not linearly independent in Hrhs (q p)[z, w]. Putting this together, we see that a subset of all distinct monomials in Hrhs (3+" 2+m) [z, w], m, n G N0 and in Hrhs (2+" 3+m) [z, w], m, n G N0 is not linearly independent.	□
Since P(z, w) = [[z, w]2, z] =0 as a function and also the polynomial Q of bidegree (4,2), Q(z, w) = [[z, w]z[z, w], z] is identically 0 as a function (as explained in the last section, Equation (4.2)), we conjecture that all zero polynomial functions not formally 0 are obtained from polynomials P and Q after multiplying them by other polynomials and inserting variables zk or wl.
Remark 4.2. The described procedure can be interpreted as a complex Fourier series analysis with respect to the complex variables z, z0 and z1 . We could have assumed that all the three variables z, z0 and z1 are unitary complex numbers, since the modulus is not relevant. In the expansion we considered, there are only 4 generators of the basis of the Fourier series in variables z0 and z1 and this is reflected in the vector functions having 4 components. With respect to the variable z, the number of the basic vector functions in question is 2p +1 if bidegree is (p, 2).
Remark 4.3. After applying the partial derivative operator dz to the generators of the kernel of Apq in Sp,q(z, w), one obtains polynomials in tridegree (p — 1, q, 1) with respect to variables z, w, h, e.g. polynomials with p — 1 copies z, q copies of w and one h. Analogous statement holds for dw.
Example 4.4. Consider a vector field X = (f(z, w), g(z, w)), where f has bidegree (3,2) and g has bidegree (2,3), and let the vector field Y be defined by Y(z, w) = X(z, w) + (P(z, w)a, P(z, w)b), (with a, b, G H), where P is the bidegree (3,2) polynomial defined in (2.2) and the polynomial P(z, w) = P(w, z) is then a bidegree (2, 3) polynomial. Obviously we have Div X(z, w)[h] = Div Y(z, w)[h] as a function since P and P are identically 0 as functions. Within this bidegree, the equivalence relation X ~ Y if [X — Y] = [0] means X — Y = (Pa, Pb) for some choice of a, b G H. After a careful study of linear independence of monomials in tridegree (2,2,1), i.e. monomials with two copies of z-s, two copies of w-s and one copy of h - which, it should be mentioned, boils down to determining the kernel of a 80 x 30 linear system !!! - it turns out that in this particular case, DivX(z, w)[h] =0 as a function if and only if X = X2,2 + (Pa, Pb), which means that the vector mapping has divergence 0 as a function if and only if it has a bidegree full representative in the sense of the above equivalence relation. Examples of bidegree full polynomial vector fields are given in (3.1).
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ORCID iDs
Jasna Prezelj © https://orcid.org/0000-0002-0384-6736 Fabio Vlacci © https://orcid.org/0000-0002-6248-8633
References
[1]	E. Andersen, Volume-preserving automorphisms of Cn, Complex Variables Theory Appl. 14 (1990), 223-235, doi:10.1080/17476939008814422.
[2]	E. Andersen and L. Lempert, On the group of holomorphic automorphisms of Cn, Invent. Math. 110 (1992), 371-388, doi:10.1007/bf01231337.
[3]	G. Gentili, C. Stoppato and D. C. Struppa, Regular Functions of a Quaternionic Variable, Springer Monographs in Mathematics, Springer, Heidelberg, 2013, doi:10.1007/ 978-3-642-33871-7.
[4]	G. Gentili and D. C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math. 216 (2007), 279-301, doi:10.1016/j.aim.2007.05.010.
[5]	R. Ghiloni and A. Perotti, Slice regular functions of several Clifford variables, AIP Conf. Proc. 1493 (2012), 734-738, doi:10.1063/1.4765569.
[6]	J. Prezelj and F. Vlacci, On a class of automorphisms in H2 which resemble the property of preserving volume, Math. Nach., in press, arXiv:1810.11412 [math.CO] .
ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 19 (2020) 209-229
https://doi.org/10.26493/1855-3974.2039.efc
(Also available at http://amc-journal.eu)
A Mobius-type gluing technique for obtaining edge-critical graphs*
Simona Bonvicini t ©
Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universita di Modena e Reggio Emilia, via Campi 213/b, 41126 Modena, Italy
Andrea Vietri ©
Dipartimento di Scienze di Base eApplicate per l'Ingegneria, Sapienza Universita di Roma, via Scarpa 16, 00161 Rome, Italy
Received 7 July 2019, accepted 15 May 2020, published online 14 November 2020
Using a technique which is inspired by topology, we construct original examples of 3-and 4-edge critical graphs. The 3-critical graphs cover all even orders starting from 26; the 4-critical graphs cover all even orders starting from 20 and all the odd orders. In particular, the 3-critical graphs are not isomorphic to the graphs provided by Goldberg for disproving the Critical Graph Conjecture. Using the same approach we also revisit the construction of some fundamental critical graphs, such as Goldberg's infinite family of 3-critical graphs, Chetwynd's 4-critical graph of order 16 and Fiol's 4-critical graph of order 18.
Keywords: Edge-colouring, critical graph, Mobius strip. Math. Subj. Class. (2020): 05C10, 05C15
1 Introduction
In the present paper, we deal with graphs that are not necessarily simple, so multiple (or parallel) edges are allowed but loops are excluded. We denote by x'(G) the chromatic index of a graph G, namely, the minimum number of colours that are needed for an edge-colouring of G. Vizing, in [12], proved that A(G) < x'(G) < A(G) + ^(G), where
*The authors are grateful to the referees, with particular regard to the constructive comments of one referee which resulted in an improved version of the manuscript. This manuscript was prepared with the funding support of Progetti di Ateneo, Sapienza Universita di Roma. t Corresponding author.
E-mail addresses: simona.bonvicini@unimore.it (Simona Bonvicini), andrea.vietri@uniroma1.it (Andrea Vietri)
Abstract
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
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A(G) and ^(G) are the maximum degree and the maximum multiplicity (the number of parallel edges for two fixed vertices) respectively. A simple graph G is said to be class 1 or 2 according to whether x'(G) is A(G) or A(G) + 1, respectively. We will restrict our attention to graphs whose chromatic index is at most A + 1. Edge-critical graphs will be our main object of study:
Definition 1.1. For a given graph G, let G - e denote the graph obtained by removing an edge e; G is A-(edge)-critical if x'(G) = A + 1 and x'(G - e) = A for any edge e.
In the literature, three small critical graphs of considerable importance appeared respectively in [9, 7] and [6]. The first graph (see the left side of Figure 10) was constructed by Goldberg as the first counterexample related to the "Critical Graph Conjecture" according to which all critical graphs should have an odd number of vertices (see [6]); such a graph had the smallest number of vertices (22) in an infinite family of graphs of even order constructed by Goldberg. The second graph - see the left side of Figure 1 - was found by Fiol as an example of critical, simple graph of smaller order, namely 18; the last graph - see the right side of the figure - is due to Chetwynd; it has order 16 but it is not simple because of one multiple edge.
It is still unknown whether a simple, critical graph of order 16 exists. As to smaller orders, such a question was settled by a number of contributions over the years. In details, Jacobsen's work (see [10]) ruled out all graphs with 4,6, 8, and 10 vertices; Fiorini and Wilson (see [8]) added the case 12 to the above list of non-admissible values; Bokal, Brinkmann, and Griinewald (see [2]) proved that also 14 is non-admissible.
In this paper, we push forward the analogy between non-orientable manifolds and class 2 graphs which was introduced in [11] and describe a new method for constructing critical graphs. We show the effectiveness of this method by constructing infinite families of critical simple graphs. The constructions cover all odd and even orders for 4-critical graphs, the odd order starting from 5, the even orders starting from 20, as well as all even orders for 3-critical graphs, including the orders of Goldberg's infinite family starting from 28
(a)
(b)
Figure 1: Two remarkable 4-critical graphs.
S. Bonvicini and A. Vietri: A Mobius-type gluing technique for obtaining edge-critical graphs 211
(the orders of Goldberg's graphs are all those numbers congruent to 8 (mod 16), and the further value 22). The 3-critical graphs of even order that we construct are not isomorphic to the graphs of Goldberg's infinite family; the graphs are simple, except the 4-critical graph of order 16. According to the literature, our constructions provide in particular the first example of an infinite family of A-critical graphs for degree 4. The present approach is expected to yield infinite families also for larger degrees, in the next future, because the key definitions can be easily exported to the general case.
Our method allows to build up critical graphs starting from class 1 graphs with an elementary and "nice" shape (see for instance Figure 2). This is innovative with respect to well-know methods that construct A-critical graphs starting from critical graphs with maximum degree not exceeding A - see Theorem 4.6 and 4.9 in [14].
Following the mentioned approach in [11], we also show that the infinite family of Goldberg's graphs disproving the "Critical Graph Conjecture" and the other two counterx-amples constructed by Fiol and Chetwynd can be obtained by a suitable identification of vertices which is pretty analogous to the topological identification yielding the Mobius strip from a rectangular strip. Details about the change of language - from topology to graph theory - can be found in [11].
Some additional terminology is required; in particular, certain distinguished vertices that play a basic role in the constructions shall be emphasised by suitable adjectives. Leaving details to the next sections, we anticipate that all the constructions will rely on particular pairs of vertices which are analogous to the extremes of a rectangular strip before the identification that leads to a Mobius strip. In our setting, any such pair will undergo a transformation which is similar to the topological identification of the extremes of the rectangular strip. The change from orientability to non-orientability, caused by the identification, is rephrased as the change from class 1 to class 2 as a consequence of the prescribed transformation.
Many standard definitions in this paper are in accordance with the textbook [3] by Bondy and Murty. As a further source, we mention the textbook [5] by Bryant. Edges like {u, v} are simply denoted by uv. We use the term t-colouring if the colour set has size t. Given a vertex v of a graph G, the palette of v, in symbols PY(v) or simply P(v), is the set of colours that a colouring 7 of G assigns to the edges containing v. In some cases, we will need to write yg so as to specify the graph we are colouring. The complementary set PY (v) or P(v) is the complementary palette of v with respect to the colour set of 7. If a colour is missing at a vertex v, we say that v lacks that colour. Finally, a vertex of degree h is an h-vertex.
For our purposes we also recall Vizing's Adjacency Lemma (VAL), concerning the structure of critical (simple) graphs, and the quite elementary, still very useful, Parity Lemma (PL):
Theorem 1.2 (VAL [13]). If uv is an edge of a A-critical graph, then u is adjacent to at least A — deg(v) + 1 A-vertices (different from v).
Lemma 1.3 (PL [1]). For any colouring of a graph G, the number of vertices that lack a given colour has the same parity as \V(G)|.
Although there exist several generalisations of VAL to multigraphs, for our purposes it suffices to consider the simple graph version (see the lines just above Remark 2.8).
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2 Fertile pairs of vertices
As hinted in the Introduction, the constructions of critical graphs that follow can be thought of as identifications of special pairs of vertices which change the colouring class from 1 to 2. Accordingly, the first step in each construction is the choice of a suitable pair of vertices which we are going to define as fertile pair. There are three kinds of fertile pairs, but after a little thought all of them can be related to the same kind - as we will soon explain. Conversely, given a critical graph, we will show that it is obtained as a suitable identification of a fertile pair which collapses to a unique vertex. In this reconstruction process, it is important to note that the identification could be arbitrarily performed on every vertex, but the choice of a particular vertex is essential both for proving criticality in a comfortable way, and for generating new critical graphs using a pattern which is readily suggested by the fertile pair.
Definition 2.1. Let u, v be vertices of a graph G. Assume that the following conditions hold:
(*) u is not adjacent to v, deg(u) + deg(v) < A and, for every A-colouring, P(u) n
P (v) = 0.
(**) For any edge e, G - e admits a A-colouring such that P(u) n P(v) = 0.
Then, u and v are said to be conflicting. Assume, instead, the following:
(*) deg(u) = deg(v) = A — 1 and, for every A-colouring, P(u) = P(v).
(**) For any edge e which does not contain u nor v, G — e admits a A-colouring such that
P(u) = P(v).
In this case, u and v are same-lacking. Finally, assume the following:
(*) deg(u), deg(v) are smaller than A and, for every A-colouring, |P(u) U P(v)| = A.
(**) For any edge e, G — e admits a A-colouring such that |P(u) U P(v)| < A.
In this last case, u and v are said to be saturating.
In all of the three cases, we say that (u, v) is a fertile pair of vertices.
Remark 2.2. After the removal of e in the same-lacking case, we equivalently require that |P(u) UP(v) | > 2; this is trivial if e contains one or both vertices u, v. Furthermore, notice that in the saturating case condition |P(u) U P(v) | = A is equivalent to P(u) n P(v) = 0.
The following lemma is the basic link between topology and graph theory in the present context, and should be considered the starting point for all the next constructions.
Lemma 2.3. Let (u, v) be a fertile pair of a graph G having x'(G) = A > 2. For each of the following cases, the corresponding operation yields a A-critical graph.
(i)	If u and v are non-adjacent and conflicting, identify u and v.
(ii)	If u and v are same-lacking, add a new vertex w and edges uw, vw.
S. Bonvicini and A. Vietri: A Mobius-type gluing technique for obtaining edge-critical graphs 213
(iii) If u and v are saturating, add the edge uv.
Proof. If we identify a pair of conflicting vertices, we obtain a graph G' having maximum degree A and no proper A-coloring, since the palettes of two conflicting vertices share at least one color; hence G' is class 2. By definition 2.1, if we remove any edge e from G', we find at lest one A-coloring of G' - e such that the two conflicting vertices have disjoint palettes with respect to it; therefore, G' is A-critical. The same-lacking and saturating cases can be managed analogously.	□
Notice that adding two pendant edges uw, vw' when u and v are same-lacking yields conflicting 1-vertices w,w'. Similarly, adding one pendant edge uw when u and v are saturating yields conflicting vertices w, v. Therefore, the above operations can be regarded as identifications of conflicting vertices in all cases. These procedures could be rephrased in terms of atlases and orientability, as explained in [11]; the prototype of this analogy is given by the odd cycle C2n+1 of any fixed length. Such a graph is the result of the identification of two conflicting vertices, namely, the extremes of the path P2n+2 having the same number of edges. The path is "orientable" (i.e. 2-colourable) but the identification of conflicting vertices increases the chromatic index and compromises orientability. More precisely, the orientation of P2n+2 starts from a "local chart" (a colouring of the 2-star containing a non-extremal vertex v), and the local chart is subsequently extended so as to cover as many edges as possible. In the case of the path, we succeed in covering all the graph (so we have a "global atlas", that is, a global 2-colouring) whereas the cycle does not allow for a global 2-colouring because one edge must be excluded (the atlas cannot be extended to the whole graph). Notice that the hypothesis (**) for conflicting vertices is crucial to prove criticality.
Remark 2.4. The 4-critical graphs in Figure 1 can be obtained in the way described in Lemma 2.3, by considering the graphs G17, G19 in Figure 8(b), 9(a), respectively, and identifying the vertices v, v'. Such vertices are conflicting, as we will show in Section 3.
Here follow some examples as a first step towards the main theorems.
Example 2.5. Let us show that the graph G5 in Figure 2(a) has saturating vertices u^ uj, with 1 < i < j < 4. For every 4-colouring the number of vertices that lack a fixed colour is odd, according to PL, whence every 3-vertex lacks a different colour; on the other hand, one can easily verify that the removal of any edge allows for a 4-colouring such that |P(uj) U P(uj)| = 3 for any pair of 3-vertices.
Example 2.6. The graphs G7 and G9 in Figure 2(c)-(d) have saturating vertices u1, u2, because PL implies that these vertices have disjoint palettes for any 4-colouring, and it remains to make routine checks after the removal of any arbitrary edge.
Example 2.7. The graph G6 in Figure 2(b) has same-lacking vertices v1, v2, because PL forces the palettes to be equal and this is no longer true if we remove any edge not containing one or both vertices v1 , v2 .
Notice that graphs with same-lacking vertices can be replicated so as to form a chain along which a color is "transmitted". Such a transmission of colour is a fundamental concept in this paper and will be described more thoroughly in the next section.
In the following remark, we consider critical graphs having at least three vertices of maximum degree. VAL implies that this property holds for every simple graph, but in the
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Figure 2: Fertile pairs of vertices: u1 and u2 are saturating, v1 and v2 are same-lacking.
presence of multiple edges the number of vertices of maximum degree might be smaller than 3. For instance, the complete graph K3 with A -1 parallel edges connecting two fixed vertices is A-critical and has only two vertices of maximum degree.
Remark 2.8. Let G be a A-critical graph having at least three vertices of maximum degree. Let u, v be adjacent vertices that are connected by h parallel edges (possibly h = 1). After deleting one of the parallel edges, u and v become saturating and the degree remains equal to A.
According to the above remark, Chetwynd's 4-critical graph can also be obtained by inserting an additional edge between the saturating vertices ui, u2.
3 Construction of graphs with fertile pairs
Graphs with fertile pairs of vertices can be obtained in several ways from smaller graphs with the same property. The methods we present here will be applied to prove the main theorems.
Lemma 3.1. Let H1 and H2 be vertex-disjoint graphs of degree A > 2 and such that x'(H1) = x'(H2) = A. Assume that v1, v2 are same-lacking in H1 and u1,u2 are same-lacking (resp. saturating) in H2. The graph H obtained from H1 and H2 by adding the edge u2v2 has again maximum degree A, chromatic index A, and has same-lacking (resp. saturating) vertices u1,v1.
Proof. Let us analyse the same-lacking case. A colouring of H can be obtained by assuming that u2 and v2 lack the same colour in two given A-colourings of H1 and H2; by the hypothesis, u1 and v1 lack that colour. If we now remove any edge, say in H1, u2v2 can be coloured with a colour which is present at u1. Such a colour is instead missing at v1. A similar argument applies to the saturating case.	□
S. Bonvicini and A. Vietri: A Mobius-type gluing technique for obtaining edge-critical graphs 215
Example 3.2. We consider two copies of G6 - see Figure 2(b) - as the graphs H1 and H2. We can actually iterate the gluing process m times, m > 1, so as to obtain a graph of order 6m, of maximum degree 4, whose 3-vertices are still fertile (same-lacking). Let us denote this graph by G™ - see Figure 3. This graph will play a basic role in the proofs of Theorem 5.1 and 5.2.
Figure 3: The graph G^ in Example 3.2 is a concatenation of graphs with same-lacking pairs.
The purpose of the next couple of definitions is twofold. On one hand, they allow to recover Chetwynd and Fiol's counterexamples in the light of our approach via transmission of colours along the edges of a graph. On the other hand, they play an important role in the construction of critical graphs of even order that will follow in the next pages. These definitions involve graphs with maximum degree 4, although they can be extended to graphs with A > 4.
Before providing the definitions, some further observations are in order. What we refer to as transmitting vertices should be regarded as terminal nodes which lend themselves to being connected to other graphs so as to yield a global graph with conflicting vertices and, eventually, a critical graph. The fundamental property of 2- or 3-colour transmitting vertices concerns the complementary palettes, that is, the colours actually missing at each vertex. For, the missing colours can be seen as the admissible colours of any edge which is added to the graph and contains that vertex. In the two definitions, it is the interplay between the colours missing at each distinguished vertex to ensure that the connecting edges, when added, will transmit some prescribed colour across the whole graph, and will eventually increase the chromatic index. Indeed, the vertices we are going to introduce are the first step towards the construction of graphs with conflicting vertices (see Propositions 3.8 and 3.12).
Let S © T denote the symmetric difference between the sets S and T.
Definition 3.3. Let G be a graph having x (G) = A = 4, and u,v,u^u2 be distinct vertices of G, where deg(u) = deg(v) = 2, deg(u1) = deg(u2) = 3. We say that G is 3-colour transmitting with respect to u, v, u1, u2 if the following conditions hold:
(1)	there exists a 4-colouring such that u1 and u2 lack distinct colours A and B, exactly one colour is missing simultaneously in u, v and this colour is either A or B;
(2)	for every 4-colouring such that u1 and u2 lack distinct colours A and B, |{A, B} U (P(u) © P(v))| = 3 (in particular, in the colouring in (1) the two other colours missing at u and v are different from A and B);
(3)	for every edge e there exists a 4-colouring of G-e with colours A, B, C, D satisfying
A € P(ui), B € PM, C € P(u) n P(Vy and the set {A, D} or {B, D} is contained in P(u) © P(v).
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If we slightly alter the above definition by setting « = u2 and deg(ui) = 2, the resulting graph is said 3-colour transmitting with respect to w, v, u1. In this case, the first requirement in (1) and (2) clearly becomes "u1 lacks colours A and B", in symbols A, B e
PR).
Definition 3.4. Let G be a graph of maximum degree A = 4 and x'(G) = 4. Let w, w1, w2 be distinct vertices of G, where deg(w) = 2, deg(w1) = deg(w2) = 3. We say that G is 2-colour transmitting with respect to w, w1, w2, if the following conditions hold:
(1)	for every 4-colouring of G the set |P(w1) U P(w2)| contains exactly two colours and coincides with P (w);
(2)	for every edge e there exists a 4-colouring of G - e with colours A, B, C such that
A e P(W1), B e P(W2) and P(w) contains {A, C} or {B, C}.
Similarly as above, if the vertices w1,w2 coincide and deg(w1) = 2, we say that the graph is 2-colour transmitting with respect to w, w1; the requirement in condition (2) becomes "w1 lacks colours A and B".
Example 3.5. The graph G12 in Figure 4(a) is 3-colour transmitting with respect to w, v, u1, u2, as we are going to explain by testing the conditions of Definition 3.3. Condition (1) holds as shown in Figure 4(a). Condition (3) can be checked by setting: P(w) C {2,3}, P(v) C {2,4}, and P(z1) C {1,4}. In the graph G12 - e, the palettes of the vertices W1,u2 take the following values: P(«1) C {1,2,3} and P(«2) C {1,3,4}; P(«1) C {2, 3,4} and P(«2) C {1,2, 3}; P(«1) C {2, 3,4} and P(«2) C {1,2,4}. Notice that P(«) C {2,3}, P(v) C {2,4} mean that 1 e P(«) n P(v) and {3,4} C P(«) © P(v), that is, colour 1 corresponds to colour C in Condition (3) and {3, 4} corresponds to one of the sets {A, D} or {B, D}, where A e P(«1), B e P(«2). Thus, for instance, if P(«1) C {1,2, 3} and P(«2) C {1,3,4}, then A = 4, B = 2 and D = 3.
It remains to prove Condition (2). By PL, the number of vertices that lack a given colour is even, and there are 6 vertices of degree smaller than 4. However, a color missing in all these vertices would make the two palettes of degree 3 equal, which is not allowed by assumption. Now let us partition the 2 • 3 + 4 • 2 colours on the above 6 vertices either as 2 + 2 + 4 + 6 or as 2 + 4 + 4 + 4, where each part counts the occurrences of a fixed colour (0 is missing, by the above discussion). Up to permutations of colours there are two colourings of the first type and three of the second type (in the table, palettes of size 4 are not present and we assume that palettes of size 3 are the same in all cases):
{1, 2, 3}	{1, 2, 3}
{1, 2,4}	{l, 2, 4}
{1, 2}	{1, 3}
{1, 2}	{1, 3}
{1, 3}	{1, 3}
{1, 4}	{1, 4}
{1, 2, 3}	{1, 2, 3}	{1, 2, 3}
{1, 2, 4}	{1, 2, 4}	{1, 2, 4}
{1, 2}	{1, 4}	{1, 3}
{1, 4}	{1, 4}	{1, 4}
{2, 4}	{2, 3}	{3, 4}
{3, 4}	{2, 4}	{3, 4}
Whatever the assignments of palettes to the 2-vertices, column 2 and column 4 satisfy (2). For the colouring 71 in the 1st column, condition |P(wi) U P(w2) U (P(w) © P(v))| = 3 is not satisfied if we choose {P(u),P(v)} = {{1, 2}, {1, 3}} or {P(w),P(v)} = {{1, 2}, {1,4}}. The permutation of colours 3 and 4 leaves y1 invariant and switches the sets {{1,2}, {1,3}}, {{1, 2}, {1,4}}. Therefore, in order to show that Condition (2)
S. Bonvicini and A. Vietri: A Mobius-type gluing technique for obtaining edge-critical graphs 217
is satisfied for the colouring 71, it suffices to show that the graph G12 cannot be coloured according to 71 by setting {P(u), P(v)} = {{1, 2}, {1, 3}}.
Suppose, on the contrary, that G12 can be coloured according to y1 by setting {P(u), P(v)} = {{1,2}, {1,3}}. The set of palettes of y1 shows that colour 1 induces a perfect matching of the graph G12. As shown in Figure 5, there are exactly four perfect matchings of G12. By the symmetry of the graph and by the fact that the sets {{1,2}, {1,3}}, {{1, 2}, {1,4}} can be obtained one from the other by a permutation of colours 3 and 4, we can consider the first two perfect matchings of Figure 5. The set of palettes of y1 also shows that colour 2 induces a matching of cardinality 5, where exactly one of the vertices u, v (respectively, z1, z2) is unmatched since we are supposing {P(u), P(v)} = {{1, 2}, {1, 3}} and {P(z1),P(z2)} = {{1,2}, {1,4}}. Figure 6 shows how to colour the edges of G12 with 1 and 2. In each of the four cases represented in Figure 6, one can see that is not possible to colour to edges of G12 according to the colouring y1 by setting {P(u), P(v)} = {{1, 2}, {1,3}}. Therefore, if G12 can be coloured by y1, then y1 satisfies Condition (2). The same can be repeated for the remaining colourings in the 3rd and 5th column. It is thus proved that every 4-colouring of G12 with |P(u1) © P(u2)| = 2 satisfies Condition (2).
Figure 4: (a): A 4-colouring of the graph Gi2 in Example 3.5 that satisfies Conditions (1) and (2) of Definition 3.3. (b): A 4-colouring of the graph H6 in Example 3.7.
Figure 5: Perfect matchings of the graph G12 that are considered in Example 3.5.
There are several methods for obtaining a 3-colour transmitting graph starting from a smaller one. For instance, in the graph G12 of Figure 4(a), we can delete the edge u1u2 and connect the remaining graph to the graph G™ in Figure 3 by adding the edges u1v1, u2v2. The resulting graph is 3-colour transmitting with respect to u, v, u1, u2. In the next example, we show a more elaborate method for obtaining a 3-colour transmitting graph starting
3
(a)
(b)
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ui	ui	ui	ui
Figure 6: The edges of the graph G12 are coloured according to the palettes
{1, 2,3,4}, {1,2,3}, {1,2,4}, {1, 2}, {1,2}, {1,3}, {1,4} by setting {P(u),P(v)} = {{1, 2}, {1, 3}} and {P(zi), P(z2)} = {{1,2}, {1,4}}; colour 1 induces a perfect matching, colour 2 induces a matching of cardinality 5, where exactly one of the vertices u, v (respectively, z1, z2) is unmatched (see Example 3.5).
from a smaller one. This method allows to find a graph that will be used to construct Fiol's 4-critical graph of order 18.
Example 3.6. Consider the graph N in Figure 7(a). Notice that P(w) = P(w1) © P(w2) for every 4-colouring of the graph N, as a straightforward consequence of PL. We denote by L the graph obtained from G12 in Figure 4 by deleting the edge m1m2. Let G16 be the graph resulting from the identification of the vertices w1 G V(N) with u1 G V(L) and of w2 G V(N) with u2 G V(L). We have that x'(L) = A = 4 (see the colouring in Figure 7(b)).
Let us show that G16 is 3-colour transmitting with respect to u, v, w by testing Definition 3.3 with u1 = u2. Condition (1) follows from the colouring in Figure 7(b). _
Condition (2) is satisfied if every 4-coloring of G16 satisfies the relation |P(w) U (P(u) © P(v))| = 3. Suppose that there exists a 4-colouring 7 of G16 such that |PY(w) U
(p7(u) © P7(v))| = 3, that is, P7(w) = {A,B}, P7(u) © P7(v) = {A, C} or {B,C}. The colouring 7 induces a colouring 7' of G12 such that Py (u1) © Py (u2) = {A, B} and P7' (u) © P7/ (v) = {A, C} or {B, C}, that is, 7' does not satisfies Condition (2) of Definition 3.3. That yields a contradiction, since G12 is 3-colour transmitting with respect
to u, v, u1, u2.
Condition (3) holds if for every edge e G E(G16) there exists a 4-colouring of G16 - e such that {A, B} Ç P (w), C G P (u) n P (v) and {A, D} Ç P (u) © P(v) where A, B, D are distinct. Assume e G E(G12). Since G12 is 3-colour transmitting with respect to u,v,u1,u2, there exists a suitable colouring which can be easily extended to the whole graph G16.
If e G E(N), we colour the edges of G16 belonging to G12 by the 4-colouring in Figure 4(a), so that P(u) = {2, 3} and P(v) = {2,4}. One can verify that the edges of N - e can be coloured in such a way that P (w) Ç {2,4}. Therefore, {1,3} Ç P (w), 1 G P (u) n P (v) and {3,4} Ç P (u) © P (v), that is, Condition (3) is satisfied if e G E(N ).
Example 3.7. The graph H6 in Figure 4(b) is 2-colour transmitting with respect to w, w1, w2. The conditions of Definition 3.4 are satisfied: Condition (1) follows from Parity Lemma; Condition (2) can be verified by coluring the edges with A, B, C, D and setting
P (w1) Ç {B, C, D}, P (w2 ) Ç {A, C, D}, P (w) Ç {A, D}.
S. Bonvicini and A. Vietri: A Mobius-type gluing technique for obtaining edge-critical graphs 219
w
(a)
4
wy 1
G
16
(b)
Figure 7: (a): The graph N. (b): A 4-colouring of the graph G16 that satisfies Conditions (1) and (2) of Definition 3.3; as proved in Example 3.6, the graph G16 is 3-colour transmitting with respect to u, v, w.
Definitions 3.3 and 3.4 are used to construct graphs having fertile vertices. The next result is a construction of graphs having fertile vertices and whose maximum degree A is 4. The construction can be extended to graphs whose maximum degree is larger than 4 and having multiple edges. In this context, we limit ourselves to consider A = 4.
We recall that a bowtie is the graph obtained by identifying two vertices belonging to two distinct 3-cycles, thus obtaining a centre of degree 4 and four 2-vertices. If the 3-cycle are (x, y1, y2) and (x', y'1 ,y'2), then we denote by B(x, y1,y2 ,y[ ,y'2) the bowtie resulting from the identification of the vertices x and x'.
Proposition 3.8. Let B = B(x, u', v', w, y) be a bowtie with centre x and 2-vertices u', v', w, y. Let K and M be graphs of maximum degree 4 and x'(K) = x'(M) = 4, with the following features. The graph K is 3-colour transmitting with respect to u, v, u1,u2, where degK(u) = degK(v) = 2, degK(u) = degK(u2) = 3; either M is 2-colour transmitting with respect to w,wi,w2, where degM(w) = 2,degM(wj) = degM(w2) = 3, or M is 2-colour transmitting with respect to w, w1, where degM (w) = degM (w1) = 2.
Let H be the graph obtained from B, K and M by identifying the vertices u' with u, w' with w and by adding the edges u1w1,u2w2 or u1w1,u2w1 according to whether M is 2-colour transmitting with respect to w, w1, w2 or with respect to w, w1, respectively. The graph H has maximum degree 4, x'(H) = 4 and the vertices v, v' are conflicting.
Proof. We identify the edge u2w2 with the edge u2w1 if w1 = w2, that is, if M is 2-colour transmitting with respect to w, w1. Since the identification of the vertices u, u' and w, w' does not increase the maximum degree of K, M and of the bowtie, the maximum degree of H is still 4. We show that x'(H) — 4. By Condition (1) of Definition 3.4, there exists a 4-colouring yM such that w1,w2 lack distinct colours A, B and these colours are missing in w (if w1 = w2, then w1 lacks both colours A, B). By Condition (1) and (2) of Definition 3.3, there exists a 4-colouring yK such that u1, u2 lack distinct colours A, B and exactly one of these two colours, say A, is missing simultaneously in u and v; the other two missing colours are different from B, that is, Py*k (u) = {A, C} Py*k (v) = {A, D}. We define a 4-colouring y* of H such that the restriction of y* to the edges of M (respectively, of K) coincides with yM (respectively, with y*k); the edges of the bowtie and u1w1,u2w2 are coloured as follows: {y* (u1w1),Y* (u2w2)} = {A, B}; Y*(wx) = A; y*(wy) = B; Y* (ux) = C; Y*(uv') = A; y(v'x) = B; and Y*(xy) = D. In conclusion x'(H) = 4.
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We prove that the vertices v, v' g V(H) are conflicting. Firstly, we show that for every 4-colouring of H, the palettes of v and v' share at least one colour. Suppose, on the contrary, that there exists a 4-colouring yi. of H such that v and v' have disjoint palettes. The restriction of yi to the edges of K (respectively, of M) is a 4-colouring yk (respectively, ym). The following relations hold: PYK(ux) = PYM(wx) = Yi(uiwi) = A; P7K (u2) = P7M (W2) = Yi(U2W2) = B (if wi = W2 then A = B and P7M (wi) = {A, B}). Moreover, PYK(v) = PY1 (v) = PY1 (v') = {y1(uv'), Y^v'x)} since we are supposing that v and v' have disjoint palettes with respect to y^ Therefore PYK (u) © P7K (v) = {Yi(uv'), Yi(ux)} © {Yi(uv'), Yi(v'x)} = {Yi(ux), Yi(v'x)}. By Condition (1) of Definition 3.4, the colours A, B are distinct and PYM (w) = {A, B}. It follows that {y1(wi), Y^wy)} = {A, B} and yi.(xy) = A, B, Y^ux), Y^v'x). Therefore, exactly one of the colours y^ux^yi.(v'x) is in {A, B}. Consequently, the set PYK (u) © PYK (v) = {yi. (ux), Y^v'x)} contains exactly one of the colours A, B. It follows that |PYK(ux) U Pyk(u2) U (PYK(u) © PYK(v))| = 3, a contradiction since K is 3-colour transmitting with respect to u, v, ux, vx. Hence, for every 4-colouring of H the palettes of the vertices v, v' share at least one colour.
We show that for every edge e g E(H) there exists a 4-colouring y' of H - e such that v and v' have disjoint palettes. We distinguish the cases: e g E(K); e g E(M); e g E(B); and e g {uxwx, u2w2}.
Case e e E(K).
By Condition (3) of Definition 3.3, there exists a 4-colouring 7 of K - e such that
A g Py (ui), B g Py (u2), C g Py (u) 0 Py(v), and the set {A, D} or {B,D} is contained in Py (u) © Py(v), where A, B, D are distinct. Without loss of generality, we can assume {A, D} Ç Py,(u) © Py(v). Now {A, D} can be contained in exactly one of the complementary palettes Py (u), Py ( v ) or in neither of them. The first case occurs only if e contains exactly one of the vertices u, v, and in this case {Py (u), Py (v)} = {{A, D, C}, {B, C}}. If, instead, e does not contain u, v, then {Py (u), Py (v)} = {{A, C}, {D, C}}.
We colour the edges of M according to an arbitrary 4-colouring ym of the graph M. By a permutation of the colours and by Condition (1) of Definition 3.4, we can assume that the colours A, B are missing in w and wx,w2 lack A, B, respectively (if wx = w2, then wx lacks both colours A, B). We define a 4-colouring y' of H - e such that the restriction of y' to K - e (respectively, to M) corresponds to the colouring Y (respectively, ym ) and y '(uxwx) = A; y'(u2w2) = B; y '(uv') = C ; y '(xy) = C. The colouring of the edges ux, v'x, wx, wy depends on the set {Py;(u),P^;(v)}. If {Py/(u),Py (v)} = {{A, C}, {D, C}}, then we set y'(wx) = B, Y'(wy) = A and the edges ux, v'x are coloured by A, D or D, A, respectively, according to whether PY/ (u) = {A, C} or Py (u) = {D, C}, respectively. If {Py (u),Py (v)} = {{A, D, C}, {B, C}}, then we set y'(wx) = A, Y'(wy) = B and the edges ux, v'x are coloured by D, B or B,D, respectively, according to whether Py (u) = {A, D, C} or Py (u) = {B, C}, respectively. Notice that Py (v') Ç PY/ (v), hence v, v' have disjoint palettes with respect to y'.
Case e e E(M).
We define a 4-colouring y' of H - e such that the edges of K are coloured according to the 4-colouring yk of K defined at the beginning of the proof. We have that Py (u) =
P7* (u) = {A, C}, P7/ (v) = P7* (v) = {A, D}. Since ux, u2 lack distinct colours A, B
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with respect to yK, we can assume that u lacks A and u2 lacks B.
By Condition (2) of Definition 3.4, we can colour the edges of M - e according to the 4-colouring yM of M such that the vertices w1, w2 lack distinct colours, say A, B, and the colours A, C are missing in w, where A, B, C are distinct (if wi = w2, then wi lacks both colours A, B). The remaining edges of H - e are coloured as follows: Y '(uiwi) = A; y' (W2W2) = B; y '(uv') = A; y'(ux) = C ; y '(v' x) = D; y'(wx) = A; Y'(wy) = C; and y'(xy) = B. The vertices v, v' have disjoint palettes with respect to y', since Py (v') = Py (v) = {A, D}.
Case e G E(B).
We define a 4-colouring y' of H - e that corresponds to the 4-colouring y* of H defined at the beginning of the proof, except on the remaining edges of B - e. The edges of B - e are coloured in such a way that Py (v') Ç {A, D}, Py (u) Ç {A, C} and {y'(wx), Y'(wy)} Ç {A, B}. The vertices v, v' have disjoint palettes with respect to y',
since Py (v') Ç Py(v) = {A, D}.
Case e G {u-iw1,u2w2}.
We define a 4-colouring y' of H - e which coincides with yK on the subgraph K. So we have that Py (u) = P7^ (u) = {A, C}, Py(v) = P7^ (v) = {a, D} and {yK(uiwi), Yk(u2w2)} = {A, B}. Without loss of generality, we can assume that the edge e that has been removed is coloured with A. By Condition (1) of Definition 3.4, we can colour the edges of M in such a way that wi, w2 lack two distinct colurs, say B, C, and these two colours are missing in w. The edges of B are coloured as follows: y'(uv') = A; Y'(ux) = C; y'(v'x) = D; y'(wx) = B; Y'(wy) = C; and Y'(xy) = A. The vertices v, v' have disjoint palettes with respect to y', since Py (v') = Py (v) = {A, D}.	□
Remark 3.9. The argument of the above proof is still valid if we assume that K is 3-colour transmitting with respect to u, v, ui, where u, v, ui have degree 2 in K.
Example 3.10. We apply Proposition 3.8 to the graphs K = Gi2 and M = H6 in Figure 4. As remarked in Example 3.5, the graph Gi2 is 3-colour transmitting with respect to u, v, ui, u2. Similarly, in Example 3.7 we have seen that H6 is 2-colour transmitting with respect to w, wi,w2. By Proposition 3.8, we obtain the graph G2i inFigure8(a). The graph G2i has order 21, maximum degree 4, and x'(G2i) = 4. The vertices v, v' G V(G2i) are conflicting. Following the proof of Proposition 3.8 we can colour the edges of G2i according to the 4-colourings yK and yM in Figure 4 by setting a =1, b = 2, c = 3 and d = 4 (or c = 4 and d = 3). This graph will be used in the proof of Theorem 5.2.
Example 3.11 (Chetwynd's counterexample). We can apply Proposition 3.8 to the graph K = Gi2 in Figure 4(a) and to the dipole M = D2 with two parallel edges even thought the dipole D2 is not 2-colour transmitting with respect to its vertices. More precisely, as remarked in Example 3.5, the graph Gi2 is 3-colour transmitting with respect to u, v, ui, u2. It is easy to see that every 4-colouring of the graph D2 satisfies conditions (1) and (2) of Definition 3.4 with wi = w2. Therefore, we can repeat the proof of Proposition 3.8 and obtain the graph Gi7 in Figure 8(b) having order 17, maximum degree 4 and x'(Gi7) = 4. The vertices v, v' G V(Gi7) are conflicting. By Lemma 2.3, the identification of the vertices v, v' yields a 4-critical graph, namely, Chetwynd's 4-critical graph in Figure 1(b).
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u
G21
u
G17
(b)
w1
Figure 8: (a): The graph G21 constructed in Example 3.10. (b): The graph G17 constructed in Example 3.11.
Proposition 3.12. Let B = B(x,u',v', w,y) be a bowtie with centre x and 2-vertices u', v', w, y. Let K and M be graphs of maximum degree 4 and x'(K ) = x' (M ) = 4 with the following features. The graph K is 3-colour transmitting with respect to u, v, u where degK(u) = degK(v) = degK(ui) = 2. The 2-vertices w, wi G V(M) are saturating and for every e G E (M ) not containing w nor w1 there exists a 4-colouring of M — e such that w, w1 lack exactly one colour simultaneously.
Let H be the graph obtained from B, K and M by identifying the vertices u' with u; w' with w; and u1 with w1. The graph H has maximum degree 4, x'(H) = 4 and the vertices v, v' are conflicting.
Proof. The argument is the same as in the proof of Proposition 3.8. It is different in the case e G E(M). We show that if we remove an edge e G E(M), then there exists a 4-colouring 7' of H — e such that v, v' have disjoint palettes with respect to it. As in the proof of Proposition 3.8, the restriction of y' to the edges of K corresponds to a 4-colouring YK of K such that P7^ (U1) = {A, B}, P7^ (u) = {A, C}, P7^ (v) = {A, D}. We set Y'(uv') = A, y'(mx) = C, y'(v'x) = D. The restriction of y' to the edges of M — e corresponds to a 4-colouring yM of M — e. Since u1 and w1 are identified, the palette of w1 with respect to y'm is contained in {A, B}. We define yM on the other edges of M — e as follows.
If e G E(M) does not contain w nor w1, then P7^ (w1) = {A, B}. By the assumptions, there exists a 4-colouring of M — e such that w, w1 lack exactly one colour simultaneously. By a permutation of the colours, we can set P7^ (w) = {A, C}. We can colour the remaining edges of H — e as follow: y'(wx) = B, y'(wy) = D, Y'(xy) = A. The colouring y' of H — e is thus defined and v, v' have disjoint palettes with respect to it, since Py (v') = Py (v) = {A, D}. We can repeat similar arguments if the edge e G E(M) contains w but not w1 .
If e G E(M) contains w1, then we can assume that P7^ (w1) = {A}. We can permute the colours in M — e so that P7^ (w) Ç {B, C} or P7^ (w) Ç {B, D}. The remaining edges of H — e are coloured as follows: y '(wx) = A, y '(xy) = B and Y'(wy) = D or C according to whether Py (w) Ç {B, C} or Py (w) Ç {B, D}, respectively. The
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colouring 7' of H - e is thus defined and v, v' have disjoint palettes with respect to it, since
Py (v') = Py (v) = {A,D}.	□
Example 3.13. The graph G25 in Figure 9(b) has order 25, maximum degree 4 and x'(G25) = 4. The vertices v, v' are conflicting. It is obtained by applying Proposition 3.12 to the graphs K = G16 in Figure 7(b) and M = G7 in Figure 2(c). The vertices w1 are identified. As remarked in Example 3.6, the graph G16 is 3-colour transmitting with respect to w, v, u1. As remarked in Example 2.6, the 2-vertices w, w1 G V(G16) are saturating. Moreover, for every e G G7 not containing w nor w1 there exists a colouring of G7 - e such that P(w1) Ç {A, B} and P(w) Ç {A, C}, that is, the assumption in Proposition 3.12 is satisfied. By Lemma 2.3, the identification of the conflicting vertices v, v' yields a 4-critical graph of order 24.
Example 3.14 (Fiol's counterexample). Proposition 3.12 is still true if we assume that M consists of exactly one vertex. For instance, consider the graph G19 in Figure 9(a) obtained from the graph G16 in Figure 7(b) and M consisting of exactly one vertex. The vertices u1 and w1 are identified. The vertices v, v' G V(G19) are conflicting (we can repeat the proof of Proposition 3.8 without considering the case e G E(M)). By Lemma 2.3, the identification of the vertices v, v' yields a 4-critical graph, namely, Fiol's 4-critical graph in Figure 1(a).
Figure 9: ui and w should be identified in both graphs. (a): The graph G19 has order 19, maximum degree 4 and x'(G19) = 4. (b): The graph G25 has order 25, maximum degree 4 and x'(G25) = 4. As shown in Example 3.14, the vertices v, v' are conflicting.
4 Counterexamples to the Critical Graph Conjecture
In 1971, Jacobsen showed that there are no 3-critical graphs of order < 10 and no 3-critical multigraphs of order < 8. This led him to formulate the Critical Graph Conjecture. As we already mentioned, the first counterexamples to the conjecture were constructed by Goldberg [9], and afterwards by Chetwynd [6] and Fiol [7]. In this section we show that
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also Goldberg's counterexample can be obtained by a Mobius-type technique. Furthermore, combining our technique with Goldberg's construction we show that for every even value value of n, n > 22, there exists a 3-critical graph of order n.
Goldberg was the first to disprove the Critical Graph Conjecture by constructing an infinite family of 3-critical graphs of even order, the smallest of which has order 22 [9]. The graph of order 22 is represented in Figure 10(a). A 3-critical graph of the infinite family can be obtained from the 3-critical graph of order 22 in Figure 10(a) by adding in pairs the graph H7 of order 7 in Figure 10(b). The result is the graph in Figure 11(a). A 3-critical graph of the infinite family has order n = 8 (mod 16), n > 24.
H
Xi	X2
Figure 10: (a): The 3-critical graph of order 22 constructed by Goldberg. (b): The graph H7 which is used to construct 3-critical graphs of order n = 8 (mod 16), n > 24.
Figure 11: (a): The infinite family of 3-critical graphs of order 8m, m > 3, m odd, constructed by Goldberg. (b): The graph H23 that yields the 3-critical graph of order 22 constructed by Goldberg by identifying the conflicting vertices u, v.
In what follows, we show that the 3-critical graphs constructed by Goldberg can be obtained by a Mobius type technique, namely, by identifying a pair of conflicting vertices in the case of the graph in Figure 10(a), or by connecting a pair of saturating vertices in
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the case of the graph in Figure 11(a). In Lemma 4.2, we will show that the vertices u, v of the graph H23 in Figure 11(b) are conflicting. We give a proof of the fact that u, v are conflicting showing that the structure of the graph H7 forces to colour the edges of the graph in Figure 10(a) in a prescribed way, thus determining which vertex has to be split into two conflicting vertices. Analogously, for the proof of Lemma 4.3. The proofs of Lemmas 4.2 and 4.3 are based on the following result.
Lemma 4.1. Every 3-colouring of the graph H7 in Figure 10(b) satisfies the following condition:
\P(xo) U P(xi) U P(xi+2)| = 3 and P(xm) = P(xi+s) = P(xr) where i = 1 or i = 2, r £ {0,i,i + 2} and the subscripts are (mod 4).
Proof. Since the colour set has cardinality 3 and PL holds, exactly three vertices of H7 lack the same colour A and the remaining 2-vertices of H7 lack distinct colours B, C, both different from A. A direct inspection on the graph shows that the vertices lacking the same colours are xi+1,xi+3 and xr, where i = 1 or i = 2 and r £ {0,i,i + 2}.	□
Lemma 4.2. The graph H23 in Figure 11(b) is class 1 and the vertices u,v £ V(H23) are conflicting.
The 3-critical graph of order 22 in Figure 10(a) constructed by Goldberg can be obtained from the graph H23 by identifying the conflicting vertices u,v £ V(H23).
Proof. It is easy to see that H23 is class 1. We show that the vertices u,v £ V(H23) are conflicting. Firstly, we prove that P(u) n P(v) = 0 for every 3-colouring of the the graph H23.
Let 7 be a 3-colouring of H23. Since 7 induces a 3-colouring of the subgraphs of H23 that are isomorphic to H7 andLemma4.1 holds, it is either \{y (x1y2), 7(x3y4), y(x0v)}\ = 3 or \{y(x2Zi),7(x4Z3),y(xov)}\ = 3. If \{7(xiy2),7(x3y4),7(xov)}\ = 3, then 7(x0v) = 7(yov), by virtue of Lemma 4.1 on the subgraph of H23 which is isomorphic to H7 and contains the vertices yi, 0 < i < 4. That yields a contradiction, hence \{7(x2z1), 7(x4z3), y(x0v)}\ = 3. Since Lemma 4.1 holds on the subgraph of H23 which is isomorphic to H7 and contains the vertices zi, 0 < i < 4, we have y(x0v) = y(z0u). It is thus proved that P(u) n P(v) = 0 for every 3-colouring of H23.
It remains to prove that for every edge e £ E(H23) there exists a 3-colouring 7' of H23 - e such that the vertices u, v have disjoint palettes with respect to it. The existence is straightforward if e is incident to u, since u has degree 1. Let {1,2, 3} be the colour set of 7'. To define 7', it suffices to define 7' on the edges in {xov,yov,zou,xiyi+1, xi+1zi : i =1, 3} and colour the remaining edges according to Lemma 4.1. For instance, if e is incident to the vertices in {xi, yi : 0 < i < 4}, e £ {xov, yov, zou, xiyi+1, xi+1Zi : i = 1,3}, then we set 7'(x^) = y'(zou) = 1; 7'(x3y4) = 7'(xov) = 2; 7'(y0v) = 3; y'(x2z1) = y'(x4z3) = a £ {1,2}. The remaining cases can be managed in a similar way. It is thus proved that u, v are conflicting. Now the assertion follows from Lemma 2.3 by identifying the vertices u, v.	□
Lemma 4.3. Let H8m, m > 3, m odd, be the graph obtained from the graph in Figure 11(a) by deleting the edge u1um. The graph is class 1 and the vertices u1,um are saturating. The 3-critical graphs of the infinite family constructed by Goldberg can be obtained by connecting a pair ofsaturating vertices.
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Proof. One can easily verify that the graph H8m is class 1. We prove that ui,um are saturating. Firstly, we show that |P(u1) U P(um)| = 3 for every 3-colouring of the graph H8m. For 1 < j < m, let Hj be the subgraph of H8m which is isomorphic to the graph H7 in Figure 10(b) and contains the vertices xj, 0 < i < 4. Every 3-colouring y of H8m induces a 3-colouring 7' of the graph H7, that is, Lemma 4.1 holds. By the symmetry of the graph, we can assume that |Py (x1) U Py (x2) U Py (x1 )| = 3 and Py (x1) = Py (x1). Consequently, P7/(x2) = Py(x|) and |Py (x0) U Py(x1) U Py(x§)| = 3. From this we deduce that |Py (x0) U Py (x2) U Py (j = 3 and Py (x1) = P7/ (x3) if j is odd, 1 < j < m; |Py (x0) U Py (x1) U Py (x3)| = 3 and Py (x2) = P7' (j if j is even, 1 < j < m. It follows that 7(x0uj) = y(x0+1Uj+1) for every 2 < j < m - 1, j even. We colour the edges of H8m by {1,2,3} and set 7(x0u2) = y(x0«3) = 3. Without loss of generality we can set y(«2«3) = 1, whence y(m1m2) = 2. One can see that {Y(xo«j), Y(«jUj+1)} = {y(x0Uj+1), Y(ujUj+1)} = {1,3} for every 2 < j < m - 1, j even. As a consequence, P(um) = {1,3}. It is thus proved that |P(u1) U P(um) | =3 for every 3-colouring of H8m, since 2 G P(u1).
We omit the routine proof that for every e G E (H8m) there exists a colouring of H8m such that |P(u1)UP(um)| < 3. It is thus proved that u1,um are saturating and the assertion follows from Lemma 2.3.	□
It is known that the 3-critical graph of order 22 constructed by Goldberg is the smallest 3-critical graph [4]. Combining our construction with that one of Goldberg, we can prove the following result.
Theorem 4.4. For every even value of n, n > 22, there exists a 3-critical graph of order n.
Proof. A critical graph of the infinite family constructed by Goldberg has order n = 8 (mod 16), n > 24. We construct a 3-critical graph of order n = 2 (mod 4), n > 26; and n = 0 (mod 4), n > 28. We define the auxiliary graphs H', K' and H'' that will be used in the construction. The graph H' is defined as follows. Consider m > 1 copies of the complete graph K4 - e; the 2-vertices of K4 - e are same-lacking. For 1 < i < m - 1, connect the ith copy of K4 - e to the (i + 1)th by adding exactly one edge joining a 2-vertex in the ith copy to a 2-vertex in the (i + 1)th copy. The resulting graph H' has exactly two 2-vertices, say v1, v2. By Lemma 3.1, the graph H' has maximum degree 3, x'(H') = 3 and the vertices v1, v2 are same-lacking. Let K' be the graph of order 6 that can be obtained from the graph G6 in Figure 2(b) by deleting the edges v1v2, v3v5, v4v6. The graph K' has maximum degree 3, x'(K') = 3 and the vertices v1, v2 are same-lacking. The graph H'' is obtained from the graphs H' and K' by connecting the vertex v2 G V(K') to the vertex v1 G V(H'). By Lemma 3.1, the graph H'' has maximum degree 3, x'(H'') = 3 and the vertices v1 ,v2 are same-lacking. Let H be the graph obtained from the graph H23 in Figure 11(b) and the graph r, where r g {H', K', H''}, by deleting the edge z0u G E(H23) and adding the edges z0v1,uv2. As remarked in Example 2.7, a graph with same-lacking vertices is able to transmit a color, therefore the graph H has maximum degree 3, x'(H) = 3 and the vertices u, v G V(H) are conflicting. Notice the following: |V(H)| = 23 + 4m > 27if r = H'; |V(H)| = 29 if r = K'; |V(H)| = 29 + 4m > 33 if r = H''. By Lemma 2.3, the identification of the conflicting vertices u, v G V(H) yields a 3-critical graph of order | V(H) | - 1. Hence, the assertion follows.	□
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The 3-critical graphs of order n = 0 (mod 4), n > 28, that are constructed in the proof of Theorem 4.4, include the orders of Goldberg's infinte family but are not isomorphic to them. In fact, Goldberg's graphs have girth larger than 3; the 3-critical graphs in the proof of Theorem 4.4 have girth 3 as K' contains a 3-cycle.
5 From graphs with fertile vertices to 4-critical graphs
We show that it is possible to obtain 4-critical graphs of order n, for every n > 5, starting from the four graphs in Figure 2, the two graphs in Figure 1 and the graph G21 in Figure 8(a); these graphs have a pair of fertile vertices.
Theorem 5.1. For every odd integer n > 5 there exists a 4-critical simple graph of order n.
Proof. For every odd integer n > 5, we exhibit a graph H of maximum degree 4, x'(H) = 4 and order n having a pair of saturating vertices u^v^ The assertion follows from Lemma 2.3 by adding the edge u1v1.
The graph H is obtained from Lemma 3.1 as follows. We take the graph G™ in Figure 3 as the graph H1 in Lemma 3.1, where m > 1. As remarked in Example 3.2, it has order 6m > 6, maximum degree 4 and the vertices v1,v2 G V(G™) are same-lacking. We define the graph H2 in Lemma 3.1 as follows: if n = 1 (mod 6), then H2 is the graph G7 in Figure 2(c); if n = 3 (mod 6), then H2 is the graph G9 in Figure 2(d); if n = 5 (mod 6), then H2 is the graph G5 in Figure 2(a). By the remarks in Examples 2.5 and 2.6, the vertices u1,u2 G V(H2) are saturating. By Lemma 3.1, the graph H obtained from H1 = G™ and H2 by adding the edge u2v2 has maximum degree 4, x'(H) = 4 and the vertices u1, v1 G V(H) are saturating. Notice that |V(H)| = 6m + |V(H2)| > 11, where m > 1 and |V(H2)| G {5,7,9}. The graph G obtained from H by adding the edge u1v1 is 4-critical, since Lemma 2.3 holds. By construction, the graph G is simple. Since |V(G)| = |V(H)|, for every odd integer n > 11 there exists a 4-critical simple graph of order n. For n = 5, 7, 9, the assertion follows from Lemma 2.3 by setting H = G5, G7, G9, respectively, and by adding the edge u 1 u2.	□
Theorem 5.2. For every even integer n > 16 there exists a 4-critical graph of order n. The graph is simple unless n is equal to 16.
Proof. For n = 16,18, we resort to the well known graphs in Figure 1. For n = 20 we consider the graph G21 in Figure 8(a). As remarked in Example 3.10, the vertices v, v' G G21 are conflicting. The existence of a 4-critical graph of order 20 follows from Lemma 2.3 by identifying the vertices v and v'. Notice that the graph is simple.
For every even integer n > 22, we exhibit a graph H of maximum degree 4, x' (H) = 4 and order n having a pair of saturating vertices u1, v1. The assertion follows from Lemma 2.3 by adding the edge u1v1. The graph H is obtained from Lemma 3.1 as follows. We take G™ in Figure 3 as the graph H1 in Lemma 3.1, where m > 1. The graph H2 in Lemma 3.1 has even order and its definition depends on the congruence class of n modulo 6.
Case n = 0 (mod 6), n > 18.
The graph H2 is obtained from the 4-critical graph of order 18 in Figure 1(a) by the deletion of the edge u1u2. Alternatively, we can consider the 4-critical graph arising from the graph G25 in Figure 9(b) by identifying the vertices v, v' (see Example 3.13); H2 can be obtained by deleting one of the two edges containing u1 .
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Case n = 2 (mod 6), n > 20.
Consider the 4-critical graph G20 of order 20 obtained from the graph G21 in Figure 8(a) by identifying the vertices v, v'. Let H2 be the graph obtained from G20 by deleting the edge u1u2.
Case n = 4 (mod 6), n > 16.
The graph H2 is obtained from the 4-critical graph of order 16 in Figure 1(b) by the deletion of one parallel edge connecting the vertices u1, u2. For each congruence class of n, the vertices u1,u2 G V(H2) are saturating, since Remark 2.8 holds. Moreover, H2 is a simple graph of maximum degree 4, x'(H2) = 4 and |V(H2)| = 18, 20,16 according to whether n = 0,2,4 (mod 6), respectively. By Lemma 3.1, the graph H obtained from H1 = Gm and H2 by adding the edge u2v2 has maximum degree 4, x'(H) = 4 and the vertices u1, v1 G V(H) are saturating. Notice that |V(H)| = 6m + |V(H2)| > 22, where m > 1 and |V(H2)| G {16,18, 20}. By Lemma 2.3, the graph G obtained from H by adding the edge u1v1 is 4-critical. Since | V(G) | = | V(H) |, for every even integer n > 22 there exists a 4-critical graph of order n. Notice that these graphs are simple. Combining this result with the remarks on the existence of 4-critical graphs of order 16,18 and 20, the assertion follows.	□
There are alternative methods for constructing 4-critical graphs. For instance, consider the 4-critical graph G of order 20 obtained from the graph G21 in Figure 8(a) by identifying the vertices v, v'. Delete the edge u1u2 g E(G) and connect the remaining graph to the graph Gm in Figure 3. For every m > 1 we obtain a 4-critical graph of order 6m + 20.
6 A concluding remark
We are confident that the present work will provide suggestions and tools for constructing infinite families of critical graphs even beyond degree 4. The next step should be inevitably the degree 5. The key definitions are compatible with the general case, and we believe that the method is versatile enough. With some effort and further investigation, new infinite families are expected to be found in the near future.
ORCID iDs
Simona Bonvicini © https://orcid.org/0000-0001-5318-7866 Andrea Vietri © https://orcid.org/0000-0002-6064-7987
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ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 19 (2020) 231-247 https://doi.org/10.26493/1855-3974.2125.7b0
(Also available at http://amc-journal.eu)
On resolving sets in the point-line incidence graph of PG(n, q)
Daniele Bartoli * ©
Dipartimento di Matematica e Informática, Universita degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
Department of Geometry and MTA-ELTE Geometric and Algebraic
Combinatorics Research Group Eotvos Lorand University, Budapest, 1117 Budapest, Pâzmâny Péter sétâny 1/C, Hungary, and FAMNIT, University of Primorska, 6000 Koper, Glagoljaska 8, Slovenia
Stefano Marcugini * ©, Fernanda Pambianco * ©
Dipartimento di Matematica e Informatica, Universita degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
Received 25 September 2019, accepted 13 July 2020, published online 15 November 2020
Lower and upper bounds on the size of resolving sets and semi-resolving sets for the point-line incidence graph of the finite projective space PG(n, q) are presented. It is proved that if n > 2 is fixed, then the metric dimension of the graph is asymptotically 2qn-1.
Keywords: Point-line incidence graph, resolving sets, finite projective spaces. Math. Subj. Class. (2020): 05B25, 05C12, 51E20
*	The research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INDAM).
t Corresponding author. The research was supported by the Hungarian National Research Development and Innovation Office, OTKA grant no. K 124950, and by the Slovenian Research Agency (research project J1-9110).
*	The research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INDAM), and by the University of Perugia (Project: "Curve, codici e configurazioni di punti", Base Research Fund 2018).
E-mail addresses: daniele.bartoli@unipg.it (Daniele Bartoli), kissgy@cs.elte.hu (Gyorgy Kiss), stefano.marcugini@unipg.it (Stefano Marcugini), fernanda.pambianco@unipg.it (Fernanda Pambianco)
György Kiss f ©
Abstract
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
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1	Introduction
For a simple, connected, finite graph r = (V, E) and x, y e V let d(x, y) denote the length of a shortest path joining x and y.
Definition 1.1. Let r = (V, E) be a finite, connected, simple graph. Two vertices v\,v2 e
V are divided by S = {si, s2,...,sr} c V if there exists sj e S so that d(v^ sj) = d(v2, Sj). A vertex v e V is resolved by S if the ordered sequence (d(v, s1), d(v, s2),..., d(v, sr)) is unique. S is a resolving set in r if it resolves all the elements of V. The metric dimension of r, denoted by ^(r), is the size of the smallest example of resolving set in it.
The study of metric dimension is an interesting problem in its own right and it is also motivated by the connection with the base size of the corresponding graph. The base size of a permutation group is the smallest number of points whose stabilizer is the identity. The base size of r, denoted by b(r), is the base size of its automorphism group Aut(r). The study of base size dates back more than 50 years, see [18]. A resolving set in r is obviously a base for Aut(r), so the metric dimension of a graph gives an upper bound on its base size. The difference ^(r) - b(r) is called the dimension jump of r. Distance-transitive graphs whose dimension jump is large with respect to the number of vertices are rare, and hence interesting objects. For more information about general results on metric dimension and base size we refer the reader to the survey paper of Bailey and Cameron [2].
Resolving sets for incidence graphs of some linear spaces were investigated by several authors [4, 9, 10, 12]. In these cases much better bounds than the general ones are known. Estimates on the size of blocking sets can be used to prove lower bounds on the metric dimension, and the knowledge of geometric properties is useful for constructions and upper bounds. It was shown by Heger and Takats [12] that the metric dimension of the point-line incidence graph of a projective plane of order q is 4q - 4 if q > 23. In a recent paper Heger et al. [11] extended this result for small values of q, too.
There are two natural generalizations of this planar result in higher dimensional spaces: one can consider either the point-hyperplane incidence graph, or the point-line incidence graph of PG(n, q). In the former case resolving sets are connected with lines in a higgledy-piggledy arrangement which were investigated by Fancsali and Sziklai [9]. Their results were recently improved by the authors of this paper [5]. The latter case is studied in the present paper. We assume that the reader is familiar with finite projective geometries. For a detailed description of these spaces we refer to [14, 16].
Let rn,q denote the point-line incidence graph of the finite projective space PG(n, q). The two sets of vertices of this bipartite graph correspond to points and lines of PG(n, q), respectively, and there is an edge between two vertices if and only if the corresponding point is incident with the corresponding line. In rn q the distance of two different lines is
2	if they intersect each other and 4 if they are skew. The distance of a point P and a line I is 1 if P is on and it is 3 if P is not on I. Finally, the distance of two different points is always 2. Hence, points cannot be resolved by other points. Considering these properties, the following definitions are natural.
Definition 1.2. A set S of points and lines of PG(n, q) is a semi-resolving set for points (lines) in rn,q if it resolves all the vertices of rn,q corresponding to points (lines).
Definition 1.3. Let S be a (semi-)resolving set in rn q. A point or a line is called inner (outer) if it is (not) in S. An outer point is called t-covered if it is incident with exactly t
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lines of S. A point or a line is called uncovered if it has empty intersection with all elements of S.
The paper is organized as follows. First, in Section 2 we provide a pure combinatorial proof of a lower bound for resolving sets in rn,q. In the second part of the section interesting constructions are presented for n > 3 which yield examples asymptotically close to the lower bound. These resolving sets are related with regular spreads of lines in projec-tive spaces. We prove that the metric dimension of rn q is asymptotically 2qn-1 and its dimension jump is roughly where v denotes the number of its vertices. In Section 3 algebraic curves and blocking sets are applied. We consider a different type of line spread in PG(3, q) and we obtain examples of resolving sets in rn,q of smaller size when q = ph, p prime and h > 1. Finally, in Section 4 computer aided results for small values of q are given.
2 General bounds
In this section we present lower and upper bounds on ^(rn,q) for all q and n > 3. Particular attention is paid to the case n = 3, since general upper bounds in any dimension depend on the 3-dimensional upper bound, see Proposition 2.15 and Theorem 2.11.
Theorem 2.1. The size of any semi-resolving set for points in rn,q is at least
2_ qn+1 -q
q2 + q - 2
Proof. Let S be a semi-resolving set for points in rn,q which consists of k lines and m points.
Count in two different ways the number of incident point-line pairs (P, with I G S. On the one hand, this number is exactly k(q +1). On the other hand, there is at most one uncovered point, the number of 1-covered points is at most k and any other outer point must be covered by at least two lines of S. Hence
/qn+^ — 1
k(q + 1) > k + 2 ---1 - k - m
q-1
so
^ 2(qn+1 - q) qm k + m > --—-- +
(q - 1)(q + 2) q + 2' This gives the required inequality at once.	□
Corollary 2.2. The size of any resolving set in r3,q is at least 2 [q2 - q + 3 - j .
Corollary 2.3. The metric dimension of r3,q is at least 2(q2 - q + 3) for q > 10.
From now on we focus on upper bounds which will be given by constructions. For our examples we need the notion of spreads, in particular line spreads.
Definition 2.4. A k-spread Sk of PG(n, q) is a set of k-dimensional subspaces with the property that each point of PG(n, q) is incident with exactly one element of Sk.
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By definition, a k-spread of PG(n, q) consists of qqkXi_\ elements. The following theorem about the existence of spreads was proved independently by several authors, see [1,7, 17].
Theorem2.5. The projective space PG(n, q) hasa k-spreadifandonly if (k+1) | (n+1).
Hence there exists a line spread in any odd dimensional projective space. Two line spreads are said to be disjoint, if they do not share any common line. Our first construction for a semi-resolving set for points in rn q is based on disjoint line spreads. We use the following theorem of Etzion [8] about the existence of disjoint line spreads.
Theorem 2.6 (Etzion). If n > 3 is odd, then there exist at least two disjoint line spreads in PG(n, q).
Theorem 2.7. If n > 3 is odd, then there exists a semi-resolving set for points in rn,q of size
q" -1 _ i
rp (n,q) = 2q2 q 2 ^ .	(2.1)
q2 — 1
Proof. Let L1 and L2 be two disjoint line spreads in PG(n, q), and lj G Lj be arbitrary lines. We claim that S = L1 U L2 \ {¿i, ¿2} is a semi-resolving set for points in rn,q.
Each point not in £j is contained in a unique pair of lines (r1, r2) G L1 x L2. Each point of ¿1 \ ¿2 is contained in a unique line of L1 U L2 \ {¿1, ¿2} and each point of ¿2 \ ¿1 is contained in a unique line of L1 U L2 \ {¿1, ¿2}. The (possible) unique point ¿1 n ¿2 is the only point of PG(n, q) not contained in any line of L1 U L2 \ {¿1, ¿2}. The size of S is
n+1_1
2 q 2 1--2, hence the statement follows.	□
q2 — 1	'
Proposition 2.8. Let E be a hyperplane and L be a line spread in PG(n, q), n > 3 odd. Then E contains exactly qq2——1 elements of L.
Proof. Any element of L is either fully contained in E, or intersects it in exactly 1 point. The elements of L partition the set of points of E. Hence, if x denotes the number of fully contained lines, then
q" — 1 ( +1) + (q"+1 — 1
-- = (q + 1)x + —----
q — 1	V q2 — 1
The claim follows from this equation at once.	□
Theorem 2.9. Let L1 be a line spread in PG(3, q). Then there exists another line spread L2 in PG(3, q) such that L1 and L2 do not share any common line.
Proof. Let f (X, Y) be an irreducible homogeneous quadratic polynomial and Hi denote the hyperbolic quadric in PG(3, q) with equation
f (Xo,Xi)+ if (X2, X3) = 0
for i = 1,2,..., q — 1. Apply a suitable linear transformation so that the images ¿1 and ¿2 of the lines : X0 = X1 =0 and 1'2: X2 = X3 =0 do not belong to L1. Let Hj denote the image of Hj, and let E and Fj denote the two reguli of lines on Hj. Then for each i at most one of Ej and Fj contains some elements of Li, because any line of Ej intersects
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any line of F and no two elements of A intersect each other. Hence we can choose the notation so that E does not contain any element of A for all i. This implies that the spread
q-1
L = u Ei
i=i
does not share any common line with L1.	□
The next proposition gives a useful recursive construction method.
Proposition 2.10. Let S be a semi-resolving set for points in rd,q of size k. Suppose that m elements of S are contained in a hyperplane Sd-1 of PG(d, q), and Sd-1 also contains the (at most one) uncovered point. Then rd+1q has a semi-resolving set for points of size (q + 1)k — qm.
Moreover, if S is a resolving set in rd,q and Sd-1 also contains the (at most one)
d— 1 1
uncovered line, then rd+1,q admits a resolving set of size (q + 1)k — qm + q q-- .
Proof. Embed Sd-1 C PG(d, q) into PG(d+1,q), and consider in PG(d+1,q) thepencil of hyperplanes with carrier Sd-1. These hyperplanes, Sd, Sd,..., Sd+1, are isomorphic to PG(d, q). Take a copy of S in Sd and denote it by S1 for i = 1,2,..., q + 1. Finally, let
_ q+1 s = u s 4.
¿=1
We claim that S is a semi-resolving set for points in rn+1q. Inner points are resolved by definition. If two outer points, P1 and P2, are in the same Sd, then they are already divided by S4. If P1 is in S4 and P2 is in Sj with i = j, then, as none of P1 and P2 is uncovered and none of them is in Sd-1, there exist distinct lines £4 G S4 through P1 and £j G Sj through P2. Hence £4 does not contain P2, so d(P1, £4) = d(P1, £4). Since the size of S is m + (q + 1)(k — m), the first part of the statement is proved.
Now suppose that S is a resolving set in rd q. Then the elements of any point-line pair are obviously divided by S. Let £1 and £2 be two lines. If at least one of them is an element of S, then they are divided by definition. From now on we assume that none of the two lines is an element of S. We distinguish three main cases and some subcases.
1.	If both of them are entirely contained in the same Sd, then they are divided by S4.
2.	If there is no Sd that contains both £1 and £2, but each of the lines is entirely contained in some Sd, say £1 C Sd1 and £2 C Sd2, then none of the lines is in Sd-1. Let Pj denote the unique point £j n Sd-1 for j = 1,2.
•	If P1 = P2, then let P3 = P1 be a point on £1. Since S41 is a semi-resolving set for points in Sd1 and P3 is not an uncovered point, either P3 G S41 or there exists at least one line £ G S11 which contains P3 but does not contain P1. In the former case d(£1,P3) = 1 = 3 = d(£2,P3). In the latter case d(£1, £) = 2 = 4 = d(£2, £), so we are done.
•	If P1 = P2, then we may assume that P1 is not an uncovered point, because there is at most one uncovered point. Again, either P1 G S41 or there exists at least one line £ G S11 which contains P1 but does not contain P2. In the former case d(£1,P1) = 1 = 3 = d(£2,P1), while in the latter case d(£1,£) = 2 = 4 = d(£2, £), so £1 and £2 are divided by S41.
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3. If i\ is not contained in any Ed, then it cannot meet Ed_i, so there exists a unique point Pi = £1 n Ed for all i = 1,2,..., q + 1.
•	If ¿2 is not contained in any Ed, then it cannot meet Ed-1, so there exists a unique point P2 = ¿2 n Ed for all i = 1, 2,..., q +1. The two lines have at most one point of intersection, hence there exist at least q superscripts so that Pi = P2\ Since Si is a semi-resolving set for points in Ed, there exists at least one element s G Si so that d(P1, s) = d(P2, s). Hence
d(^, s) = d(Pi, s) + 1 = d(P2\ s) + 1 = d(^2, s),
so the lines are divided by Si.
•	If ¿2 is contained in a unique Ed, then it is not contained in Ed_ 1, so there exists a unique point P2 = ¿2 n Ed-1. Let j = i and consider Ej which contains both Pj and P2. Since Sj is a semi-resolving set for points in Ej there exists at least one element s G Sj so that d(Pj, s) = d(P2, s). Hence
d(^, s) = d(Pj, s) + 1 = d(P2, s) + 1 = d(^2, s),
the claim is proved.
•	Finally, suppose that ¿2 is contained in Ed_ 1. Then and ¿2 are not necessarily
divided by S. Suppose that S consists of lines only. Then ¿2 and Pi are divided
by Si, but it could happen that a line of Si intersects ¿2 if and only if it contains
Pi. If it holds for all i, then and ¿2 have the same distance sequence with
- - d-l_ 1
respect to S. We can handle this problem by extending S with all the q _
points of a hyperplane in Ed-1. Then ¿2 contains at least one of these points
and ¿4 does not contain any of them. Hence the two lines are divided.
The size of the constructed resolving set is (q + 1)k - qm + q _ , the statement is proved.	□
Theorem 2.11. If n > 4 is even, then there exists a semi-resolving set for points in rnq of size
rp(n, q) = 2qn-1 + 2qn_2 + 2(qn_4 + qn_6 + • • • + q2).	(2.2)
Proof. We apply Proposition 2.10 for d = n - 1. Let S be the semi-resolving set for points in r„_1jq which was constructed in Theorem 2.7. Its size is
k=2 ^.
q2 - 1
By Proposition 2.8, we can choose the hyperplane En_2 so that it contains
' q"_2 - 1 \ qn_2 - q2 m = 2 —^--1 =2-
q2 - 1	q2 - 1
elements of S. Thus we get from Proposition 2.10 that there exists a semi-resolving set for points in rn q of size
qn — q2 qn_2 — q2
rp(n, q) = 2(q + 2—r - 2q—2—r~
q2 - 1	q2 - 1
= 2qn_1 + 2qn_2 + 2(qn_4 + qn_6 + • • • + q2).	□
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Now we turn to semi-resolving sets for lines. Let us start with a simple, but very useful observation.
Lemma 2.12. Let E be a hyperplane in PG(n, q), S be a semi-resolving set for points in E and t\ and t2 be two distinct lines in PG(n, q). Suppose that none of the lines is contained in E and the points P® = E n t\ and P2 = E n ¿2 are distinct. Then the lines t\ and t2 are divided by S in rn,q.
Proof. Since S is a semi-resolving set for points in E, there exists at least one element s € S so that d(Pi, s) = d(P2\ s). Hence
d(^, s) = d(Pi, s) + 1 = d(Pl, s) + 1 = d(l2, s), the statement follows.	□
Theorem 2.13. For all n > 3 and q > 2n — 1 there exists a semi-resolving set for lines in rn,q of size rL(n, q) = 2nrp (n — 1, q), where rp follows (2.1) or (2.2) depending on the parity of n — 1.
Proof. Let H = {Ei, E2,..., E2n} be a subset of 2n hyperplanes of the (q + 1)-element set formed by the dual hyperplanes of points on a normal rational curve. Then these hyperplanes are in general position, no n +1 of them have a point in common. Let S® be a semi-resolving set for points in E®. We claim that S = |J2=i S® is a semi-resolving set for lines in rn,q.
Let li and l2 be two distinct lines in PG(n, q). We may assume that ¿j is contained in the intersection of mj elements of H for j = 1,2, and mi > m2. The elements of H are in general position, so n — 1 > mj, hence 2n — mi — m2 > 2. We may assume without loss of generality that j intersects E® in a single point, denoted by P®, for i = 1,2,..., 2n — mi — m2 and j = 1,2. It could happen, that Pf = P®2 = • • • = P®k for some indices, but k < n — m2, otherwise the point would be a common point of at least mi + (n — m2 + 1) > n elements of H. So we may assume that Pi = P>12. As contains at most one point of we may also assume that P/ is not on Then, by Lemma 2.12, ¿4 and are divided by Si.
By Theorems 2.7 and 2.11, the size of S is at most 2nrp (n — 1, q) for n > 3, thus the theorem is proved.	□
The union of a semi-resolving set for points and a semi-resolving set for lines is a resolving set. Thus Theorems 2.7, 2.11 and 2.13 give our first general upper bound.
Corollary 2.14. For all n > 3 and q > 2n — 1 there exists a resolving set in rn,q of size
r(n, q) = 2qn- i + (4n + 1 + ( — 1)n)qn-2 + gn(q),
where gn is a polynomial of degree n — 3 whose coefficients depend only on n.
In this bound the coefficient of the second highest degree term depends on the dimension. In the next part, by a more sophisticated construction, we prove an upper bound in which the coefficient of the second highest degree term is a constant.
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Proposition 2.15. Let q = ph, pprime. Suppose that there exists a resolving set S3 in r3,q of size 2q2 + aq + g3(p), where a G R, g3 is a polynomial of degree s < h — 1, and S3 contains the 2q2 + 2 elements of two disjoint line spreads. Then there exists a resolving set
S4m+3 in r4m+3,q ofsize
2qn—1 + aqn-2 + g4m+3 (p) where g4m+3 is a polynomial of degree at most (n — 3)h + s.
Proof. As (4m + 3) + 1 is divisible by 3+1, there existsa 3-spreadin PG(4m + 3, q). This
4(m + 1)_i	10
3-spread contains t = q q4-1— elements, say S3, S3,..., S3, each of them is isomorphic to PG(3, q). By the assumption of the theorem, in each S3 there exists a resolving set S3 of size 2q2 + aq + g3 (p). We claim that
t
s = U S3
¿=1
is a resolving set in T4m+3,q. The elements of any pair of points and any point-line pair are obviously divided by S. Let and be two lines. If at least one of them is contained in a S3, then they are divided by S3. If none of them is contained in any S3, then we may assume without loss of generality that n S3 is a point P which is not on Let s1 and s2 be the two elements of the disjoint line spreads in S3 which are incident with P. Then s1) = s2) = 2. As is not contained in S3, it cannot intersect both s1 and s2. Hence at least one of the distances d(^2, s1) and d(^2, s2) is 4. Thus and are divided by Sf c S.
The size of S is
qn+1 _ 1
(2q2 + aq + g3(p)) qq4 — 1 = 2qn—1 + aqn-2 + g4m+3(p),
where the degree of g4m+3 is (n — 3)h + deg g4m+3 = (n — 3)h + s < (n — 2)h — 1, so we are done.	□
Theorem 2.16. Let q = ph, p prime. Suppose that there exists a resolving set in r3,q of size 2q2 + aq + g3 (p) where g3 is a polynomial of degree s < h — 1. Then for n > 3 there exists a resolving set in rn,q ofsize
r(n,q) =
'2qn—1 + (a + 2)qn—2 + g„,o(p),	if n = 0	(mod 4),
2qn-1 + (a + 2)qn—2 + g„,i(p),	if n = 1	(mod 4),
2qn—1 + (a + 4)qn-2 + gn,2(p),	if n = 2	(mod 4),
^2qn—1 + aqn-2 + g„,3(p),	ifn = 3	(mod 4),
where g„,j (i = 0,1, 2, 3) is a polynomial of degree (n — 3) h + s whose coefficients depend only on n.
Proof. We prove it by induction on the dimension modulo 4. For n = 3 (mod 4) the statement follows from Proposition 2.15.
If n = 0 (mod 4), then we apply Proposition 2.10 for d = n — 1 with k = rp (n — 1, q) and m = 0. Therefore by the induction hypothesis
qn-2 — 1
rp(n q) < (q + 1)rp(n — 1 q) +--
q — 1
= 2qn—1 + (a + 2)qn—2 + (q + 1)gn-1,3(p) + (a + 1)qn-3 + qn-4 + • • • + 1.
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Thus gn,o(p) = (q + 1)g„-i,3(p) + (a + 1)qn-3 + qn-4 +----+ 1, hence its degree is
(n - 3)h + s < (n - 2)h - 1.
If n = 1 (mod 4), then n — 2 = 3 (mod 4), hence we can apply Proposition 2.10 for d = n - 1 so that £d-1 contains a resolving set constructed in Proposition 2.15. Then
k = 2qn-2 + (a + 2)qn-3 + gn-i,o(p) and m = 2qn-3 + aqn-4 + gn-2,s(p). Hence
q"-2 — 1
rp(n, q) = (q + 1)k - qm + q-— = 2qn-1 + (a + 2)qn-2 + g„,i(p),
q - 1
where
gn,l(p) = (q + 1)gn-1,0 (p) - qgn-2,3 (p) + 3q"-3 + q"-4 + • • • + 1,
so its degree is (n - 3)h + s < (n - 2)h - 1.
Finally, if n = 2 (mod 4), then n - 3 = 3 (mod 4). Hence we cannot do better than apply Proposition 2.10 for d = n - 1 so that £d-1 contains entirely only elements of a (d - 2)-dimensional resolving set constructed in Proposition 2.15. Now k = 2qn-2 + (a + 2)qn-3 + gn-1,1 (p) and m = 2qn-4 + aqn-5 + gn-3,3(p). This gives
qn-2 _ 1
rp(n, q) = (q + 1)k - qm + q-— = 2qn-1 + (a + 4)qn-2 + gn,2(p),
q - 1
where
gn,2(p) = (q + 1)gn-1,1 (p) - qgn-3,3(p) + (a + 1)q"-3 + q"-4 + • • • + 1,
thus its degree is (n - 3)h + s < (n - 2)h - 1 again. The theorem is proved.	□
Let us remark that the polynomials gn,j can be determined exactly. We omit the long, but straightforward calculations, because their coefficients do not play any role in the rest of the paper.
In the next part of the section semi-resolving sets for lines in rn,q are investigated. In their constructions double blocking sets and their duals play an important role. For the relevant definitions and estimates on their sizes we refer to the paper of Ball and Blokhuis [3].
Theorem 2.17. For all q > 3 there exists a semi-resolving set for lines in r3,q of size
rL(3, q) = min{12q - 22, 4r2(q) - 10}, where t2 (q) denotes the size of the smallest minimal double blocking set in PG(2, q).
Proof. First, we construct two sets of lines in PG(2, q) which are semi-resolving sets for points.
1. Let E1, E2, and E3 be the vertices of a triangle, ^ denote the line Ej Ek and P be the pencil of lines with carrier Ej. Let
S = P1 UP2 UP3 \ {¿1,^2,4,^,
where I G P1, ¿2 = I = ¿3. Then S is a semi-resolving set for points in r2,q, because U = I n is a unique uncovered point, every point in the set U ¿2 U ¿3 \ {E1, E2, E3, U} is 1-covered and all other points are at least 2-covered, hence resolved. The size of S is 3q - 4.
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2. Let D be a dual double blocking set in PG(2, q). Then, by definition, each point is incident with at least two lines of D. Thus if we delete an arbitrary line I from D, then the set of lines D \ {¿} is still a semi-resolving set for points and, by the Principle of Duality, its size is at most T2(q) - 1.
Hence, for all q > 3 there is a set of lines in PG(2, q) of size min{3q — 4, r2(q) — 1} which is a semi-resolving set for points. Let Hi, H2, H3, and H4 be the faces of a tetrahedron K in PG(3, q). Let Tj be a semi-resolving set for points in Hi which consists of lines only. We can choose Tj so that each edge of K belongs to both corresponding semi-resolving sets, because the full collineation group of PG(2, q) acts transitively on triangles. We claim that S = U4=1Tj is a semi-resolving set for lines in r3,q.
The edges of K belong to S, thus they are resolved by definition. Let ¿1 and ¿2 be lines such that none of them is an edge of K. Then each of them is contained in at most one face of K, so we may assume without loss of generality that ¿1 intersects Hi in a single point, denoted by Pi, for i = 1,2, 3. We distinguish two main cases.
1.	If Pi = P2 = Pf, then this point is a vertex K of K.
•	If ¿2 also contains K, then H4 n ¿4 = H4 n ¿2, hence, by Lemma 2.12, the two lines are divided by T4.
•	If ¿2 does not contain K, then we may assume that H2 n ¿2 is a single point P22. Since P22 = K, by Lemma 2.12, the two lines are divided by T2.
2.	If none of ¿1 and ¿2 contains any vertex, then we may assume that P/ = P2.
•	If ¿2 is not contained in neither H1 nor H2, then it intersects Hi in a single point, denoted by P2, for i = 1,2. Since ¿1 n ¿2 contains at most one point, we may assume that P1 = P21. Then, by Lemma 2.12, the two lines are divided by T1.
•	Finally, if ¿2 is contained in one of H1 and H2, then we may assume that
¿2 C H1 and ¿2 n H2 in a single point P22. Then P22 is in H1, so P22 = Pf, because otherwise ¿1 C H1. Hence, by Lemma 2.12, the two lines are divided by T2.
Since S has the required size, we are done.	□
Remark 2.18. Let us remark that if the double blocking set D in the proof of Theorem 2.17 is the disjoint union of two dual blocking sets, then not only one, but two lines can be deleted without violating the semi-resolving set property. We will consider this case in Section 3, Theorem 3.1.
Unfortunately, the exact value of t2 (q) is not known in general. It is known that t2 (q) = 2q + 2^q + 2 for q is a square and q > 16 [3, Theorem 3.1], and for some small values of q. In the latter case for the known values T2(q) > 3q — 3 always holds. Combining the semi-resolving set for points constructed in Corollary 2.7 and the semi-resolving set for lines of Theorem 2.17, we get the following upper bound on y«(r3,q).
Theorem 2.19. The metric dimension of r3,q satisfies the inequality
) < 2q2 + 12q — 24
for all q > 3.
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Proof. For q > 3 let SL be a semi-resolving set for lines of size 12q - 22 constructed in Theorem 2.17. Let L1 be a regular line spread of PG(3, q) which contains two skew (non-intersecting) elements of SL. Such spread exists, because the collineation group of PG(3, q) acts transitively on the pairs of skew lines. Create a semi-resolving set for points Sp which contains L1 as we did it in Corollary 2.7. Then S = SL U Sp is a resolving set in r3,q and its size is 2q2 + 12q - 24. This proves the inequality.	□
By combining Theorem 2.16, with s = 0, and Theorem 2.19, we get the following bounds.
Corollary 2.20. Let n > 3 and q > 3. Then the metric dimension of rn,q satisfies the inequality
{2qn-1 + 14qn-2 + hn,2(q), if n = 0 or n = 1 (mod 4), 2qn-1 + 16qn-2 + hn,3(q), if n = 2 (mod 4), 2qn-1 + 12qn-2 + hnj1(q), if n = 3 (mod 4),
where hnji (i = 1,2, 3) is a polynomial of degree at most n — 3 whose coefficients depend only on n.
The metric dimension of r2,q for q > 23 was determined by Heger and Takats [12]. For higher dimensions we do not know the exact value, but Theorems 2.1, 2.19, and Corollary 2.20 imply the following result.
Corollary 2.21. For all n > 2 and q > 3
|M(r„,q) — 2qn—11 = O(qn—2).
This means that M(rn,q) is asymptotically 2qn-1. The number of vertices in rn,q is v = q q--1 +	1), so its metric dimension is roughly The automorphism
group of PG(n, q) is PrL(n + 1, q) and it is well-known that its base size is n + 1 if q is a prime, and it is n + 2 if q = with h > 1. Hence the dimension jump of rn,q is roughly 2^v.
3 Bounds for q = ph, h > 2
In this section we consider the case q = ph, h > 1. In the case h even, we will present a better bound on the size of a semi-resolving set for points in r3 q using small dual double blocking sets in PG(2, q). When h > 2, then we will show that a particular type of spread of lines in PG(3, q) can be used to resolve the lines. In fact, for a regular spread, there exist many pairs of lines of the spaces intersecting the same set of elements of the spread. We now investigate a different type of spread, called aregular, and we determine all the lines of the space intersecting the same set of elements of the spread; see Theorem 3.6. The main goal is to construct a set of lines of PG(3, q) which resolves all the lines of the spaces; see Theorem 3.7.
Theorem 3.1. If q is a square, then the metric dimension of r3,q satisfies the inequality
M(r3,q) < 2q2 + 8q + vq — 8.
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Proof. The union of the sets of lines of two disjoint Baer subplanes is a dual double blocking set in PG(2, q) and its size is 2q + + 2. This set is the disjoint union of two dual blocking sets. Hence, by a result of Heger and Takats [12, Proposition 22], we can delete two of its lines so that the remaining set is still a semi-resolving set for points in PG(2, q); see also Remark 2.18.
Thus we can construct a semi-resolving set for lines SL of size 8q + - 6 by the method applied in the proof of Theorem 2.17. Finally, we can extend it to a resolving set of size 2q2 + 8q + - 8 in the same way as we did in the proof of Theorem 2.19. □
By combining Theorem 2.16, with s = 0, and Theorem 2.19, we get the following bounds.
Corollary 3.2. If q is a square and n > 3, then the metric dimension of rn,q satisfies the inequality
{2qn-1 + 10qn-2 + hn,2(q), if n = 0 or 1 (mod 4), 2qn—1 + 12qn—2 + h„,3(q), if n = 2 (mod 4), 2qn-1 + 8qn-2 + hn,1(q), if n = 3 (mod 4),
where hn,i (i = 1,2, 3) is a polynomial of degree at most n — 3 whose coefficients depend only on n.
Theorem 3.3 ([13, Theorem 17.3.3]). Let q = ph, h > 1, and choose b, c G F* such that the polynomial tp+1 — tb + c has no roots in Fq. Let
Ab,c = {ta,p : a, £ G Fq} U {Z = T = 0},
where ta,p is the line through the points (a : £ : 1 : 0) and (c£p : ap + b£p : 0 : 1). Then Ab,c is a spread, called the aregular spread.
In what follows, we will associate to each line r of the space an algebraic curve Cr : F (X, Y, T) = 0 such that taJj intersects r if and only if F (a, £, 1) = 0. We distinguish four types of lines.
1. Lines rx,y,£,m through the points (x : y : 0 : 1) and (^ : m : 1 : 0). Note that if
x = cmp and y = £p + bmp then rXyV^m coincides with t^,m G Ab,c. From now on we consider (x, y) = (cmp, lp + bmp). A line ta^ intersects r = rx y im if and only if for some A G Fq the points
(x + A^ : y + Am : A : 1), (a : £ : 1 : 0), (c£p : ap + b£p : 0 : 1)
are collinear, that is
x + A^ — c£p = Aa, y + Am — (ap + b£p) = A£.
This implies
A(^ — a) = c£p — x, A(m — £) = ap + b£p — y,
and therefore
(c£p — x)(m — £) = (ap + b£p — y)(^ — a). In this case, Fr (X, Y, T) = (cYp — xTp)(mT — Y) — (Xp + bYp — yTp)(^T — X).
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2.	Lines s = sx,y,z,£ through the points (x : y : z : 1) and : 1 : 0 : 0). A line intersects sx,y,z,£ if and only if for some A G Fq the points
(x + A^ : y + A : z : 1), (a : £ : 1 : 0), (c,0p : ap + b,0p : 0 : 1)
are collinear, that is
x + A^ - = za, y + A - (ap + = z£.
This implies
A^ = -x + + za, A = -y + + ap +
and therefore
-x + + za = ¿(-y + + ap + So, Fs(X, Y, T) = -¿Xp + (c - ¿b)Yp + zXTp-1 - ¿zYTp-1 + (¿y - x)Tp.
3.	Lines u = ux,y,z through the points (x : y : z : 1) and (1 : 0 : 0 : 0). In this case,
Fu(X, Y, T) = Xp + bYp + zYTp-1 - yTp.
4.	Lines v = vx,y,z contained in the planes T = 0 and xX + yY + zZ = 0. Then,
Fv (X, Y, T) = xX + yY + zT.
Such a curve is absolutely irreducible if z = 0, otherwise it collapses into a single line.
Proposition 3.4. Consider the curves Cr, Cs, Cu, and Cv. Then
1.	Cr is absolutely irreducible;
2.	Cs is either absolutely irreducible or a line repeated p times;
3.	Cu is either absolutely irreducible or a line repeated p times.
Proof.
1. Now we prove that Cr is absolutely irreducible. Let y(X, Y, T) = (X + x0T, Y + yoT, T) with xp = y - bx/c, yP = x/c. Then
Fr (y(X, Y, T))
= (c(Y + yoT)p - xTp)(mT - Y - yoT)
-	((X + xoT)p + b(Y + yoT)p - yTp)^T - X - xoT) = (cYp + cypTp - xTp)(mT - Y - yoT)
-	(Xp + xpTp + bYp + ypTp - yTp)^T - X - xoT) = (cYp + xTp - xTP)(mT - Y - yoT)
-	(Xp + yTP - bx/cTp + bYP + bx/cTP - yTP)^T - X - xoT) = cYp(mT - Y - yoT) - (Xp + bYp)^T - X - xoT)
= Gr (X, Y, T).
Finally
Gr(X, 1, Y) = c(mY - 1 - yoY) - (Xp + b)^Y - X - xoY),
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that is the curve Cr is Fq-isomorphic to
c - X (Xp + b)
C': Y =
cm - cyo + (xo - ¿)(Xp + b)'
which is an irreducible rational curve with q +1 Fq-rational points (note that (x, y) =
(cmp, ¿p+bmp) yields m = y0 or ¿ = x0). This means that the curve Cr is absolutely irreducible.
2.	First, note that the homogeneous term — ¿Xp + (c - ¿b)Yp cannot vanish otherwise c = 0, a contradiction.
•	If (¿, z) = (0,0), Cs is a line of type b0Y + c0T = 0 repeated p times.
•	If ¿ = 0 and z = 0, then Fs (X, Y, T) reads cYp + zXTp-1 - xTp and Cs is absolutely irreducible.
•	If ¿ = 0 and z = 0 then Cs is a (repeated) line a0X + b0Y + c0T = 0, where
ap0 = -¿, bp = (c - ¿b), cP = (¿y - x).
•	If ¿ = 0 and z = 0 then consider y(X, Y, T) = (X + ^¿bp Y, Y, T) and so
Gs(X,Y,T) = Fs(^(X,Y,T)) =
- ¿Xp + zXTP-1 + z( V(c-¿b)7i - ¿)YTp-1 + (¿y - x)TP.
By our assumption of b, c, there is no ¿ G Fq such that ^(c - ¿b^ - ¿ = 0. The curve Gs (X, Y, T) = 0 is rational and irreducible and it is Fq-isomorphic to Cs.
3.	Clear.	□
Proposition 3.5. Let q = ph, h > 2. Two lines of the same type (r,s,u,v) do not intersect the same set of lines of the aregular spread Abc.
Proof. The assumption h > 2 implies q +1 > (p + 1)2. The curves Cr, Cs, Cu, Cv have degree at most p and they have q +1 Fq -rational points (corresponding to the lines of the spread intersecting them). By Proposition 3.4, such curves are either absolutely irreducible or they consist of a repeated line. Thus, if two curves attached to the lines w1 and w2 of the same type share q +1 Fq-rational points, the corresponding polynomials must be proportional. By direct computations, this yields w1 = w2.	□
Theorem 3.6. Let q = ph, h > 2. If two lines in PG(3, q) intersect the same set of lines of the spread Ab,c then one of them lies on the plane Z = 0 and the other on the plane T = 0.
Proof. The reduced (absolutely irreducible) curves associated with the different types of lines (r, s, u, v) have degree p +1, degree p or 1, degree p or 1, and degree 1, respectively. They can share q +1 Fq-rational points only in the following cases:
•	both Cs and Cu have degree p;
•	both Cs and Cu have degree 1;
•	both Cs and Cv have degree 1;
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• both Cu and Cv have degree 1.
The first case is not possible. The second case would imply c = 0, a contradiction. Recall that the lines v = vxyz are contained in the plane T = 0 of PG(3, q). The claim follows observing that if Cs or Cu have degree 1, then s = sx,y,o/ or u = ux,yi0. So, both s and u are contained in the plane Z = 0.	□
Theorem 3.7. Let q = ph, h > 2. Then there exists a set of q2 +3 lines resolving all the lines of PG(3, q).
Proof. Consider the aregular spread Abc with b,c e F* and such that the polynomial tp+1 — tb + c has no roots in Fq. We already know by Theorem 3.6 that lines of PG(3, q) intersecting the same set of elements of Ab,c are contained in the planes Z = 0 or T = 0. Note that two lines in a fixed plane cannot intersect the same elements of Abc. Consider two distinct extra lines w1 and w2 contained in Z = 0 intersecting the line Z = T = 0 at
two distinct points. It is readily seen that Ab,c U{wi, w2} resolves all the lines of PG(3, q).
' □
Corollary 3.8. If q = ph, h > 2, then there exists a resolving set in r3iq of size 2q2 + 2.
Proof. Consider the set of q2 + 3 lines from Theorem 3.7 and use the argument of Theorem 2.9. One of the two extra lines could be an element of the other spread. Finally, delete one line from the modified regular spread.	□
Finally, the following bounds are obtained combining again Theorem 2.16, with s = 0, and Theorem 2.19.
Corollary 3.9. If q = ph, h > 2, then the metric dimension of rn,q satisfies the inequality
{2qn-1 + 2qn-2 + hn,2(q), if n = 0 or 1 (mod 4), 2qn—1 + 4qn—2 + hn^(q), if n = 2 (mod 4), 2qn—1 + hn,i(q),	ifn = 3 (mod 4),
where hnji (i = 1,2, 3) is a polynomial of degree at most n — 3 whose coefficients depend only on n.
4 Resolving sets for small q
We performed a computer search to obtain sets of lines that are semi-resolving sets for lines in PG(3, q) for small q. We used Magma, a computer algebra system for symbolic computation developed at the University of Sydney; see [6]. We started classifying all set of lines of a certain size k. Then we extended the non-equivalent sets of size k using a backtracking algorithm.
In PG(3,2) there are 35 lines, so a semi-resolving set for lines must contain at least six elements. We found that there are 165 non-equivalent sets of lines of size six. Forty-eight of them are semi-resolving sets for lines in PG(3,2). An example is the following set of six lines:
{((1	0	0	0), (0	1	0	0)), ((0	1	0	0), (0	0	0	1)),
((0	0	1	0), (0	0	0	1)), ((0	0	0	1), (1	1	0	0)),
((0	1	0	0), (1	1	1	1)), ((1	1	0	0), (0	0	1	1))}
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In PG(3,3) there are 130 lines, so a semi-resolving set for lines must contain at least eight elements. We found that there are 10681 non-equivalent sets of lines of size seven. An exhaustive search by backtracking has proved that no set of lines of size eight or nine is a semi-resolving set for lines in PG(3,3). There exist semi-resolving sets for lines of size ten. An example is the following set of ten lines:
{((1	0	0	0), (0	1	0	0)), ((1	0	0	0), (0	0	0	1)),
((0	1	0	0), (0	0	1	0)), ((0	1	0	0), (1	0	0	1)),
((1	0	1	2), (0	1	1	0)), ((1	0	0	2), (0	0	1	1)),
((1	0	0	0), (0	1	1	2)), ((1	0	0	0), (0	1	1	0)),
((0	1	1	1), (1	2	0	0)), ((1	1	1	0), (0	1	2	0))}
ORCID iDs
Daniele Bartoli © https://orcid.org/0000-0002-5767-1679 Gyorgy Kiss © https://orcid.org/0000-0003-3312-9575 Stefano Marcugini © https://orcid.org/0000-0002-7961-0260 Fernanda Pambianco © https://orcid.org/0000-0001-5476-5365
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ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 19 (2020) 249-275 https://doi.org/10.26493/1855-3974.2128.ccf
(Also available at http://amc-journal.eu)
Building maximal green sequences via component preserving mutations*
Eric Bucher
Department of Mathematics, Xavier University, Cincinnati, OH 45207, U.S.A.
John Machacek
Department of Mathematics and Statistics, York University, Toronto, Ontario M3J1P3, Canada
Evan Runburg, Abe Yeck, Ethan Zewde
Department ofMathematics, Michigan State University, East Lansing, MI 48824, U.S.A.
Received 27 September 2019, accepted 16 July 2020, published online 16 November 2020
We introduce a new method for producing both maximal green and reddening sequences of quivers. The method, called component preserving mutations, generalizes the notion of direct sums of quivers and can be used as a tool to both recover known reddening sequences as well as find reddening sequences that were previously unknown. We use the method to produce and recover maximal green sequences for many bipartite recurrent quivers that show up in the study of periodicity of T-systems and Y-systems. Additionally, we show how our method relates to the dominance phenomenon recently considered by Reading. Given a maximal green sequence produced by our method, this relation to dominance gives a maximal green sequence for infinitely many other quivers. Other applications of this new methodology are explored including computing of quantum dilogarithm identities and determining minimal length maximal green sequences.
Keywords: Cluster algebra, maximal green sequence, direct sum. Math. Subj. Class. (2020): 13F60
* The authors would like to thank the anonymous referees for this paper. Their insightful feedback has helped strengthen the paper.
E-mail addresses: buchere1@xavier.edu (Eric Bucher), machacek@yorku.ca (John Machacek), runburge@msu.edu (Evan Runburg), yeckmarc@msu.edu (Abe Yeck), zewdeeth@msu.edu (Ethan Zewde)
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
Abstract
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1 Introduction
Quiver mutation is the fundamental combinatorial process which determines the generators and relations in Fomin and Zelevinsky's cluster algebras [15]. Cluster algebras have arisen in a variety of mathematical areas including Poisson geometry, Teichmuller theory, applications to mathematical physics, representation theory, and more. Quiver mutation is a local procedure that alters a quiver and produces a new quiver. Understanding how a quiver mutates is essential to understanding the corresponding cluster algebra. We will consider the problem of explicitly constructing sequences of mutations with some special properties.
1.1 Some history of the problem
A maximal green sequence, and more generally a reddening sequence, is a special sequence of quiver mutations related to quantum dilogarithm identities which was introduced by Keller [28, 29]. Such sequences of mutations do not exist for all quivers and determining their existence or nonexistence is an important problem. For a good introduction to the study of maximal green and reddening sequences see the work of Briistle, Dupont, and Perotin [3]. In addition to the role they play in quantum dilogrithm identities, these sequences of mutations are a key tool utilized in other cluster algebra areas. For example, the existence of a maximal green sequence allows one to categorify the associated cluster algebras following the work of Amiot [2]. Also the existence of a maximal green sequence is a condition which plays a role in the powerful results of Gross, Hacking, Keel, and Kont-sevich [25] regarding canonical bases. These results use the notion of scattering diagrams to prove the positivity conjecture for a large class of cluster algebras. Additionally the existence of a reddening sequence is thought to be related to when a cluster algebra equals its upper cluster algebra [5, 34]. Maximal green sequences are also related to representation theory [3] and in the computation of BPS states in physics [1]. Our notion of a component preserving sequence of mutations, which will be defined in Section 3, is closely related to what has been called a factorized sequence of mutations [9, 10, 12] in the physics literature where particular attention has been paid to ADE Dynkin quivers. Our definition is more general which allows for use with both maximal green sequences and reddening sequences. Being able to work with reddening sequences is desirable since the existence of a reddening sequence is mutation invariant while the existence of a maximal green sequence is not [35]. Hence, the existence of a reddening sequences ends up being a invariant of the cluster algebra as opposed to just the quiver.
In general it can be a difficult problem to determine if a quiver admits a maximal green or reddening sequence. These sequences have been found or shown to not exist in the case of finite mutation type quivers by the work of a variety of authors [1, 4, 7, 33, 40] leaving the question of existence only to quivers that are not of finite mutation type. This makes finding these sequences particularly difficult as the exchange graph for such quivers can be very complicated. Additionally there are branches of the exchange graph, in which no amount of mutations can lead to a maximal green sequence; meaning random computer generated mutations are extremely unlikely to produce maximal green sequences for these quivers. In addition to finite mutation type quivers, headway has been made on specific families of quivers such as minimal mutation-infinite quivers [32] and quivers which are associated to reduced plabic graphs [17]. This gives us many quivers for which we know reddening or maximal green sequences for. This provides a foundation to produce redden-
E. Bucher et al.: Building maximal green sequences via component preserving mutations 251
ing and maximal green sequences for quivers which are built out of these.
When a quiver does admit a maximal green or reddening sequence it is desirable to have an explicit and well understood construction of the sequence. Having the specific sequence of mutations and understanding the corresponding c-vectors gives us a product of quantum dilogarithms [28, 29] and an expression for the Donaldson-Thomas transformation of Kontsevich and Soibelman [31]. The method which we present in this paper allows one to explicitly produce the sequence so that it can be used to for the corresponding computation.
Work by Garver and Musiker [22], as inspired by [2] and [1], and later by Cao and Li [8] looked at using what has been called direct sums of quivers to produce maximal green and reddening sequences when the induced subquivers being summed exhibit the appropriate sequences. This heuristic approach of building large sequences of mutations from subquiv-ers is essentially the direction we want to expand upon in this paper. Component preserving mutations are a way of taking known maximal green and reddening sequences for induced subquivers (which we will call components) and combining them together to obtain a maximal green or reddening sequence for the whole quiver. The direct sum procedure becomes a particular instance of the theory of component preserving mutations.
The methodology presented has an assortment of applications. It can be used to produce maximal green sequences for bipartite recurrent quivers, recover known results regarding admissible source mutation sequences for acyclic quivers, and show that the existence of a maximal green or reddening sequence is an example of a certain dominance phenomena in the sense of recent work by Reading [37].
1.2 Summary of the methodology
The goal of this paper is to develop a methodology which allows one to use reddening sequences of subquivers of a given quiver to build reddening sequences for larger quivers. Since mutation is a local procedure, only affecting neighboring vertices, this is a natural approach. Moreover, it is known that when a quiver has a maximal green or reddening sequence, then the same is true for any induced subquiver [35]. Hence, developing a method to produce a maximal green or reddening sequence from induced subquivers is a type of converse to this fact.
The method starts by breaking the quiver, Q, into subquivers which we call components; each of which has a known reddening sequence. The components will partition the vertices of the quiver, giving a partitioned quiver (Q, n), where n := n/n2/ • • • /n is a partition of the vertices of Q. We label the components Qi. We start with the framed quiver, where we partition all of the frozen and mutable pairs into the same parts. We call this quiver the framed partition quiver (Q, n). We then try to shuffle the respective reddening sequences together to see if they form a reddening sequence for the entire quiver. It is not the case that one can always find a shuffle which works on the entire quiver. To guarantee that they do build a reddening sequence, we must check that at each mutation step the mutation vertex satisfies the component preserving condition which will be given in Definition 3.6. If this condition holds the main result of this paper shows that you have constructed a reddening sequence for the larger quiver.
Theorem 1.1 (Main Result). Let (Q, n) be a framed partition quiver where for each Qi we have a reddening sequence ai. Let t be a shuffle of the ai such that at every mutation step of the sequence t we have that k is component preserving with respect to n. Then t is a reddening sequence for Q.
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This main result is proven in Section 3 where is it restated in Theorem 3.11. This approach gives one a starting point as to where to search for reddening sequences given an arbitrary quiver. First break the quiver into subquivers you are comfortable constructing reddening sequences for; and then attempt to shuffle these sequences. This approach may initially seem overwhelming as you could consider any partition of the quiver into sub-quivers along with any shuffle of reddening sequences. However, as we explored utilizing this technique what we realized was that there are often very natural shuffles and partitions present in many commonly studied quivers. For instance, this concept generalizes the idea of direct sums of quivers where the shuffle takes the particular simple form of concatenation. Additionally, it can be used to give short and effective constructions of maximal green sequences for bipartite recurrent quivers, and many more examples where some well behaved properties of a specific quiver provides the recipe for how to shuffle and partition the vertices.
This article is structured in the following way. Section 2 will give some preliminaries for quiver mutation and the study of reddening sequences. In Section 3 we will present the main results of the paper outlining how the component preserving procedure can produce new maximal green and reddening sequences from induced subquivers. Within Section 3 we present a large amount of examples to try and illustrate how this procedure works. In the sections following this we look at some applications of this procedure to produce interesting and new results. Results related to dominance phenomena are in Section 4 and bipartite recurrent quivers are considered in Section 5. In Section 6 we consider the computation of Donaldson-Thomas invariants and minimal length maximal green sequences. We have added a large amount of examples to the article in an effort to try and give the reader an opportunity to become familiar with how one uses this method in a hands-on manner. This is intentional, as from exploring these methods it appears that many reddening sequences are built in this manner from small set of "basic reddening sequences." The intuition of the authors is that there may be a way to describe a list of "basic reddening sequences" from which any reddening sequence can be built. It is our hope that this paper is the first step in building the concrete theory behind this intuition.
2 Preliminaries
A quiver Q is a directed multigraph with vertex set V(Q) and whose edge set E(Q) contains no loops or 2-cycles. Elements of E(Q) will typically be referred to as arrows. An ice quiver is a pair (Q, F) where Q is a quiver, F C V(Q), and Q contains no arrows between elements of F. Vertices in F are called frozen while vertices in V(Q) \ F are called mutable. The framed quiver associated to a quiver Q, denoted Q, is the ice quiver whose vertex set, edge set, and set of frozen vertices are the following:
V(Q) := V(Q) U{i' | i e V(Q)}, E(Q) := E(Q) U{i ^ i' | i e V(Q)}, F = {i' | i e V(Q)}.
The framed quiver corresponds to considering a cluster algebra with principal coefficients.
Given an ice quiver (Q, F) for any mutable vertex i, mutation at the vertex i produces a new quiver denoted by (^(Q), F) obtained from Q by doing the following:
(1) For each pair of arrows j ^ i, i ^ k such that not both i and j are frozen add an
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arrow j ^ k.
(2)	Reverse all arrows incident on i.
(3)	Delete a maximal collection of disjoint 2-cycles.
Mutation is not allowed at any frozen vertex. Since mutation does not change the set of frozen vertices we will often abbreviate an ice quiver (Q, F) by Q and (pi(Q), F) by ^(Q) where the set of frozen vertices is understood from context. We will be primarily focused on framed quivers and quivers which are obtained from a framed quiver by a sequence of mutations. In fact, whenever we have an ice quiver with a nonempty set of frozen vertices we will assume it is obtainable from a framed quiver by some sequence of mutations. So, the set of frozen vertices will be of a very particular form.
A mutable vertex is green if it there are no incident incoming arrows from frozen vertices. Similarly, a mutable vertex is red if there are no incident outgoing arrows to frozen vertices. If we start with an initial quiver Q and perform mutations at mutable vertices of the framed quiver Q, then any mutable vertex will always be either green or red. The result is known as sign-coherence and was established by Derksen, Weyman,and Zelevin-sky [13]. For each vertex i in a quiver obtained from Q by some sequence of mutations, the corresponding c-vector is defined by its jth entry being the number of arrows from i to j' (with arrows j' to i counting as negative). In these terms sign-coherence says a c-vector's entries are either nonnegative or nonpositive. Notice also that all vertices are initially green when starting with Q. Keller [28, 29] has introduced the following types of sequences of mutations which will be our main interest. A sequence mutations is called a reddening sequence if after preforming this sequence of mutations all mutable vertices are red. A maximal green sequence is a reddening sequence where each mutation occurs at a green vertex. When a sequence of mutations is a reddening sequence we may say it is a reddening sequence for either Q or Q. In terms of being a reddening sequence or not, the quiver Q and the framed quiver Q are equivalent data.
We may write a maximal green or reddening sequence as either a sequence of vertices (read from left to right) or as a composition of mutations (read from right to left as is usual with composition of functions). For a quiver Q we will let green(Q) denote the set of maximal green sequences for Q. If we consider the quiver Q = 1 ^ 2 there are exactly two maximal green sequences and we can record them either as
green(Q) = {(1, 2), (2,1, 2)} in sequence of vertices notation or as
green(Q) = {^2^1 ,M2MiM2}
in composition notation.
We will need to modify and combine sequences of vertices when producing maximal green and reddening sequences. This is done by shuffling mutation sequences together.
Definition 2.1. A shuffle of two sequences (a1, a2,..., ak) and (b1,b2,... ,b£) is any sequence whose entries are exactly the elements of {a1, a2,..., ak} U {b1, b2,..., b^} (considered as a multiset) with the relative orders of (a1, a2,..., ak) and (b1, b2,..., b£) are preserved.
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For example there are 6 shuffles of the sequences (1,2) and (a, b). They are the sequences (1, 2, a, b), (1, a, 2, b), (1,a, b, 2), (a, 1, 2,b), (a, 1,b, 2), and (a, b, 1, 2). In the next section we will define component preserving mutations and show how by checking for the component preserving property you can create shuffles of reddening sequences on induced subquivers whose result is a reddening sequence for a larger quiver.
3 Component preserving mutations
We start by establishing some basic definitions and notation of what we mean by a component of the quiver.
Definition 3.1. Let Q be an ice quiver with vertex set V. Then let n = ni/n2/ • • • /n be a set partition of V. Then let Qj be the induced subquiver of Q obtained by deleting every vertex v G nj. We will call the Qj the components of Q and the pair (Q, n) a partitioned quiver.
Definition 3.2. When (Q,n) is a partitioned quiver with n = n1/n2/ • • • /n, we will define 7? as the partition of V? where each n? = {v, 7? | v g n.j}. Then (Q, 7?) will be called a partitioned ice quiver.
Remark 3.3. In other words, for each mutable vertex v, the frozen copy of a vertex, 7?, lies in the same component as v. It is straight forward to see that (Qj) = (Q)».
Definition 3.4. Mutation of a partitioned ice quiver is defined as the following:
Mfc((Q,n)) := (Mfc(Q),n).
Definition 3.5. Let (Q,n) be a partitioned ice quiver. A bridging arrow a ^ b is any arrow in Q in which a and b are in different components.
Now we can talk about the definition that is crucial to all the results in the rest of the paper. This is the notion of component preserving vertices and component preserving mutations.
Definition 3.6. A vertex k g Qj is component preserving with respect to n when one of the following occurs:
•	If 3 k ^ j' for a frozen vertex j', then V a ^ k we have a g V(Qj); or
•	If 3 j' ^ k for a frozen vertex j', then V k ^ a we have a g V(Qj).
Remark 3.7. Another way of thinking about component preserving mutations is in terms of sign-coherence. One can think of a component preserving vertex, k, as a vertex where freezing each mutable vertex outside of its component results in an ice quiver in which the extended exchange matrix is still sign-coherent with respect to this larger set of frozen vertices. In this way one can think of component preserving mutations as being a type of locally sign-coherent mutation.
Remark 3.8. Another observation to make is that whenever one starts from a framed quiver, mutation at component preserving vertices does not result in creating bridging arrows that involve frozen vertices. This means that any quiver which is the result of a sequence of component preserving mutations starting from a framed quiver has the support
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Figure 1: An illustration of a component preserving vertex k e Qi on the left with arrow k ^ j' and on the right with arrow j' ^ k.
of all of its c-vectors contained entirely within a component. In terms of the quiver, this means that the sequence of component preserving mutations results in a quiver in which all arrows involving frozen vertices are between mutable vertices and frozen vertices within the same component.
The choice of terminology is because performing mutation at a component preserving vertex, k, does not affect Qi unless k e ni. We will prove this fact and then show how one can use this fact to shuffle maximal green sequences together if at every mutation step you mutated at a component preserving vertex.
3.1 Preservation proof
Now that we have the language to talk about components of the quiver, we want to set up a condition on a vertex, k, which forces to only affect the component which contains k and none of the other induced subquivers. This is exactly the property that component preserving vertices have.
Lemma 3.9. Let (Q, n) be a partitioned ice quiver. If k is a component preserving vertex then ^k(Q)i = Mfc(Qi) v 1 < i < £
Proof. First notice that these are in fact two ice quivers on the same set of vertices. To check that the lemma holds we need to see that each step of mutation has the same effect on the subquivers ^k(Q)i and ^k(Qi) for each i. The key step of mutation to check is where new arrows are created, which is step one in our definition of mutation. There are two cases to consider:
Case 1: k e ni. Let a ^ b be an arrow in ^k (Qi) created by mutation at vertex k. Then since Qi is the quiver Q restricted to the component ni we know that a, b along with k are elements of V(Qi). Therefore the arrows a ^ k and k ^ b are elements of E(Qi).
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Therefore all of these arrows are present in Q and hence the arrow a ^ b is present in Mk(Q). Since both endpoints of the arrow are in ni the arrow a ^ b is also created in the mutation Mk(Q)i.
We will now show this is a biconditional relationship. Assume a ^ b is an arrow in Pk(Q)i which is created from mutation. This occurs if and only if a ^ k ^ b is present in Q and a,b e ni. Since we have assumed that k e ni we know that a,b,k e ni and the arrow a ^ b is also created in ^k(Qi).
Case 2: k G ni. Since k is not a vertex in Qi we will not be able to mutate the quiver Qi in direction k. Therefore mk(Qi) = Qi. Now what we must check is that no arrow a ^ b is created in pk(Q)i by step (1) of mutation.
By way of contradiction, assume that a ^ b in mk(Q)i is created by the composition of mutation and restriction. Then a ^ k ^ b is present in Q and also a,b e ni. But since k is not in the same component as a and b, arrows a ^ k and k ^ b are bridging arrows in opposite directions. This is a contradiction since each component preserving vertex is incident to bridging arrows in at most one direction.	□
3.2 Applications to reddening sequences and maximal green sequences
We have seen that if k is a component preserving vertex, then mk only affects arrows in Qi and possibly bridging arrows. This can be extremely useful in the context of reddening sequences. The goal is to utilize reddening sequences on each component to create a reddening sequence for the larger quiver. This turns out to be possible if at each mutation step you are performing a component preserving mutation. The following is a useful consequence which follows directly from the sign-coherence of c-vectors as presented in [13] and Remark 3.8 on the support of c-vectors.
Lemma 3.10. Let (Q, n) be a partitioned framed quiver. Let a be any sequence of component preserving mutations. Also, let v be a vertex in the component ni. Then the color of a vertex v in (Q) is the same as the color of the vertex v in (Qf)i.
Theorem 3.11. Let (Q,n) be a framed partition quiver where for each Qi we have a reddening sequence ai. Then let t be a shuffle of the ai such that at every mutation step of the sequence t we have that k is component preserving with respect to n. Then t is a reddening sequence for Qf.
Proof. Let (Q, if ) be a framed partition quiver. Then since each mutation in t is component preserving you have from the Lemma 3.9 that
Mt (Q)i = Mr (Qi ) = Mn (Qi).
Meaning that for each i any vertex v e n is red in mt (Q)i since it is the result of running a reddening sequence. It then follows from Lemma 3.10 that v is red in the larger quiver mt (Q).	□
Corollary 3.12. Furthermore if additionally you have that each ai is a maximal green sequence for the component Qi then you have that t is a maximal green sequence for Q.
Proof. By Theorem 3.11 we know we have a reddening sequence. By Lemma 3.10 and Lemma 3.9 to decide if a mutation step occurred at a green vertex we only need to look at
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the component containing that vertex. Then we consider that each ai is a maximal green sequence and it follows from the same equation:
Mt (Q)i = Mt (Qi) = Mo* (Qi).	1=1
This can be quite useful. In practice what it tells you is that if you partition your quiver up into components, and you know a reddening (or maximal green) sequence for each component then you can try and shuffle the sequences together. If every mutation in the shuffle is component preserving, then you have successfully created a reddening (or maximal green) sequence for the larger quiver. In the sections that follow we will show some of the applications of using this approach to find maximal green and reddening sequences for a variety of quivers. Before showing new applications of the component preserving mutation method, we first provide some examples of previously known maximal green sequences that come from component preserving mutations. These known examples serve to show that our framework unifies many known maximal green sequences. Also the following examples aim to demonstrate that applications of Corollary 3.12 occur "in nature" and thus Definition 3.6 is not too restrictive as it includes many naturally occurring examples.
3.3	Example: Admissible source sequences
A sequence of vertices (i1,i2,..., in) of a quiver Q with n vertices is called an admissible numbering by sources if {i1,i2,..., in} = V(Q) and j is a source of Mj-i ◦ • • • ◦ Mi1 (Q). It is well known that any acyclic quiver Q admits an admissible numbering by sources and that any such admissible numbering by sources (ii, i2,..., in) is a maximal green sequence [3, Lemma 2.20]. In terms of component preserving mutations, (i1, i2,... ,in) being an admissible numbering by sources means that t = Min ◦ Min-1 ◦ • • • ◦ Mi1 is a component preserving sequence of mutations with respect to the partition {i1}/{i2}/ ••• /{in} of V(Q) into singletons. Corollary 3.12 states (i1, i2,..., in) is amaximal green sequence in this special case. Figure 2 shows an example of an acyclic quiver where (4,1,2,3,5) is a maximal green sequence from an admissible numbering by sources with the vertices as labeled in the figure.
3
t
4 —- 1 —-2
I
5
Figure 2: An acyclic quiver with maximal green sequence (4,1,2, 3, 5).
3.4	Example: Direct sum
A direct sum of quivers A and B is any quiver Q with
V(Q) = V(A) u V(B) E(Q) = E(A) U E(B) U E
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where E is any set of arrows such which has for any i ^ j £ E implies i £ V (A) and j £ V(B). In other words, a direct sum of quivers simply takes the disjoint union of the two quivers then adds additional arrows between the quivers with the condition that all arrows are directed from one quiver to the other. We can take the partition V(A)/V(B) of V(Q) and the consider the concatenation t = tbta for any reddening sequence ta of A and tb of B. Then t will be component preserving and hence a reddening sequence by Theorem 3.11
An example of a direct sum of quivers A and B where V(A) = {1,2} and V(B) = {4, 5,6} is given in Figure 3. We can take the maximal green sequences (2,1,2) and (4,6, 5) on the components and obtain maximal green sequence (2,1,2,4, 6, 5) on the direct sum. We will not prove that such sequences of mutations are component preserving since proofs for maximal green sequences and reddening sequences of direct sums are already in the literature [22, Theorem 3.12], [8, Theorem 4.5].
Figure 3: A direct sum of quivers with maximal green sequence (2,1, 2,4,6,5).
3.5 Example: Square products
The square product of two Dynkin quivers is considered by Keller in his work on periodicity [30]. For two type A quivers the square product is a grid with all square faces oriented in a directed cycle. In Figure 4 we show a square product of type (A2, An). Consider the partition n = B/B' of the quiver in Figure 4 where B is the set of vertices in the top row and B' is the set of vertices in the bottom row. Then the quiver restricted to either B or B' is an alternating path which has a maximal green sequence of repeatedly applying sink mutations. A component preserving shuffle for these quivers can be found by alternating between mutations in B and B' until you have completed both maximal green sequences. This example generalizes to many other quivers in a family called bipartite recurrent quivers. Maximal green sequences for bipartite recurrent quivers will be investigated in more depth in Section 5.
4
6
Figure 4: An arbitrary length square product of type (A2, An).
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3.6 Example: Dreaded torus
Let Q be the quiver shown in Figure 5 which comes from a triangulation of the torus with one boundary component and a single marked point on the boundary. With vertices as labeled in the figure we can take the partition {1,4}/{2,3} and the maximal green sequences (1,4,1) and (3, 2,3) on the two components. The sequence (1,3,4, 2,1,3) is component preserving and hence a maximal green sequence by Corollary 3.12. The quiver Q is an example of a quiver which admits a maximal green sequence, and hence a reddening sequence, but is not a member of the class P of Kontsevich and Soibelman [31]. So, Q should be included in a solution to a question posed by the first two authors which seeks to identify a collection of quivers which generate all quivers with reddening sequences by using quiver mutation and the direct sum construction [5, Question 3.6].
4
1-- 2
Figure 5: The quiver for the torus with one boundary component and one marked point. A maximal green sequence for this quiver is (1, 3,4,2,1, 3).
3.7 Example: Cremmer-Gervais
In the Gekhtman, Shapiro, and Vainshtein approach to cluster algebras with Poisson geometry there is an exotic cluster structure on SLn known as the Cremmer-Gervais cluster structure [23, 24]. The mutable part of the quiver defining this cluster structure for the case n = 3 is shown in Figure 6. The cluster algebra has the interesting property that whether or not it agrees with its upper cluster algebra is ground ring dependent [6, Proposition 4.1]. A maximal green sequence for the quiver in Figure 6 is (2, 3,4,1,5,1, 6,3) which can be obtained by considering the partition {1, 2, 5}/{3,6}/{4} along with maximal green sequences (2,1,5,1), (3,6, 3), and (4). The authors believe it would be interesting to try the technique of component preserving maximal green sequences on quivers for the Cremmer-Gervais cluster structure for larger values for n.
¡3^6 1 ^ 2 -— 5
Figure 6: The mutable part of the quiver defining the Cremmer-Gervais cluster structure.
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4 Applications to quiver dominance
One natural question that arises when discussing any algebraic object is to ask questions about what information can be extracted from considering the smaller sub-objects inside your larger object. The methods we have presented thus far give a way of producing reddening sequences on larger quivers by considering reddening sequences on quivers with fewer vertices. In this section we will give a way of producing reddening sequences on larger quivers by considering reddening sequences on quivers with fewer arrows but the same number of vertices.
Component preserving mutations give rise to a dominance phenomenon of quivers. In terms of matrices dominance is given by the following definition. One obtains a definition of dominance in quivers by considering its skew-symmetric exchange matrix.
Definition 4.1. Given n x n exchange matrices B = [bj] and A = [a^], we say B dominates A if for each i and j, we have bj aj > 0 and |bj | > |aj |.
An initiation of a systematic study of dominance for exchange matrices was put forth by Reading [37]. Dominance had previously been considered by Huang, Li, and Yang [26] as part of their definition of a seed homomorphism. One instance of the dominance phenomenon observed by Reading is the following observation about scattering fans.
Phenomenon 4.2 ([37, Phenomenon III]). Suppose that B and B' are exchange matrices such that B dominates B'. In many cases, the scattering fan of B refines the scattering fan of B'.
Remark 4.3. Following [25] to any quiver one can associate a cluster scattering diagram inside some ambient vector space. Reddening sequences and maximal green sequences then correspond to paths in the ambient vector space subject to certain restrictions coming from the scattering diagram. A cluster scattering diagram partitions the ambient vector into a complete fan called the scattering fan [38]. Hence, the phenomenon that the scattering fan of B often refines the scattering fan of B' when B dominates B' means that it should be more difficult to find a reddening sequence for B since the scattering diagram of B has additional walls imposing more constraints. However, we will find certain conditions for when a reddening sequence for B' will still work as a reddening sequence for B.
In this section we will apply the results of Section 3 to show that the existence of a reddening (maximal green) sequence passes through the dominance relationship in many cases. The interesting aspect of this result is it appears to go in the wrong direction; the property is passed from the dominated quiver to the dominating quiver. Let B dominate A. If A has a reddening (maximal green) sequence then, we wish to produce a reddening (maximal green) sequence for B. This is not a true statement in general, but if we put some restrictions on how B dominates A and extra conditions on the reddening or maximal green sequence this turns out to be true. Going forward we will consider dominance in terms of the quivers instead of exchange matrices. A reformulation of dominance is the following.
Definition 4.4. Given quivers B and A on the same vertex set we say that B dominates A if:
• for every pair of vertices (i, j) any arrows between i and j in A are in the same direction as any arrow between i and j in B; and
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• for every pair of vertices (i, j) the number of arrows in B involving vertices i and j is greater than or equal to the number of arrows in A involving i and j.
For an example of quiver dominance see Figure 7 where multiplicity of an arrow greater than 1 is denoted by the number next to the arrow. We now need to establish the notion of n-dominance. This is a restrictive form of dominance, where we the quivers A and B have the same component subquivers with respect to a partition n but have the multiplicity of the bridging arrows altered in a consistent way.
2	3	4	5
Figure 7: An example where the quiver on the right dominates the quiver on the left.

Definition 4.5. Let (A, n) and (B, n) be two partitioned ice quivers with the same vertex set and same set partition n. We say that B n-dominates1 A if:
•	the component quivers Ai = Bi for each i;
•	for all u e Bj and v e Bj with i = j we have the #(u ^ v in B) is equal to dij x #(u ^ v in A), where djj is a positive integer that is the same for the entire i-th and j-th components.
The dij are called the dominance constants associated to (B, n) and (A, n). As usual in Definition 4.5 arrows in the opposite direction are counted as negative. A practical way of thinking about n-dominance is that B is obtained from the A by scaling up the multiplicity of the bridging arrows between components by the appropriate dominance constant. Notice that the dominance constants are always positive, and hence bridging arrows are always in the same direction after scaling by the dominance constants. An example of n-dominance can be seen in Figure 8. This example has the type (A2, A4) square product on the left side and the Q-system quiver of type A4 on the right side.
2 2 2 2
Figure 8: This is n-dominance where the components are the horizontal rows of the quiver. The right hand quiver n-dominates the left hand quiver and d\2 = 2.
Theorem 4.6. Let k be a component preserving vertex in (A, n) and (B, n) be an ice quiver which n-dominates A with dominance constants dij. Then (B) dominates ^k(A)
with dominance constants dij.
Proof. Since k is a component preserving vertex in (A, n) we know that k is also a component preserving vertex in (B, n) since the direction of the bridging arrows is unchanged by scaling by the multiple dij. Also as k is component preserving in both A and B we know
1This is a more restrictive version of the dominance phenomena presented by Reading. In general, not all quivers B which dominate a quiver A will n-dominate the quiver.
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by Lemma 3.9 that	= ^k(Aj) = (Bj) = (B)j. Therefore we only need to
consider the bridging arrows between components.
The bridging arrows incident to k are only affected by the step of mutation which reverses arrows incident to k. Therefore dominance is preserved for these arrows because they are reversed by mutation at k in both A and B.
Now we must check the number of bridging arrows created during mutation for both ^fc(B) and ^k(A). For some nonnegative integer a, we will use the notation i A j to denote that there are a arrows from i to j in a quiver.
Assume s A k A t is present in A with a, p > 0. Then mutation will create arrows from s a t with multiplicity ap. Since we need only consider bridging arrows we will assume the ap many arrows from s to t created are bridging arrows. In the case that k is green we know that s must be in the same component as k because k is component preserving. Assume k, s G V(Aj) and t G V(Aj) for i = j. We now will show that (B)
creates djj ap arrows from s to t. The presence of s A k A t in A implies that there
is s A k A t in B. Therefore mutation at k in B creates dj ap arrows s a t. Now we can consider the multiplicity of bridging arrows resulting from cancellation of 2-cycles mutation. In ^k (A) the multiplicity of the arrows from s to t is ap + 7, where 7 is the number of arrows from s to t in A (here we allow 7 to be negative if there are arrows from t to s). In ^k (B) the multiplicity of arrows from s to t is djjap + djj7 since there are djj7 arrows from s to t in B by the assumption that B n-dominates A. Therefore there are exactly djj (ap + 7) arrows from s to t in (B) which is exactly the condition needed to say that (B) n-dominates ^k(A).
The case where k is red is very similar. In this case t must be in the same component as
k because k is component preserving. The presence of s A k A t in A now implies that
there is s "A k A t in B. Again mutation at k in B creates djj ap arrows s a t and the rest of the argument follows the case where k was green.	□
We can now state our main result regarding dominance, that certain reddening sequences can be passed from a quiver A to a n-dominating quiver B.
Corollary 4.7. Let (A, n) be a partitioned quiver, with n = ni/n2/ • • • /n^. Let cti, ct2, ..., ct^ be reddening sequences for A1, A2,..., A^ respectively. If A admits a reddening sequence, t, which is a component preserving shuffle of ct1, ct2, ..., ct and B n-dominates A, then t is also a reddening sequence for B. Moreover, if t is a maximal green sequence for A, then t is a maximal green sequence for B.
Proof. Theorem 4.6 shows that each component preserving mutation in A is also a component preserving mutation in B. Therefore the mutation sequence t is a component preserving sequence for B since it is a component preserving sequence for A. The definition of n-dominance tells us that A1 = B1,A2 = B2,...,A^ = B^. Therefore since ct1, ct2, ... ct^ are reddening sequences for A1, A2,..., A^, they are also reddening sequences for B1, B2,..., B^. Then by Theorem 3.11 and Corollary 3.12 we have that they are in fact reddening sequences and additionally maximal green in the case where each CTj is a maximal green sequence.	□
Now we are equipped to use n-dominance to produce reddening and maximal green sequences for the dominating quivers by having well behaved sequences on the dominated
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quiver. We conclude this section with a few examples each providing a family of applications of Corollary 4.7.
4.1 Examples of applying Corollary 4.7
Corollary 4.7 applies to any case where one can produce a maximal green or reddening seqeunce using component preserving mutations. Thus, this result can be applied in many cases to produce infinite families of examples. In this section we highlight a few examples.
Example 4.8 (Dreaded torus). Previously much attention has been paid to maximal green sequences for finite mutation type quivers (see [33]). In Section 3.6 we saw one example of a maximal green sequence for a finite mutation type quiver using component preserving mutations. Now we revisit this example, except we can scale the bridging arrows between the components and leave the case of finite mutation type. By Corollary 4.7 we know that the original maximal green sequence for the dreaded torus will also be a maximal green sequence for all n-dominating quivers. Therefore (1,3,4, 2,1,3) is a maximal green sequence for all of the quivers in Figure 9, where a is a positive integer. This is an example of a quiver where the shuffle is not one that can be obtained from direct sum results as the partition does not form a direct sum of either the original quiver or the n-dominating quivers.
Figure 9: For each positive integer a, Corollary 4.7 produces a maximal green sequence for the quiver, which was the maximal green sequence from the dreaded torus. The maximal green sequence is (1,3,4, 2,1,3).
Example 4.9 (The cycle). Another example of finite mutation type quiver is the directed cycle quiver with vertex set {1,2,..., n} and arrow set {i ^ (i + 1) : 1 < i < n} U {n ^ 1}. In [4, Lemma 4.2] it is shown this quiver has the maximal green sequence
which can be seen to be component preserving with respect to the partition {1,2,... ,n - 3, n - 2, n}/{n - 1}. By applying Corollary 4.7 we then obtain maximal green sequences for many quivers of infinite mutation type. The case n = 6 is shown in Figure 10.
Example 4.10 (Q-systems). Consider Figure 11 when a = 2 in which we can produce a maximal green sequence for the Q-system quiver of type A4 by utilizing the maximal green sequence from the square product quiver of type (A2, A4). This technique also produces
4
a
(1, 2, .. ., n - 2, n - 1, n, n - 2, n - 3,. .., 2,1)
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6-► 1
4 <-3
Figure 10: A quiver dominating the cycle which has the maximal green sequence
(1, 2, 3, 4, 5, 6,4, 3, 2,1).
maximal green sequences for other Q-system quivers (see [14, 27]) which are dominating quivers of square products. The next section will focus on producing maximal green sequences for a variety of bipartite recurrent quivers.
Figure 11: This is n-dominance where the components are the horizontal rows of the quiver. The square product quiver on the left has a maximal green sequence compatible with a n component preserving shuffle of (2,3,6,7,1,4,5,8,2, 3, 6,7,1,4, 5, 8,2,3, 6, 7). Corollary 4.7 shows that the quiver on the left where a is any positive integer admits the same maximal green sequence.
3
4
6
2
4
6
5 Bipartite recurrent quivers
In this section we consider certain quivers arising in the setting of T-systems and Y-systems. An early application of cluster algebras was Fomin and Zelevinsky's proof of periodicity for Y-systems associated to root systems [16] which was conjectured by Zamolod-chikov [42]. This has lead to many more applications of cluster algebra theory in periodicity for T-systems and Y-systems. We will focus on work of Galashin and Pylyavskyy on bipartite recurrent quivers [18,19,20]. For certain bipartite recurrent quivers we will produce maximal green sequences in Theorem 5.3. An important ingredient in our constructions of maximal green sequences will be an extension of Stembridge's bigraphs [41]. The pattern for the maximal green sequences produced in this section was originally observed by Keller in the case of square products [30]. For a quantum field theory perspective on the results in this section we refer the reader to [10] where some of the same mutation sequences we construct are also considered. The main contribution of this section is to demonstrate how component preserving mutation neatly establishes the existence of a maximal green sequence for all quivers in Galashin and Pylyavskyy's classification of Zamolodchikov periodic quivers [18] as well as for some additional bipartite recurrent quivers.
We call a quiver Q bipartite if there exists a map e: V(Q) ^ {0,1} such that e(i) = e(j) for every arrow i ^ j of Q. The choice of such a map e when it exists for a quiver Q is called a bipartition. Given a bipartition e for Q a vertex i G V(Q) will be called white if e(i) = 0 and black if e(i) = 1. Let ii, i2,..., i^ denote the white vertices and Q and
E. Bucher et al.: Building maximal green sequences via component preserving mutations 265
ji,j2,... ,jm denote the black vertices. We then let
Mo = Mil ° ◦ • • • ◦
and
M. = Mjl ° Mj2 ◦•••◦ Mjm
denote the mutations at all white vertices or black vertices respectively. Since the quiver is bipartite no white vertex is adjacent to any other white vertex and so the order of mutation among the white vertices in Mo does not matter. Similarly the order among the black vertices in m. does not matter. A bipartite quiver Q is recurrent if both mo(Q) = Qop and m. (Q) = Qop where Qop denotes the quiver obtained from Q by reserving the direction of all arrows. Thus for a bipartite recurrent quiver we have m.(mo(Q)) = Q and Mo(M.(Q)) = Q.
A bigraph is a pair (r, A) of undirected graphs on the same underlying vertex set with no edges in common. Let Ar and Aa denote the adjacency matrices of r and A respectively. Given any bipartite quiver Q with bipartition e we obtain a bigraph (r(Q), A(Q)) on vertex set V(Q) where r(Q) has an edge {i, j} for each arrow i ^ j in Q with e(i) = 0 and A(Q) has an edge {i, j} for each arrow i ^ j of Q with e(i) = 1. By abuse of notation we may also think of r(Q) and A(Q) as directed graphs with the direction of edge inherited from the quiver. Galashin and Pylyavskyy have shown that a bipartite quiver Q is recurrent if and only if Ar(Q) and Aa(q) commute [18, Corollary 2.3]. A bigraph (r, A) is called an admissible ADE bigraph if every component of both r and A is an ADE Dynkin diagram and the adjacency matrices of r and A commute. In the case of an admissible ADE bigraph, each connected component of r, and similarly of A, will be an ADE Dynkin diagram will the same Coxter number [41, Corollary 4.4]. More generally, we wish to also consider what we will refer to as half-finite bigraphs where for at least one of r or A each connected component is a ADE Dynkin diagram. Note the half-finite case includes both the admissible ADE bigraph case (which are exactly those quivers which are Zamolodchikov periodic [18]) as well as the affine K finite case in the classification of Galashin and Pylyavskyy [20]. An example of a bipartite recurrent quiver is shown in Figure 12. Let Q denote the bipartite recurrent quiver in Figure 12. The edges of r(Q) correspond to the thick red arrows while the edges of A(Q) correspond to the thin blue arrows.
Figure 12: An example of a bipartite recurrent quiver.
For an ADE Dynkin diagram A we denote its Coxeter number by h(A) and its number of positive roots by |$+(A)|. These quantities will be important in the maximal green sequences we construct. Table 1 shows the values for h(A) and |$+(A)| for each ADE Dynkin diagram A. We now present a result due to Galashin and Pylyavskyy generalizing the result for admissible ADE bigraphs.
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Table 1: Coxeter numbers and number of positive roots for ADE types.
A	An	Dn	Ee	E7	Eg
h(A)	n + 1	2n — 2	12	18	30
|$+(A)|	C+1)	n2 — n	36	63	120
Lemma 5.1 ([20, Corollary 1.1.9]). If (T, A) is a half-finite bigraph so that each component of r is an ADE Dynkin diagram, then the Coxeter number of each component of V will be the same.
If Q is an orientation of an ADE Dynkin diagram r, then the length of the longest possible maximal green sequence is |$+(A)| which has been shown in [3, Theorem 4.4] and [36, Proposition 7.3]. A quiver Q is an alternating orientation of an ADE Dynkin diagram A if it is an orientation of A so that every vertex is either a source or sink. In the case we have an alternating orientation, we will be interested in a certain maximal green sequence of length |$+(A)| coming from bipartite dynamics. We may assume we have a bipartition of Q such that all sinks are the white vertices and all sources are the black vertices. The maximal green sequence in the following lemma was first observed by Keller [29].
Lemma 5.2 ([29]). Let Q be an alternating orientation of an ADE Dynkin diagram with Coxeter number h. If h = 2k, then	is a maximal green sequence. If h = 2k + 1,
then	is a maximal green sequence.
We are ready to state and prove our theorem which gives a maximal green sequence for any half-finite bipartite recurrent quiver. Notice the assumption that r(Q) consists of connected components which are all ADE Dynkin diagrams can easily be exchanged for the assumption that A(Q) consists of connected components which are all ADE Dynkin diagrams. Also the assumption on white vertices is only to allow us to explicitly state the maximal green sequences. An easy modification gives the correct statement of the theorem with the roles of black and white vertices reversed.
Theorem 5.3. Let Q be a half-finite bipartite recurrent quiver. Assume that r(Q) consists of connected components which are all ADE Dynkin diagrams. Further assume that with the orientation induced by Q the white vertices are sinks in r(Q) and sources is A(Q). Let h be the Coxeter number of some component of r(Q). If h = 2k is even, then (^.^0)fc is a maximal green sequence of Q. If h = 2k +1 is odd, then	is a maximal green
sequence of Q.
Proof. We will construct a maximal green sequence for Q via component preserving mutations where components are given by the connected components of r(Q). By construction within each component every vertex will be either a source or sink. Under our assumptions white vertices are initially sinks while black vertices are initially sources within each component. Since Q is a bipartite recurrent quiver ^o(Q) = Qop and p>,(p0(Q)) = Q. Initially, mutation at any white vertex will be component preserving as each white vertex is a sink within its component and thus all arrows to other components will be outgoing. Mutation at a given white vertex will not change the fact another white vertex is component preserving. For the same reason mutation at any black vertex is component preserving
E. Bucher et al.: Building maximal green sequences via component preserving mutations 267
in Qop. It follows that	and	are component preserving sequences
of mutations for any m. By Lemma 5.1 each component has the same Coxeter number. Lemma 5.2 says that we do indeed have maximal green sequences on each component and therefore the theorem is proven by appealing to Corollary 3.12.	□
6 Other applications
In this section we provide a variety of uses of the technique of component preserving mutations.
6.1 Quantum dilogarithms
We will review Keller's [28] association of a product of quantum dilogarithms with a sequence of mutations. We will then consider properties of such products of quantum dilogarithms which come from component preserving mutations. Let q1 be an indeterminant. We define the quantum dilogarithm as
1	n2 n
E( ) = i + q2y + +_q2 y_+
E(y) 1 + q - 1 + • • • + (qn - 1)(qn - q) • • • (qn - qn-1) + • • •
which is consider as an element of the power series ring Q(q1 )[[y]]. Keller has shown how reddening sequences give identities of quantum dilogarithms in a certain quantum algebra determined by a quiver.
Given a quiver Q with vertex set V and skew-symmetric adjacency matrix B = (buv) we obtain a lattice A = ZV with basis {ev}veV. There is a skew-symmetric bilinear form A: A x A ^ Z defined by
A(eu ev ) — buv •
The completed quantum algebra of the quiver Q, denoted by Aq, is then the noncommu-tative power series ring modulo relations defined as
Aq := Q(q2)«ya,a € A : y V = q1 ya+l3»• For any sequence a = (i1, i2,..., iN) of vertices in Q we define
Qa,t := Mit ° Mit-1 O • • • O (Q)
for 0 < t < N where Qctj0 = Q. We then define the product Eqjct € Aq as
Eqjct := E(y£1ft )£1 E(y£2fef2 ••• E(y£N^)eN
where pt is the c-vector corresponding to vertex it in QCTjt-1 and et € {±1} is the common sign on the entries of ,0t. If a is a reddening sequence, then Eqjct is known as the combinatorial Donaldson-Thomas invariant of the quiver Q. If a and a' are two reddening sequences, then we have the quantum dilogarithm identity Eqjct = Eqjct' [29, Theorem 6.5].
In the case that a = J2ie/ ei where I = {i1, i2,..., i^} we may write yi1i2...ie in place of ya. Using this abbreviated notation, the well known pentagon identity is
E(y1)E(y2) = E(y2)E(y^)E(y1)	(6.1)
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and can be seen by looking at the two maximal green sequences for the quiver Q = (1 ^ 2). Now consider the quiver in Figure 13 which is an alternating orientation of the Dynkin diagram A3. The two maximal green sequences
(2,1, 3)
and
(1, 3, 2,1, 3, 2)
give the quantum dilogarithm identity
E(y2)E(y1)E(y3) = E(yOE(y3)E(ym)E(y23)E(y12)E(y2).	(6.2)
Reineke [39] has given quantum dilogarithm identities associated to any alternating orientation of an ADE Dynkin diagram which generalize Equations (6.1) and (6.2). Using cluster algebra theory, Keller [29] has further generalized these identities to square products associated to pairs of ADE Dynkin diagrams. Even more general identities follow from Theorem 5.3 since we have now produced two maximal green sequences for any Zamolodchikov periodic quiver.
1 -«— 2 —3
Figure 13: An alternating orientation of the Dynkin diagram A3.
Let us give a few properties of quantum dilogarithm products coming from component preserving mutations. For a = £i aiei e A we define its support to be Supp(a) := {i : ai = 0}. Consider a quiver Q, a subset of vertices C C V(Q), and a sequence of vertices a = (i1, i2,... ,iN). Define a|C to be the restriction of a to C (i.e. a where all vertices not in C have been deleted). Again write
Eqjct = E(yeit31 )eiE(ye2132p ••• E(y£N)eN
and define (Eqjct)|c to be the product Eqjct (taken in the same order) with the terms E(y£tft removed whenever it e C. We now provide a proposition which tells us that when a reddening sequence of component preserving mutations is performed, there is a restriction on the support of the c-vectors occurring in the combinatorial Donaldson-Thomas invariant. The proposition follows readily from the definitions and Remark 3.8. When n is a set partition of a set X and x e X is an element of that set, we will use n(x) to denote the block of the set partition n which contains x.
Proposition 6.1. Let (Q, n) be a partitioned quiver so that a = (i1, i2,... ,iN) is a component preserving sequence of vertices. If C = Qj is some component, then Eq^\c = (Eqjct)|c. Moreover, we have that Supp(ftt) C n(it) for each 1 < t < N.
When Q is such that (r(Q), A(Q)) is an admissible ADE bigraph we can obtain a second maximal green sequence from Theorem 5.3 by exchanging the roles of r(Q) and A(Q). A square product of two ADE Dynkin diagrams produces a quiver Q such that (r(Q), A(Q)) is an admissible ADE bigraph. For square products of ADE Dynkin diagrams Keller [29] has previously produced the maximal green sequences in Theorem 5.3. The square product of A3 and A4 is shown in Figure 12. Stembridge's classification [41] of
E. Bucher et al.: Building maximal green sequences via component preserving mutations 269
admissible ADE bigraphs includes more than just those bigraphs encoding square products of ADE Dynkin diagrams. Thus, Theorem 5.3 provides new quantum dilogarithm identites which can be thought of as generalizations of the pentagon identity. An infinite family examples of quivers which are not square products are the twists of an ADE Dynkin diagrams [41, Example 1.4]. The quiver Q which is the twist of A3 is shown in Figure 14. On the left of Figure 14 the quiver is pictured to indicated the bigraph (r(Q), A(Q)), and on the right we show the quiver with vertex labels. The two expressions of the combinatorial Donaldson-Thomas invariant of Q obtain from the maximal green sequences constructed in Theorem 5.3 are
E(yi)E(y3)E(y4)E(ye)E(yi23)E(y456)E(y23)E(yi2)E(y56)E(y45)E(y2)E(y4) (6.3) and
E(y2)E(y5)E(yi5)E(y35)E(y24)E(y26)E(y246)E(yi35)E(yi)E(y3)E(y4)E(y6). (6.4)
These expressions are equal and give one example of the quantum dilogarithm identities obtained from Theorem 5.3. Looking at supports we can verify Proposition 6.1 in this example. Expression (6.3) comes from considering {1,2, 3} and {4,5, 6} as components while Expression (6.4) comes from considering {1,3,5} and {2,4, 6} as components. The maximal green sequences corresponding to the products of quantum dilogarithms in Equations (6.3) and (6.4) are
(1, 3,4, 6, 2, 5,1, 3,4, 6, 2, 5)
and
(2, 5,1, 3, 4, 6, 2, 5,1, 3, 4, 6)
respectively.
6.2 Minimal length maximal green sequences
There has been recent interest in finding maximal green sequences of minimal possible length for a given quiver [11, 21]. We will now show how minimal length maximal green sequences can be constructed with component preserving mutations. In additional to being a natural question to ask about maximal green sequences, it has been observed by Garver, McConville, and Serhiyenko that the minimal possible length of a maximal green sequence may be related to derived equivalence of cluster tilted algebras (see [21, Question 10.1]). The following result is a component preserving generalization of [21, Proposition 4.4] which considers the direct sum case.
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Lemma 6.2. Let (Q, n) be a partitioned quiver with n = ni/n2/ • • • /n^. Also let ai be a minimal length maximal green sequence for Qi for each 1 < i < t. If t is a component preserving shuffle of a1, a2,..., then t is a minimal length maximal green sequence for Q.
Proof. Let Li be the length of a minimal length maximal green sequence of Qi for each 1 < i < t and let L = L1 + L2 + • • • + L^. By Corollary 3.12 we know that t is a maximal green sequence and will have length L. So, we now need to show that there are no shorter maximal green sequences. Consider any maximal green sequence t' for Q. By [21, Theorem 3.3] it follows that for each 1 < i < t there is a subsequence of mutations in t' at vertices in Qi which is a maximal green sequence of Qi. This means t' must mutate at vertices of Qi at least Li times for each 1 < i < t. Since n is a partition, Qi and Qj share no vertices when i = j. It follows that t' has length at least L = L1 + L2 + • • • + L^. □
To illustrate a use of Lemma 6.2, let Q be the quiver2 in Figure 15. We will take the set partition {v1, v2, v3, v4, v5}/{m1, u2, «3, u4}. A minimal length maximal green sequence for Q is then
(«1, «2, «3, V1, V2, V3, V4, V5, V3, V2, V1, «4)
which is a shuffle of (v1, v2, v3, v4, v5, v3, v2, v1) and (u1, u2, «3, u4). The first is a maximal green sequence for the cycle by [4, Lemma 4.2] and is of minimal length by [21, Theorem 6.1]. The second is a maximal green sequence coming from an admissible numbering by sources.
«4 -«- «3 -«- «2 - «1
V5 -V1
t J
V4	V2
V3
Figure 15: A quiver where a minimal length maximal green sequence can be found by component preserving mutations.
6.3 Exponentially many maximal green sequences for Dynkin quivers
In [3, Remark 4.2 (3)] the authors observe that the number of maximal green sequences of the lineary oriented Dynkin quiver of type An seems to grow exponentially with n. The main result of this section will affirm this observation. A Dynkin quiver of type An is any orientation of the Dynkin diagram of type An. The linearly oriented Dynkin quiver of type An has vertex set {i : 1 < i < n} and arrow set {i ^ i + 1:1 < i < n}. Figure 16 shows the linearly oriented Dynkin quiver of type A5. We will show that the number of maximal green sequences of arbitrarily oriented Dynkin quiver of type An is at least expontential. We give a simple and explicit proof of an exponential lower bound to | green(Q) | where Q
2 The use of Lemma 6.2 readily generalizes to quivers similar to Q with longer cycle or longer path.
E. Bucher et al.: Building maximal green sequences via component preserving mutations 271
is any Dynkin quiver of type An. After we will provide an improved bound in the case Q is a linearly oriented Dynkin quiver of type An.
1 —2 —3 —-4 —«-5 Figure 16: The linearly oriented Dynkin quiver A5.
Recall the Fibonacci numbers are defined by the recurrence F1 = 1, F2 = 2, and
Fn = Fn-1 + Fn-2 for n > 2. A closed form expression for Fn is
-
Fn
V5 where
1 + V5 . 1 - V5 * = -T" ^ = —T~.
Proposition 6.3. If Q is a Dynkin quiver of type An for any n > 1, then | green(Q)| > Fn+1.
Proof. It can be easily checked that | green(Q)| = 1 = F2 for n =1 and | green(Q)| = 2 = F3 for n = 2. For n > 3 assume inductively that | green(Q)| > Fm+1 for all 1 < m < n. We first consider components of Q coming from the set partition C/C' where C = {i : 1 < i < n — 1} and C' = {n}. Here Q is isomorphic to a direct sum of a Dynkin quiver of type 1 and a Dynkin quiver of type A1. Hence, Q has at least | green(Q | c) | maximal green sequences by considering any maximal green sequence on Q|C with (n) either appended or prepened depending of whether (n — 1) ^ n e Q or n ^ (n — 1) e Q.
Next consider components of Q coming from the set partition D/D' where D = {i : 1 < i < n — 2} and D' = {n — 1, n}. Now Q is isomorphic to a direct sum of Dynkin quiver of type An-2 and a Dynkin quiver of type A2. Thus, Q has at least | green(Q|D)| maximal green sequences by considering any maximal green sequence on D with:
•	(n, n — 1, n) appended if (n — 2) ^ (n — 1), (n — 1) ^ n e Q.
•	(n, n — 1, n) prepended if (n — 1) ^ (n — 2), (n — 1) ^ n e Q.
•	(n — 1, n, n — 1) appended if (n — 2) ^ (n — 1), n ^ (n — 1) e Q.
•	(n — 1, n, n — 1) prepended if (n — 1) ^ (n — 2), n ^ (n — 1) e Q.
We see that the set of maximal green sequences for Q coming from green(Q | C) are disjoint from those coming from green(Q|D). In the former n is mutated at only once and is either mutated first or last in the sequence. In the latter n is either mutated at twice or otherwise is neither the first nor the last mutation. It follows that
| green(Q)| > | green(Q|c)| + | green(Q|D)| > Fn + Fn_1 = Fn+1
and the proposition is proven.	□
For a linearly oriented Dynkin quiver Q of type An, we have the maximal green sequence
(n, n — 1, . . . , 1, n, n — 1, . . . , 2, . . . , n, n — 1, n)
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which we will call the long sequence.3 As an example in the case n = 4 the long seqeunce is
(4, 3, 2,1,4, 3, 2, 4, 3,4).
The long sequence is a maximal green sequence coming from a reduced factorization of the longest element in the corresponding Coxeter group.
Proposition 6.4. If Q is the linearly oriented Dynkin quiver of type An for any n > 1, then
| green(Q) | > 2n—1.
Proof. For n =1 we have | green(Q)| = 1 and for n = 2 and | green(Q)| = 2. Given n > 3, assume inductively that | green(Q)| > 2m—1 for all 1 < m < n. Consider components from the set partition C(fc)/D(fc) where C= {1, 2,..., k} and = {k +1, k + 2,..., n} for 0 < k < n. For each k, our quiver Q has at least | green(Q | C(k)) | many maximal green sequences by appending the long sequence of Q | D(k) to any maximal green sequence of Q|C(k). Here we count one maximal green sequence, the long sequence for Q, when k = 0. In the long sequence for Q|D(k) vertex n is mutated at n - k times, and thus the maximal green sequences coming from green(Q|C(fcl)) and green(Q|C(k2)) are disjoint for k1 = k2. So,
n—1	n—1
| green(Q)| > £ | green(Q|C«)| > 1 + £ 2k—1 = 2n—1
fc=0 fc=1
and the proposition follows.	□
Let green(An) denote the set of maximal green sequences of a linearly oriented type An quiver. Proposition 6.4 is constructive starting from knowing green(A1) = {(1)} and green(A2) = {(1,2), (2,1,2)}. The method in the proof of Proposition 6.4 produces
{(1, 2, 3), (2,1, 2, 3), (1, 3, 2, 3), (3, 2,1, 3, 2, 3)} C green(A3),
and we show in Table 2 the 8 maximal green sequences in green(A4) constructed by applying the proof of Proposition 6.4 one more time. The maximal green sequences in Table 2 are arranged according to the set partition C/D(k).
Table 2: Maximal green sequences in green(A4) constructed in proof of Proposition 6.4 according to set partition C(fc)/D(fc).
k	Maximal green sequences
0	(4, 3, 2,1,4, 3, 2, 4, 3,4)
1	(1,4, 3, 2,4, 3,4)
2	(1, 2,4, 3,4), (2,1, 2, 4, 3,4)
3	(1, 2, 3,4), (2,1, 2, 3,4), (1, 3, 2, 3, 4), (3, 2,1, 3, 2, 3,4)
3There are many possible maximal green sequences of this maximal length. So, we should perhaps say a long sequence instead of the long sequence. However, we wish to emphasize that in this section we will be using only this particular sequence of mutations.
E. Bucher et al.: Building maximal green sequences via component preserving mutations 273
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/^creative ^commor
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 19 (2020) 277-295 https://doi.org/10.26493/1855-3974.2348.f42
(Also available at http://amc-journal.eu)
ARS MATHEMATICA CONTEMPORANEA
The Cayley isomorphism property
'5 x Cp
for the group C5 x Cp
Grigory Ryabov * ©
Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia, and Novosibirsk State University, 1 Pirogova st., 630090, Novosibirsk, Russia
Received 28 May 2020, accepted 29 July 2020, published online 17 November 2020
Abstract
A finite group G is called a DCI-group if two Cayley digraphs over G are isomorphic if and only if their connection sets are conjugate by a group automorphism. We prove that the group Cf x Cp, where p is a prime, is a DCI-group if and only if p = 2. Together with the previously obtained results, this implies that a group G of order 32p, where p is a prime, is a DCI-group if and only if p = 2 and G = Cf x Cp.
Keywords: Isomorphisms, DCl-groups, Schur rings. Math. Subj. Class. (2020): 05C25, 05C60, 20B25
1 Introduction
Let G be a finite group and S C G. The Cayley digraph Cay(G, S) over G with connection set S is defined to be the digraph with vertex set G and arc set {(g, sg) : g G G, s g S}. Two Cayley digraphs over G are called Cayley isomorphic if there exists an isomorphism between them which is also an automorphism of G. Clearly, two Cayley isomorphic Cayley digraphs are isomorphic. The converse statement is not true in general (see [3, 10]). A subset S C G is called a Cl-subset if for each T C G the Cayley digraphs Cay(G, S) and Cay(G, T) are isomorphic if and only if they are Cayley isomorphic. A finite group G is called a DCI-group (Cl-group, respectively) if each subset of G (each inverse-closed subset of G, respectively) is a CI-subset.
*The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.
The author would like to thank Prof. Istvan Kovacs for the fruitful discussions on the subject matters, Prof. Pablo Spiga and the anonymous referee for valuable comments which help to improve the text significantly.
E-mail address: gric2ryabov@gmail.com (Grigory Ryabov)
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
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The investigation of DCI-groups was initiated by Adam [1] who conjectured, in our terms, that every cyclic group is a DCI-group. This conjecture was disproved by Elspas and Turner in [10]. The problem of determining of finite DCI- and Cl-groups was suggested by Babai and Frankl in [5]. For more information on DCI- and Cl-groups we refer the readers to the survey paper [21].
In this paper we are interested in abelian DCI-groups. The cyclic group of order n is denoted by Cn. Elspas and Turner [10] and independently Djokovic [8] proved that every cyclic group of prime order is a DCI-group. The fact that Cpq is a DCI-group for distinct primes p and q was proved by Alspach and Parsons in [3] and independently by Klin and Poschel in [17]. The complete classification of cyclic DCI-groups was obtained by Muzychuk in [23, 24]. He proved that a cyclic group of order n is a DCI-group if and only if n = k or n = 2k, where k is square-free.
Denote the class of all finite abelian groups where every Sylow subgroup is elementary abelian by E. From [18, Theorem 1.1] it follows that every DCI-group is the coprime product (i.e. the direct product of groups of coprime orders) of groups from the following list:
Cpk, C4, Q8, A4, H x (z),
where p isaprime, H is a group of odd order from E, |z| G {2,4}, and hz = h-1 for every h G H. One can check that the class of DCI-groups is closed under taking subgroups. So one of the crucial steps towards the classification of all DCI-groups is to determine which groups from E are DCI.
The following non-cyclic groups from E are DCI-groups (p and q are assumed to be distinct primes): Cp [2, 14]; Cp [2, 9]; C4, Cf [7]; Cp, where p is odd [15] (a proof for Cp with no condition on p was given in [22]); Cp, where p is odd [13]; Cp x Cq [18]; Cp x Cq [27]; Cp4 x Cq [20]. The smallest example of anon-DCI-group from E was found by Nowitz [28]. He proved that C6 is non-DCI. This implies that C^ is non-DCI for every n > 6. Also C3n is non-DCI for every n > 8 [33] and C^ is non-DCI for every prime p and n > 2p + 3 [32].
In this paper we find a new infinite family of DCI-groups from E which are close to the smallest non-DCI-group from E. The main result of the paper can be formulated as follows.
Theorem 1.1. Let p be a prime. Then the group Cf x Cp is a DCI-group if and only if p = 2.
Theorem 1.1 extends the results obtained in [18, 20, 27] which imply that the group Cp x Cq is a DCI-group whenever p and q are distinct primes and k < 4. Note that the "only if" part of Theorem 1.1, in fact, was proved by Nowitz in [28]. The next corollary immediately follows from [18, Theorem 1.1] and Theorem 1.1.
Corollary 1.2. Let p be a prime. Then a group G of order 32p is a DCI-group if and only ifp = 2 and G = Cf x Cp.
To prove Theorem 1.1, we use the S-ring approach. An S-ring over a group G is a subring of the group ring ZG which is a free Z-module spanned by a special partition of G. If every S-ring from a certain family of S-rings over G is a CI-S-ring then G is a DCI-group (see Section 4). The definition of an S-ring goes back to Schur [31] and Wielandt [34]. The usage of S-rings in the investigation of DCI-groups was proposed by
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Klin and Poschel [17]. Most recent results on DCI-groups were obtained using S-rings (see [15, 18, 19, 20, 27]).
The text of the paper is organized in the following way. In Section 2 we provide definitions and basic facts concerned with S-rings. Section 3 contains a necessary information on isomorphisms of S-rings. In Section 4 we discuss CI-S-rings and their relation with DCI-groups. We also prove in this section a sufficient condition of CI-property for S-rings (Lemma 4.4). Section 5 is devoted to the generalized wreath and star products of S -rings. Here we deduce from previously obtained results two sufficient conditions for the generalized wreath product of S-rings to be a CI-S-ring (Lemma 5.5 and Lemma 5.8). Section 6 and 7 are concerned with p-S-rings and S-rings over a group of order pk, where p is a prime and GCD(p, k) = 1, (so-called non-powerful order) respectively. In Section 8 we provide properties of S-rings over the groups C^, n < 5, and prove that all S-rings over these groups are CI. The material of this section is based on computational results obtained with the help of the GAP package COCO2P [16]. Finally, in Section 9 we prove Theorem 1.1.
Notation. Let G be a finite group and X Ç G. The element J2xeX x of the group ring ZG is denoted by X.
The set {x-1 : x G X} is denoted by X-1.
The subgroup of G generated by X is denoted by (X};we also set rad(X) = {g G G : gX = Xg = X}.
Given a set X Ç G the set {(g, xg) : x G X, g G G} of arcs of the Cayley digraph Cay(G, X) is denoted by A(X).
The group of all permutations of G is denoted by Sym(G).
The subgroup of Sym(G) consisting of all right translations of G is denoted by Gright.
The set {K < Sym(G) : K > Gright} is denoted by Sup(Gright).
For a set A Ç Sym(G) and a section S = U/L of G we set AS = {fS : f G A, Sf = S}, where Sf = S means that f permutes the L-cosets in U and fS denotes the bijection of S induced by f.
If K < Sym(Q) and a G Q then the stabilizer of a in K and the set of all orbits of K on Q are denoted by Ka and Orb(K, Q) respectively.
If H < G then the normalizer of H in G is denoted by NG(H).
The cyclic group of order n is denoted by Cn.
The class of all finite abelian groups where every Sylow subgroup is elementary abelian is denoted by E.
2 S-rings
In this section we give a background of S-rings. In general, we follow [20], where the most part of the material is contained. For more information on S-rings we refer the readers to [6, 25].
Let G be a finite group and ZG the integer group ring. Denote the identity element of G by e. A subring A C ZG is called an S-ring (a Schur ring) over G if there exists a partition S(A) of G such that:
(1)	{e} G S(A),
(2)	if X G S(A) then X-1 G S(A),
(3) A = SpanZ{X : X G S (A)}.
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The elements of S(A) are called the basic sets of A and the number rk(A) = |S(A)| is called the rank of A. If X, Y e S(A) then XY e S(A) whenever |X | = 1 or |Y | = 1.
Let A be an S-ring over a group G. A set X C G is called an A-set if X e A. A subgroup H < G is called an A-subgroup if H is an A-set. From the definition it follows that the intersection of A-subgroups is also an A-subgroup. One can check that for each A-set X the groups (X} and rad(X) are A-subgroups. By the thin radical of A we mean the set defined as
Oe(A) = {x e G : {x} e S(A)}. It is easy to see that O6 (A) is an A-subgroup.
Lemma 2.1 ([11, Lemma 2.1]). Let A be an S-ring over a group G, H an A-subgroup of G, and X e S(A). Then the number |X n Hx| does not depend on x e X.
Let L < U < G. A section U/L is called an .A-section if U and L are A-subgroups. If S = U/L is an A-section then the module
AS = SpanZ {Xn : X e S(A), X C U} ,
where n: U ^ U/L is the canonical epimorphism, is an S-ring over S.
3 Isomorphisms and schurity
Let A and A' be S-rings over groups G and G' respectively. A bijection f: G ^ G' is called an isomorphism from A to A' if
{A(X)f : X e S(A)} = {A(X') : X' e S(A')},
where A(X)f = {(gf, hf) : (g, h) e A(X)}. If there exists an isomorphism from A to A' then we say that A and A' are isomorphic and write A = A'.
The group of all isomorphisms from A onto itself contains a normal subgroup
{f e Sym(G) : A(X)f = A(X) for every X e S(A)}
called the automorphism group of A and denoted by Aut(A). The definition implies that Gright < Aut(A). The S-ring A is called normal if Gright is normal in Aut(A). One can verify that if S is an A-section then Aut(A)S < Aut(AS). Denote the group Aut(A) n Aut(G) by AutG(A). It easy to check that if S is an A-section then AutG(A)S < AutS (AS). One can verify that
AutG(A) = (NAut(A)(Gright))e.
Let K e Sup(Gright). Schur proved in [31] that the Z-submodule
V(K, G) = SpanZ{X : X e Orb(Ke, G)},
is an S-ring over G. An S-ring A over G is called schurian if A = V(K, G) for some K e Sup(Gright). One can verify that given Ki, K2 e Sup(Gright),
if Ki < K2 then V(Ki,G) > V(K2,G).	(3.1)
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If A = V (K, G) for some K e Sup(Gright) and S is an A-section then AS = V (KS, S). So if A is schurian then AS is also schurian for every A-section S. It can be checked that
V(Aut(A), G) > A	(3.2)
and the equality is attained if and only if A is schurian.
An S-ring A over a group G is defined to be cyclotomic if there exists K < Aut(G) such that S(A) = Orb(K, G). In this case we write A = Cyc(K, G). Obviously, A = V(GrightK, G). So every cyclotomic S-ring is schurian. If A = Cyc(K, G) for some K < Aut(G) and S is an A-section then AS = Cyc(KS, S). Therefore if A is cyclotomic then AS is also cyclotomic for every A-section S.
Two permutation groups Ki and K2 on a set Q are called 2-equivalent if Orb(K, Q2) = Orb(K2, Q2) (here we assume that K1 and K2 act on Q2 componentwise). In this case we write K1 «2 K2. The relation «2 is an equivalence relation on the set of all subgroups of Sym(Q). Every equivalence class has a unique maximal element with respect to inclusion. Given K < Sym(Q), this element is called the 2-closure of K and denoted by K(2). If A = V(K, G) for some K e Sup(Gright) then K(2) = Aut(A). An S-ring A over G is called 2-minimal if
{K e Sup(Gright) : K «2 Aut(A)} = {Aut(A)}.
Two groups K1,K2 < Aut(G) are said to be Cayley equivalent if Orb(K1,G) = Orb(K2, G). In this case we write K1 «Cay K2. If A = Cyc(K, G) for some K < Aut(G) then AutG (A) is the largest group which is Cayley equivalent to K. A cyclotomic S-ring A over G is called Cayley minimal if
{K < Aut(G) : K «cay AutG(A)} = {AutG(A)}.
It is easy to see that ZG is 2-minimal and Cayley minimal.
4 CI-S-rings
Let A be an S-ring over a group G. Put
Iso(A) = {f e Sym(G) : f is an isomorphism from A onto an S-ring over G}.
One can see that Aut(A) Aut(G) C Iso(A). However, the converse statement does not hold in general. The S-ring A is defined to be a CI-S-ring if Aut(A) Aut(G) = Iso(A). It is easy to check that ZG and the S-ring of rank 2 over G are CI-S-rings. Put
Sup2 (Gright) = {K e Sup(Gright) : K(2) = K}.
The group M < Sym(G) is said to be G-regular if M is regular and isomorphic to G. Following [15], we say that a group K e Sup(Gright) is G-transjugate if every G-regular subgroup of K is K-conjugate to Gright. Babai proved in [4] the statement which can be formulated in our terms as follows: a set S C G is a CI-subset if and only if the group Aut(Cay(G, S)) is G-transjugate. The next lemma provides a similar criterion for a schurian S-ring to be CI.
Lemma 4.1. Let K e Sup2(Gright) and A = V (K, G). Then A is a CI-S-ring if and only if K is G-transjugate.
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Proof. The statement of the lemma follows from [15, Theorem 2.6].	□
Let Ki,K2 G Sup(Gright) such that K < K2. Then K is called a G-complete subgroup of K2 if every G-regular subgroup of K2 is K2-conjugate to some G-regular subgroup of Ki (see [15, Definition 2]). In this case we write Ki <G K2. The relation <G is a partial order on Sup(Gright). The set of the minimal elements of Sup2(Gright) with respect to is denoted by Supmin(Gright).
Lemma 4.2 ([20, Lemma 3.3]). Let G be a finite group. If V(K, G) is a CI-S-ring for every K G Supmin(Gright) then G is a DCI-group.
Remark 4.3. The condition that V(K, G) is a Cl-S-ring for every K G Supmin(Gright) is equivalent to, say, that every schurian S-ring over G is a CI-S-ring. The latter condition means that every 2-closed overgroup of Gright is G-transjugate. However, 2-closed over-group of Gright may not be the automorphism group of a Cayley digraph over G. So the condition that the automorphism group of every Cayley digraph over G is G-transjugate or, equivalently, that G is a DCI-group, seems weaker than the condition of Lemma 4.2. It is a natural question whether there exists a DCI-group for which the condition of Lemma 4.2 does not hold.
We finish the section with the lemma that gives a sufficient condition for an S-ring to be a CI-S-ring. In order to formulate this condition, we need to introduce some further notations. Let A be a schurian S-ring over an abelian group G and L a normal A-subgroup of G. Then the partition of G into the L-cosets is Aut(A)-invariant. The kernel of the action of Aut(A) on the latter cosets is denoted by Aut(A)G/L. Since Aut(A)G/L is a normal subgroup of Aut(A), we can form the group K = Aut(A)G/LGright. Clearly, K < Aut(A). From [15, Proposition 2.1] it follows that K = K(2).
Lemma 4.4. Let A be a schurian S-ring over an abelian group G, L an A-subgroup of G, and K = Aut(A)G/LGright. Suppose that both AG/L and V(K, G) are CI-S-rings and Ag/l is normal. Then A is a Cl-S-ring.
Proof. Firstly we prove that the group Aut(A)G/L is G/L-transjugate. Suppose that F is a G/L-regular subgroup of Aut(A)G/L. The S-ring AG/L is a CI-S-ring by the assumption of the lemma. So Lemma 4.1 implies that the group Aut(AG/L) is G/L-transjugate. Since
F < Aut(A)G/L < Aut(AG/L), we conclude that F and (G/L)right are Aut(AG/L)-conjugate. However, AG/L is normal and hence F = (G/L)right. Therefore Aut(A)G/L is G/L-transjugate.
Now let us show that K <G Aut(A). Let H be a G-regular subgroup of Aut(A). Then HG/L is abelian transitive subgroup of Aut(A)G/L and hence HG/L is regular on G/L. Therefore HG/L ^ (G/L)right = (Gright)G/L. There exists 7 G Aut(A) such that (Hg/l)7g/l = (G/L)right = (Gright)G/L because Aut(A)G/L is G/L-transjugate. This yields that H7 < K. Thus, K ^g Aut(A).
Finally, let us prove that Aut(A) is G-transjugate. Again, let H be a G-regular subgroup of Aut(A). Since K ^g Aut(A), there exists 7 G Aut(A) such that H7 < K. The S-ring V(K, G) is a CI-S-ring by the assumption of the lemma. So K is G-transjugate by Lemma 4.1. Therefore H7 and Gright are K-conjugate and hence H and Gright are Aut(A)-conjugate. Thus, Aut(A) is G-transjugate and A is a CI-S -ring by Lemma 4.1.	□
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It should be mentioned that the proof of Lemma 4.4 is similar to the proof of [20, Lemma 3.6].
5 Generalized wreath and star products
Let A be an S-ring over a group G and S = U/L an A-section of G. An S-ring A is called the S-wreath product or the generalized wreath product of Au and AG/L if L < G and L < rad(X) for each basic set X outside U. In this case we write A = Au ls AG/L and omit S when U = L. The construction of the generalized wreath product of S-rings was introduced in [12].
The S-wreath product is called nontrivial or proper if L = {e} and U = G. An S-ring A is said to be decomposable if A is the nontrivial S-wreath product for some A-section S of G; otherwise A is said to be indecomposable. We say that an A-subgroup U < G has a gwr-complement with respect to A if there exists a nontrivial normal A-subgroup L of G such that L < U and A is the S-wreath product, where S = U/L.
Lemma 5.1 ([19, Theorem 1.1]). Let G G E, A an S-ring over G, and S = U/L an A-section of G. Suppose that A is the nontrivial S-wreath product and the S-rings Au and Ag/l are CI-S-rings. Then A is a CI-S-ring whenever
Auts (As ) = Autu (Au )S A^g/l(Ag/l )S.
In particular, A is a CI-S-ring if
Auts (As) = Autu (Au )s or Auts (As) = A^g/l(Ag/l )S.
Lemma 5.2 ([19, Proposition 4.1]). In the conditions of Lemma 5.1, suppose that As = ZS. Then A is a CI-S-ring. In particular, if U = L then A is a CI-S-ring.
Lemma 5.3 ([20, Lemma 4.2]). In the conditions of Lemma 5.1, suppose that at least one of the S-rings Au and AG/L is cyclotomic and As is Cayley minimal. Then A is a CI-S-ring.
Lemma 5.4. Let A be an S-ring over an abelian group G. Suppose that A is the nontrivial S = U/L-wreath product for some A-section S = U/L and Li is an A-subgroup containing L. Then B = V(K, G), where K = Aut(A)G/LlGr;ght, is also the S-wreath product.
Proof. Since K < Aut(A), from Equations (3.1) and (3.2) it follows that
B = V(K, G) > V(Aut(A), G) > A.
So U and L are also B-subgroups.
Let G = ZU ls Z(G/L). The S-rings Gu and CG/L are schurian and Gs is 2-minimal because Cs = ZS. So G is schurian by [26, Corollary 10.3]. This implies that
G = V(Aut(G), G).	(5.1)
Every element from Aut(C)e fixes every basic set of G and hence it fixes every L-coset. Since Li > L, every element from Aut(C)e fixes every Li-coset. We conclude that
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Aut(C)e < Aut(A)G/Ll and hence Aut(C) < K. Now from Equations (3.1) and (5.1) it follows that
e = V(Aut(e), G) > V(K, G) = B.	(5.2)
The group U is a B- and a C-subgroup. Due to Equation (5.2), every basic set of B which lies outside U is a union of some basic sets of C which lie outside U .So L < rad(X) for every X G S(B) outside U. Thus, B is the S-wreath product.	□
Lemma 5.5. In the conditions of Lemma 5.1, suppose that: (1) every S-ring over U is a CI-S-ring; (2) AG/L is 2-minimal or normal. Then A is a CI-S-ring.
Proof. Let B = V(K, G), where K = Aut(A)G/LGright. From Lemma 5.4 it follows that B is the S-wreath product. Since Li = L, the definition of B implies that BG/L = Z(G/L) and hence BS = ZS. Clearly, BG/L is a CI-S-ring. The S-ring Bv is a CI-S-ring by the assumption of the lemma. Therefore B is a CI-S-ring by Lemma 5.2. The S-ring AG/L is a CI-S-ring by the assumption of the lemma. Thus, A is a CI-S-ring by [20, Lemma 3.6] whenever AG/L is 2-minimal and by Lemma 4.4 whenever AG/L is normal.	□
Let V and W be A-subgroups. The S-ring A is called the star product of AV and AW if the following conditions hold:
(1)	V n W < W;
(2)	each T g S(A) with T C (W \ V) is a union of some V n W-cosets;
(3)	for each T g S(A) with T C G \ (V U W) there exist R G S(Av) and S G S(Aw) such that T = RS.
In this case we write A = AV * AW. The construction of the star product of S-rings was introduced in [15]. The star product is called nontrivial if V = {e} and V = G. If V n W = {e} then the star product is the usual tensor product of AV and AW (see [11, p. 5]). In this case we write A = AV ( AW. One can check that if A = AV ( AW then Aut(A) = Aut(AV) x Aut(AW). If V n W = {e} then A is the nontrivial V/(V n W)-wreath product. Indeed, let T g S(A) such that T £ V .If T C W \ V then V n W < rad(T) by Condition (2) of the definition. If T C G \ (V U W) then T = RS for some R G S(AV) and some S g S(Aw ) such that S C W \ V by Condition (3) of the definition. Since V n W < rad(S), we obtain V n W < rad(T).
Lemma 5.6. Let G G E and A a schurian S-ring over G. Suppose that A = AV * AW for some A-subgroups V and W of G and the S-rings AV and AW/(VnW) are CI-S-rings. Then A is a CI-S-ring.
Proof. The statement of the lemma follows from [18, Proposition 3.2, Theorem 4.1]. □
Lemma 5.7 ([13, Lemma 2.8]). Let A be an S-ring over an abelian group G = Gi x G2. Assume that G1 and G2 are A-groups. Then A = AGl ( AG2 whenever AGl or AG2 is the group ring.
Lemma 5.8. In the conditions of Lemma 5.1, suppose that |G : U | is a prime and there exists X G S(Ag/l) outside S with |X| = 1. Then A is a CI-S-ring.
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Proof. Let X = {x} for some x G G/L. Due to G G E, we conclude that |(x)| is prime. So |(x) n S| = 1 because x lies outside S. Since |G : U| is a prime, G/L = (x) x S. Note that A^ = Z(x). Therefore
AG/L = Z(x) <g> As
by Lemma 5.7.
Let ^ G Auts(As). Define ^ G Aut(G/L) in the following way:
= x^ = x.
Then ^ g AutG/L(AG/L) because AG/L = Z(x) <g> AS. We obtain that
AutG/i(AG/i)S > Auts(As), and hence AutG/L(AG/L)S = Auts(As). Thus, A is a Cl-S-ring by Lemma 5.1. □
6	p-S-rings
Let p be a prime. An S-ring A over a p-group G is called a p-S-ring if every basic set of A has a p-power size. Clearly, if |G| = p then A = ZG. In the next three lemmas G is a p-group and A is a p-S-ring over G.
Lemma 6.1. If B > A then B is a p-S-ring.
Proof. The statement of the lemma follows from [29, Theorem 1.1].	□
Lemma 6.2. Let S = U/L be an A-section of G. Then As is a p-S-ring.
Proof. From Lemma 2.1 it follows that for every X G S(A) the number A = |X n Lx| does not depend on x g X .So A divides |X | and hence A is a p-power. Let n: G ^ G/L be the canonical epimorphism. Note that |n(X )| = |X |/A and hence |n(X )| is a p-power. Therefore every basic set of As has a p-power size. Thus, As is a p-S-ring.	□
Lemma 6.3 ([13, Proposition 2.13]). The following statements hold:
(1)	|Oe(A)| > 1;
(2)	there exists a chain of A-subgroups {e} = G0 < Gi < • • • < Gs = G such that |Gj+i : Gj| = p for every i G {0,..., s - 1}.
Lemma 6.4. Let G be an abelian group, K G Supmin(Grlght), and A = V(K, G). Suppose that H is an A-subgroup of G such that G/H is a p-group for some prime p. Then Ag/h is a p-S-ring.
Proof. The statement of the lemma follows from [18, Lemma 5.2].	□
7	S-rings over an abelian group of non-powerful order
A number n is called powerful if p2 divides n for every prime divisor p of n. From now throughout this section G = H x P, where H is an abelian group and P = Cp, where p is a prime coprime to | H|. Clearly, |G| is non-powerful. Let A be an S-ring over G, Hi a maximal A-subgroup contained in H, and Pi the least A-subgroup containing P. Note that Hi Pi is an A-subgroup.
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Lemma 7.1 ([20, Lemma 6.3]). In the above notations, if H1 = (Hi-Pi)p/, the Hall p'-subgroup of H1P1, then AHlPl = AHl * APl.
Lemma 7.2 ([27, Proposition 15]). In the above notations, if AHlPl/Hl = ZCp then
AHi Pi = Ahi * APi.
Lemma 7.3 ([11, Lemma 6.2]). In the above notations, suppose that H1 < H. Then one of the following statements holds:
(1)	A = Ahi I AG/Hi with rk(AG/Hi) = 2;
(2)	A = Ah1p1 Is AG/Pl, where S = Hi Pi/Pi and Pi < G.
8 S-rings over CJ,1, n < 5
All S-rings over the groups C2\ where n < 5, were enumerated with the help of the GAP package COCO2P [16]. The list of all S-rings over these groups is available on the web-page [30] (see also [35]). The next lemma is an immediate consequence of the above computational results (see also [11, Theorem 1.2]).
Lemma 8.1. Every S-ring over Clf, where n < 5, is schurian.
To prove Theorem 1.1, we will show that every schurian S-ring over Cf x Cp is CI. Since the most of schurian S-rings over Cf x Cp are generalized wreath or star products of S-rings over its proper subgroups, we need to check that all schurian S-rings over proper subgroups of Cf x Cp are CI. In this section we will do it for G = C2, where n < 5. Note that G is a DCI-group by [2, 7] but this does not imply that every S-ring over G is CI (see Remark 4.3). We will describe 2-S-rings over G using computational results and check that all S-rings over G are CI. Until the end of the section G is an elementary abelian 2-group of rank n and A is a 2-S-ring over G.
Lemma 8.2. Let n < 3. Then A is cyclotomic. Moreover, A is Cayley minimal except for the case when n = 3 and A = ZC21 ZC21ZC2.
Proof. The first part of the lemma follows from [20, Lemma 5.2]; the second part follows from [20, Lemma 5.3].	□
Analyzing the lists of all S-rings over C2 and Cf available on the web-page [30], we conclude that up to isomorphism there are exactly nineteen 2-S-rings over G if n = 4 and there are exactly one hundred 2-S-rings over G if n = 5. It can can be established by inspecting the above 2-S-rings one after the other that there are exactly fifteen decomposable and four indecomposable 2-S-rings over G if n = 4 and there are exactly ninety six decomposable and four indecomposable 2-S-rings over G if n = 5.
Lemma 8.3. Let n G {4,5} and A indecomposable. Then A is normal. If in addition n = 5 then A = ZC2 (g> A', where A' is indecomposable 2-S-ring over C2.
Proof. Let n = 4. One can compute | Aut(A)| and |NAut(A)(Gright)| using the GAP package COCO2P [16]. It turns out that for each of the four indecomposable 2-S-rings over G the equality
| Aut(A)| = |NAut(A)(Gright)|
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is attained. So every indecomposable 2-S-ring over G is normal whenever n = 4.
Let n = 5. The straightforward check for each of the four indecomposable 2-S-rings over G yields that A = AH ( ZL, where H = C2, L = C2, and AH is indecomposable 2-S-ring. Clearly, ZL is normal. By the above paragraph, AH is normal. Since Aut(A) = Aut(AH) x Aut(AL), we obtain that A is normal.	□
Note that if p > 2 then Lemma 8.3 does not hold. In fact, if p > 2 then there exists an indecomposable p-S-ring over Cp which is not normal (see [13, Lemma 6.4]).
Lemma 8.4. Let n < 5. Then A is normal whenever one of the following statements holds:
(1)	A is indecomposable;
(2)	|G : Og (A)| = 2;
(3)	n = 4 and A = (ZC21 ZC2) <g> (ZC2 \ ZC2).
Proof. If n < 3 and A is indecomposable then A = ZG by [20, Lemma 5.2]. Clearly, in this case A is normal. If n G {4, 5} and A is indecomposable then A is normal by Lemma 8.3. There are exactly n — 1 2-S-rings over G for which Statement (2) of the lemma holds. For every A isomorphic to one of these 2-S-rings and for A = (ZC21ZC2) ( (ZC2 I ZC2) one can compute | Aut(A)| and |NAut(A)(Gright)| using the GAP package COCO2P [16]. It turns out that in each case the equality | Aut(A)| = |NAut(A)(Gright)| holds and hence A is normal.	□
Lemma 8.5. Let n = 4. Then A is cyclotomic.
Proof. If A is decomposable then A is cyclotomic by [20, Lemma 5.6]. If A is indecomposable then A is normal by Lemma 8.3. This implies that
Aut(A)e = (NAut(A)(Gright))e < Aut(G).
The S-ring A is schurian by Lemma 8.1. So from Equation (3.2) it follows that A =
V(Aut(A), G) and hence A = Cyc(Aut(A)e, G).	□
Lemma 8.6. Let n = 5. Suppose that A is decomposable and |Og (A)| = 8. Then A is cyclotomic.
Proof. Let A be the nontrivial S-wreath product for some A-section S = U/L. Note that |U| < 16, |G/L| < 16, and |S| < 8. The S-rings Au, AG/L, and As are 2-S-rings by Lemma 6.2. So each of these S-rings is cyclotomic by Lemma 8.2 whenever the order of the corresponding group is at most 8 and by Lemma 8.5 otherwise. Since | Og (A) | = 8, we conclude that |S| < 4 or |S| = 8 and |Og(AS) | > 4. In both cases AS is Cayley minimal by Lemma 8.2. This implies that
Autu (Au )S = AutG/i(AG/i)S = Auts (As ).
Now from [20, Lemma 4.3] it follows that A is cyclotomic.
□
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In the next two lemmas we establish some properties of decomposable 2-S-rings over G = C| whose thin radical is of size 2 or 4. These properties will be used in the proof of Theorem 1.1. The statements of Lemma 8.7 and Lemma 8.8 can be verified by analysis of computational results obtained with the help of the GAP package COCO2P [16]. For every decomposable 2-S-ring A with |Og(A) | G {2,4} over G (see the list [30]), we compute all A-subgroups, automorphism groups, and Cayley automorphism groups of some restrictions and quotients.
Lemma 8.7. Let n = 5. Suppose that A is decomposable and | Og (A) | = 4. Then one of the following statements holds:
(1)	there exists an A-subgroup L < Og (A) of order 2 such that A = ZOg (A) ls AG/L, where S = Og (A)/L;
(2)	| AutG(A)| > | Auty(Au)| for every A-subgroup U with |U| = 16 and U > Og (A);
(3)	A is normal;
(4)	there exist an A-subgroup L < Og(A) and X G S(A) such that |L| = |X| = 2, L = rad(X), and AG/L is normal.
Lemma 8.8. Let n = 5. Suppose that A is decomposable, |Og(A)| = 2, and there exists X G S(A) with |X| > 1 and | rad(X)| = 1. Then |X| = 4 and one of the following statements holds:
(1)	A = B l ZC2, where B is a 2-S-ring over C|;
(2)	| AutG(A)| > | Autu(Au)| for every A-subgroup U with |U| = 16;
(3)	there exists an A-subgroup L such that |L| G {2,4} and AG/L is normal.
Lemma 8.9. Let D G E such that every S-ring over a proper section of D is CI, D an S-ring over D, and S = U/L a D-section. Suppose that D is the nontrivial S-wreath product. Then D is a CI-S-ring whenever D/L = Ck for some k < 4 and Dd/l is a 2-S-ring.
Proof. The S-ring Dd/l is cyclotomic by Lemma 8.2 whenever |D/L| < 8 and by Lemma 8.5 whenever |D/L| = 16. The S-ring Ds is a 2-S-ring by Lemma 6.2. If Ds £ ZC2 I ZC2 l ZC2 then Ds is Cayley minimal by Lemma 8.2. The S-rings Dy and Dd/l are CI-S-rings by the assumption of the lemma. So D is a CI-S-ring by Lemma 5.3. Assume that
Ds = ZC21 ZC21 ZC2.
In this case |D/L| = 16, |S| = 8, and there exists the least Ds-subgroup A of S of order 2. Every basic set of Dd/l outside S is contained in an S-coset because D(D/L)/s = ZC2. So rad(X) is a Ds-subgroup for every X G S(Dd/l) outside S. If | rad(X)| > 1 for every X g S (Dd/l ) outside S then Dd/l is the S/A-wreath product because A is the least Ds-subgroup. This implies that D is the U/n-1(A)-wreath product, where n: D ^ D/L is the canonical epimorphism. One can see that |D/n-1(A)| < 8 and |U/n-1(A)| < 4. The S-rings DD/n-i(A) and Dy/n-i(A) are 2-S-rings by Lemma 6.2. The S-ring DD/n-i(A) is cyclotomic by Lemma 8.2 and the S-ring Dy/n-i(A) is Cayley minimal by
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Lemma 8.2. The S-rings Du and DD/n-i(A) are CI-S-rings by the assumption of the lemma. Thus, D is a Cl-S-ring by Lemma 5.3.
Suppose that there exists a basic set X of Dd/l outside S with | rad(X)| = 1. If Dd/l is decomposable then
AutD/L(DD/L)S = Auts (Ds )
by [20, Lemma 5.8]. Therefore D is a CI-S-ring by Lemma 5.1.
If Dd/l is indecomposable then Dd/l is normal by Lemma 8.3. So all conditions of Lemma 5.5 hold for D. Thus, D is a CI-S-ring.	□
Lemma 8.10. Let n < 5. Then every S-ring over G is a CI-S-ring.
Proof. Every S-ring over G is schurian by Lemma 8.1. So to prove the lemma, it is sufficient to prove that B = V(K, G) is a CI-S-ring for every K e Supmm(Gright) (see Remark 4.3). The S-ring B is a 2-S-ring by Lemma 6.4. If n < 4 then B is CI by [20, Lemma 5.7]. Thus, if n = 4 then the statement of the lemma holds.
Let n = 5. Suppose that B is indecomposable. Then the second part of Lemma 8.3 implies B = ZC2 <E>B', where B' is indecomposable 2-S-ring over C|. Since B is schurian by Lemma 8.1 and every S-ring over an elementary abelian group of rank at most 4 is CI by the above paragraph, we conclude that B is a CI-S-ring by Lemma 5.6.
Now suppose that B is decomposable, i.e. B is the nontrivial S = U/L-wreath product for some B-section S = U/L. Clearly, |G/L| < 16. The S-ring BG/L is a 2-S-ring by Lemma 6.2. Since every S-ring over an elementary abelian group of rank at most 4 is CI, B is a CI-S-ring by Lemma 8.9.	□
9 Proof of Theorem 1.1
Let G = H x P, where H = Cf and P = Cp, where p is a prime. These notations are valid until the end of the paper. If p = 2 then G is not a DCI-group by [28]. So in view of Lemma 4.2, to prove Theorem 1.1, it is sufficient to prove the following theorem.
Theorem 9.1. Let p be an odd prime and K e Supmm(Gright). Then A = V (K, G) is a CI-S-ring.
The proof of Proposition 9.1 will be given at the end of the section. We start with the next lemma concerned with proper sections of G.
Lemma 9.2. Let S be a section of G such that S = G. Then every schurian S-ring over S is a CI-S-ring.
Proof. If S = C2n for some n < 5 then we are done by Lemma 8.10. Suppose that S = C2n x Cp for some n < 4. Then the statement of the lemma follows from [20, Remark 3.4] whenever n < 3 and from [20, Remark 3.4, Theorem 7.1] whenever n = 4.	□
A key step towards the proof of Theorem 9.1 is the following lemma.
Lemma 9.3. Let A be an S-ring over G and U an A-subgroup with U > P. Suppose that P is an A-subgroup, A is the nontrivial S-wreath product, where S = U/P, |S| = 16, and Ag/p is a 2-S-ring. Then A is a CI-S-ring.
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Proof. Firstly we prove two lemmas concerned with some special cases of Lemma 9.3.
Lemma 9.4. Suppose that S has a gwr-complement with respect to AG/P. Then A is a CI-S-ring.
Proof. The condition of the lemma implies that there exists an AG/P-subgroup A such that AG/P is the nontrivial S/A-wreath product. This means that A is the nontrivial U/n-1(A)-wreath product, where n: G ^ G/P is the canonical epimorphism. Note that
|G/n-1(A)| < 16 and AG/n-i(A) = A(G/P)/A is a 2-S-ring by Lemma 6.2. Therefore A is a CI-S-ring by Lemma 9.2 and Lemma 8.9.	□
Lemma 9.5. Suppose that S does not have a gwr-complement with respect to AG/P. Then
| AutG/P(Ag/p)s| = | AutG/P(Ag/p)|. Proof. To prove the lemma it is sufficient to prove that the group
(AutG/p(Ag/p))s = [f € AutG/p(Ag/p) : fS = ids}
is trivial. Let f € (AutG/P(AG/P))s. Put C = Cyc((f), G/P). Clearly, (f) < Aut(AG/P). So from Equations (3.1) and (3.2) it follows that C > AG/P. Lemma 6.1 yields that C is a 2-S-ring. Since f s = ids, we conclude that Oe(C) > S.
If C = Z(G/P) then Oe(C) = S. Therefore C = ZS Z((G/P)/A) for some C-subgroup A by Statement (i) of [19, Proposition 4.3]. This implies that AG/P = As A((G/P)/A) because C > AG/P and S is both AG/P, C-subgroup. We obtain a contradiction with the assumption of the lemma. Thus, C = Z(G/P) and hence f is trivial. So the group (AutG/p(Ag/p))s is trivial.	□
If Ag/p is indecomposable then AG/P is normal by Lemma 8.3. So A is a CI-S-ring by Lemma 9.2 and Lemma 5.5. Further we assume that AG/P is decomposable. Due to Lemma 9.4, we may assume also that
S does not have a gwr-complement with respect to AG/P.	(9.1)
If there exists X € S(AG/P) outside S with |X | = 1 then A is a CI-S-ring by Lemma 9.2 and Lemma 5.8. So we may assume that
Oe (Ag/p) < S.	(9.2)
Note that |Oe(AG/P)| > 1 by Statement (1) of Lemma 6.3 and |Oe(AG/P)| < 16 by Equation (9.2). So |Oe(AG/P)| € [2,4, 8,16}. We divide the rest of the proof into four cases depending on | Oe(AG/P) |.
Case 1: |Oe(Ag/p)| = 16.
Due to Equation (9.2), we conclude that As = ZS. So A is a CI-S-ring by Lemma 9.2 and Lemma 5.2.
Case 2: |Oe(Ag/p)| = 8.
Since Ag/p is decomposable, Lemma 8.6 implies that AG/P is cyclotomic. The S-ring As is a 2-S-ring by Lemma 6.2. In view of Equation (9.2), we obtain that |Oe (As )| = 8. So Statement (ii) of [19, Proposition 4.3] yields that the S-ring As is Cayley minimal. Thus, A is a CI-S-ring by Lemma 9.2 and Lemma 5.3.
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Case3: |Oe(Ag/p)| = 4.
In this case one of the statements of Lemma 8.7 holds for AG/P. If Statement (1) of Lemma 8.7 holds for AG/P then we obtain a contradiction with Equation (9.1).
If Statement (2) of Lemma 8.7 holds for AG/P then | AutG/P (AG/P )| > | Auts (As )|. From Lemma 9.5 it follows that | AutG/P(AG/P)S| = | AutG/P(AG/P)| and hence
| AutG/P(Ag/p)S| > | Auts(As)|.
Since AutG/P(AG/P)s < Auts(As), we conclude that AutG/P(AG/P)s = Auts(As). Thus, A is a CI-S-ring by Lemma 9.2 and Lemma 5.1.
If Statement (3) of Lemma 8.7 holds for AG/P then AG/P is normal. In this case A is a CI-S-ring by Lemma 9.2 and Lemma 5.5.
Suppose that Statement (4) of Lemma 8.7 holds for AG/P, i.e. there exists an AG/P-subgroup A < Oe (AG/P) of order 2 and X = {x^x2} G S(AG/P) suchthat A(G/P )/A is normal and A = rad(X). Let L = n-1(A), where n: G ^ G/P is the canonical epimorphism, and B = V(N, G), where N = Aut(A)G/LGright.
Prove that B is a CI-S-ring. Lemma 5.4 implies that B is the S-wreath product. From Equations (3.1) and (3.2) it follows that B > A. So BG/P > AG/P and hence BG/P is a 2-S-ring by Lemma 6.1. We obtain that B and U satisfy the conditions of Lemma 9.3. One can see that X is a BG/P-set and
Oe(Bg/p) > Oe(Ag/p)	(9.3)
because BG/P > AG/P. The definition of B yields that every basic set of B is contained in an L-coset and hence every basic set of BG/P is contained in an A-coset. Therefore
{xi}, {x2}G S(Bg/p)	(9.4)
because X is a BG/P-set and A = rad(X). Now from Equations (9.3) and (9.4) it follows that
|Oe(Bg/p)| > 8.	(9.5)
If Bg/p is indecomposable then BG/P is normal by Lemma 8.3 and hence B is CI by Lemma 9.2 and Lemma 5.5. If S has a gwr-complement with respect to BG/P then B is CI by Lemma 9.4. If Oe(BG/P) ^ S then B is CI by Lemma 9.2 and Lemma 5.8. Suppose that none of the above conditions does not hold for B. Then, in view of Equation (9.5), B satisfies all conditions from one of the Cases 1 or 2. Therefore, B is CI.
Clearly, AG/L = A(G/P)/A and hence AG/L is normal. Also AG/L is CI by Lemma 9.2. The S-ring B is CI by the above paragraph. Thus, A is CI by Lemma 4.4.
Case 4: |Oe(Ag/p)| = 2.
Let A = Oe(Ag/p). Clearly, A is the least AG/P-subgroup. If | rad(X)| > 1 for every X G S(AG/P) outside S then A < rad(X) for every X G S(AG/P) outside S and we obtain a contradiction with Equation (9.1). So there exists X G S(Ag/p) outside S with | rad(X)| = 1. From Equation (9.2) it follows that |X | > 1. Lemma 8.8 implies that |X | =4. The number A = |X n Ax| does not depend on x G X by Lemma 2.1. If A = 2 then A < rad(X), a contradiction. Therefore
A = 1.
(9.6)
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One of the statements of Lemma 8.8 holds for AG/P. If Statement (1) of Lemma 8.8 holds for AG/P then there exists Y e S(AG/P) with |Y| = 16 and | rad(Y)| = 16. Since |S| = 16, we conclude that Y lies outside S and hence Y = (G/P) \ S. This means that S is a gwr-complement to S with respect to AG/P. However, this contradicts Equation (9.1).
If Statement (2) of Lemma 8.8 holds for AG/P then | AutG/P (AG/P) | > | AutS (AS) |. So Lemma 9.5 implies that AutG/P(AG/P)S = AutS(AS). Therefore, A is CI by Lemma 9.2 and Lemma 5.1
Suppose that Statement (3) of Lemma 8.8 holds for AG/P, i.e. there exists an Ag/p-subgroup B such that |B| e {2,4} and A(G/P)/B is normal. Let L = n-i(B), where n: G ^ G/P is the canonical epimorphism, and B = V(N, G), where N =
Aut(A)G/LG right.
We prove that B is a CI-S-ring. As in Case 3, B is the S-wreath product by Lemma 5.4 and B > A by Equations (3.1) and (3.2). So BG/P > AG/P and hence BG/P is a 2-S-ring by Lemma 6.1. Therefore B and U satisfy the conditions of Lemma 9.3.
Note that X is a BG/P-set and Equation (9.3) holds because BG/P > AG/P. By the definition of B, every basic set of B is contained in an L-coset and hence every basic set of Bg/p is contained in a B -coset. The set X isa BG/P-set with |X | =4 and | rad(X )| = 1. So there exists X1 e S(BG/P) such that
Xi C X and |Xi| e {1,2}.
If |Xi| = 1 then Xi C Oe (BG/P). If |Xi| = 2 then Xi isacosetbya BG/P-subgroup A1 of order 2. Clearly, Ai C O^(BG/P). In view of Equation (9.6), we have Ai = A. Thus, in both cases (BG/P) ^ A. Together with Equation (9.3) this implies that
O(Bg/p)|> 4.	(9.7)
If Bg/p is indecomposable then BG/P is normal by Lemma 8.3 and hence B is CI by Lemma 9.2 and Lemma 5.5. If S has a gwr-complement with respect to BG/P then B is CI byLemma9.4. If Oe (BG/P) ^ S then B is CI by Lemma 9.2 and Lemma 5.8. Suppose that none of the above conditions does not hold for B. Then, in view of Equation (9.7), B satisfies all conditions from one of the Cases 1, 2 or 3. Therefore, B is CI.
The S-ring AG/L is normal because it is isomorphic to A(G/P)/B. The S-rings AG/L and B are CI by Lemma 9.2 and the above paragraph respectively. Thus, A is CI by Lemma 4.4.
All cases were considered.	□
Proof of Theorem 9.1. Let Hi be a maximal A-subgroup contained in H and Pi the least A-subgroup containing P.
Lemma 9.6. If Hi = H then A is a CI-S-ring.
Proof. The S-ring AG/H is ap-S-ring over G/H = by Lemma 6.4. So AG/H = ZCp. Clearly, G = HPi. Therefore A = AH * APl by Lemma 7.2. Since H and Pi/(H n Pi) are proper sections of G, the S-rings AH and APl/(HnPl) are CI by Lemma 9.2. Thus, A is CI by Lemma 5.6.	□
Lemma 9.7. If Hi < H and HiP^i = G then A is a CI-S-ring.
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Proof. Since H = (H1P1 )p> = H, Lemma 7.1 implies that A = AHl * APl. The S-rings AHl and APl/(HnPl) are CI by Lemma 9.2 because H1 and P1/(H1 n P1) are proper sections of G. Therefore A is CI by Lemma 5.6.	□
In view of Lemma 9.6, we may assume that H1 < H. Then one of the statements of Lemma 7.3 holds for A. If Statement (1) of Lemma 7.3 holds for A then
A = Ahi I Ag/H1 ,
where rk(AG/Hl) = 2. If H1 is trivial then rk(A) = 2. Obviously, A is CI in this case. If H1 is nontrivial then A is CI by Lemma 9.2 and Lemma 5.2. Assume that Statement (2) of Lemma 7.3 holds for A, i.e.
A = Au is Ag/p1 ,
where U = H1P1, S = U/P^ and P1 < G. In view of Lemma 9.7, we may assume that H1P1 < G, i.e. A is the nontrivial S-wreath product. The group G/P1 is a 2-group of order at most 32 because P1 > P. Lemma 6.4 implies that AG/Pl is a 2-S-ring. If |G/P1| < 16 then A is CI by Lemma 9.2 and Lemma 8.9. So we may assume that |G/P11 = 32. Clearly, in this case
P1 = P.
In view of Statement (2) of Lemma 6.3, we may assume that
|S | = 16.
Indeed, if |S | < 16 then S is contained in an AG/P-subgroup S' of order 16 by Statement (2) of Lemma 6.3. Clearly, A = AU/ iS> AG/P, where U' = n-1(S') and n: G ^ G/P is the canonical epimorphism. Replacing S by S', we obtain the required.
Now all conditions of Lemma 9.3 hold for A and U. Thus, A is CI by Lemma 9.3. □
ORCID iD
Grigory Ryabov © https://orcid.org/0000-0001-5935-141X References
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ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 19 (2020) 297-309 https://doi.org/10.26493/1855-3974.2103.e84
(Also available at http://amc-journal.eu)
On the divisibility of binomial coefficients
Silvia Casacuberta * ©
Harvard University, 1 Oxford Street, Cambridge, MA, USA
Received 30 August 2019, accepted 16 August 2020, published online 18 November 2020
Shareshian and Woodroofe asked if for every positive integer n there exist primes p and q such that, for all integers k with 1 < k < n - 1, the binomial coefficient (k) is divisible by at least one of p or q. We give conditions under which a number n has this property and discuss a variant of this problem involving more than two primes. We prove that every positive integer n has infinitely many multiples with this property.
Keywords: Binomial coefficients, divisibility, primorials. Math. Subj. Class. (2020): 11B65, 05A10
1 Introduction
Binomial coefficients display interesting divisibility properties. Conditions under which a prime power pa divides a binomial coefficient (k) are given by Kummer's Theorem [10] and also by a generalized form of Lucas' Theorem [5, 13].
Still, there are problems involving divisibility of binomial coefficients that remain unsolved. In this article we investigate the following question, which was asked by Shareshian and Woodroofe in [16].
Question 1.1. Is it true that for every positive integer n there exist primes p and q such that, for all integers k with 1 < k < n — 1, the binomial coefficient (k) is divisible by p or q?
As in [16], we say that n satisfies Condition 1 if such primes p and q exist for n. In this article we discuss sufficient conditions under which an integer n satisfies Condition 1. In Sections 2 and 3 we prove a variation of the Sieve Lemma from [16] and use it to show that
*I am indebted to my mentor Oscar Mickelin for his guidance throughout this research and to Prof. Russ Woodroofe for correspondence and kind suggestions. This work was carried out during the Research Science Program at MIT in the summer of 2017 and was supported by the Center for Excellence in Education, the MIT Mathematics Department, and the Youth and Science Program of Fundacio Catalunya La Pedrera (Barcelona). E-mail address: scasacubertapuig@college.harvard.edu (Silvia Casacuberta)
Abstract
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
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n satisfies Condition 1 if certain inequalities hold. In Section 5 we infer that every positive integer has infinitely many multiples for which Condition 1 is satisfied.
The collection of numbers for which Condition 1 is not known to hold has asymptotic density 0 assuming the truth of Cramer's conjecture (as first shown in [16]) and includes most primorials • • • Pi, where pi,..., pi are the first i primes, namely those primorials such that (pip2 • • • pi) - 1 is not a prime.
In addition, we introduce the following variant of Condition 1:
Definition 1.2. A positive integer n satisfies the N-variation of Condition 1 if there exist N different primes pi;..., pN such that if 1 < k < n - 1 then (k) is divisible by at least one of pi,... ,pN.
For example, it follows from Kummer's Theorem or from Lucas' Theorem that a positive integer n satisfies the 1-variation of Condition 1 if and only if n is a prime power, and every integer n satisfies the m-variation of Condition 1 if n = p^1 • • • p^T where pi;..., pm are distinct primes. In Section 4 we discuss upper bounds on N so that a given n satisfies the N-variation of Condition 1.
2 An extended Sieve Lemma
Our results in this section will be based on Lucas' Theorem: Theorem 2.1 (Lucas [13]). Let p be a prime and let
n = nr pr + nr-ipr-i + • • • + nip + no k = kr pr + kr-ipr-i + • • • + kip + ko
be base p expansions of two positive integers, where 0 < ni < p and 0 < ki < p for all i, and nr = 0. Then
() - n (lij p).
By convention, a binomial coefficient (k*) is zero if ni < ki. Hence, if any of the digits of the base p expansion of n is 0 whereas the corresponding digit in the base p expansion of k is nonzero, then (k) is divisible by p. A s a particular case, if a prime power pa with a > 0 divides n and does not divide k, then (k) is divisible by p.
Observe that, if n satisfies Condition 1 with two primes p and q, then at least one of these primes has to be a divisor of n, because otherwise (i) would not be divisible by any of them. The next two results are elementary consequences of Lucas' Theorem.
Proposition 2.2. If n = pa + 1 with p a prime and a > 0, then n satisfies Condition 1 with p and any prime dividing n.
Proof. If n -1 is a prime power then the two summands in the left-hand term of the equality
n - 1	n - 1	n
k - 1 + k = k
are divisible by p by Lucas' Theorem if 2 < k < n - 2, and hence (k) is also divisible by p. If k =1 or k = n - 1, then (k) = n, so any prime factor of n divides (k).	□
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Proposition 2.3. If a positive integer n is equal to the product of two prime powers pf and p2 with a > 0, b > 0, and pi = p2, then n satisfies Condition 1 with pi and p2.
Proof. The base pi expansion of n ends with a zeroes and the base p2 expansion of n ends with b zeroes. Because a positive integer k smaller than n cannot be divisible by both pf and p2, it is not possible that k ends with a zeroes in base pi and b zeroes in base p2. Consequently, we can apply Lucas' Theorem modulo pi if pf does not divide k or modulo p2 if p2 does not divide k.	□
Proposition 2.3 generalizes as follows.
Proposition 2.4. If pi;... ,pm are distinct primes and n = pf1 • • • pm with aj > 0 for all i, then n satisfies the m-variation of Condition 1 with pi... ,pm.
Proof. If 1 < k < n - 1, then the base pj expansion of k ends with less zeroes than the base pj expansion of n for at least one prime factor pj of n.	□
The following result extends [16, Lemma 4.3]. It is the starting point of our discussion of Question 1.1 in the next sections. By symmetry, we only need to consider those values of k with k < n/2. Moreover, we may restrict our study further to those values of k that are multiples of pf, since otherwise (k) is divisible by p.
Theorem 2.5. Let n be a positive integer and suppose that pf divides n where p is a prime and a > 0. Suppose that there is a prime q with n/(d +1) < q < n/d, where d > 1, and let k n /2. Then (£) is divisible by p or q except possibly when k is a multiple ofpf belonging to one of the intervals [cq, cq + p] with P = n — dq and 0 < c < (d + 1)/2.
Proof. Since q < n/d, the number p = n — dq is positive. If k < p then k is in the interval [0, p], which is the case c = 0 in the statement of the theorem.
The assumption that n/(d +1) < q is equivalent to assuming the inequality n — dq < q, which implies that the last digit in the base q expansion of n is equal to p. Hence, if P < k < q then we may infer from Lucas' Theorem that (k) is divisible by q.
The remaining range of values of k to be considered is q < k < n/2. In this case we look at the last digit of the base q expansion of k. If this last digit is bigger than p, then (k) is again divisible by q. Thus the undecided cases are those in which the residue of k modulo q is smaller than or equal to p. This happens when cq < k < cq + p for some positive integer c, and if cq < k < n/2 then c < n/(2q) < (d + 1)/2.	□
By the Bertrand-Chebyshev Theorem [2], for every integer n > 2 there exists a prime q such that n/2 < q < n. This yields the following particular instance of Theorem 2.5, which is also a special case of [16, Lemma 4.3].
Corollary 2.6. For a positive integer n, suppose that pf divides n where p is a prime and a > 0. If q is a prime such that n/2 < q < n and n — q < pf, then n satisfies Condition 1 with p and q.
Proof. Pick d =1 in Theorem 2.5.	□
Note that, under the assumptions of Corollary 2.6, the equality n — q = pf cannot hold, since p divides n and p = q because q does not divide n. Hence there remains to study the case when n — q > pf and q is the largest prime smaller than n while pf is the largest
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prime power dividing n. In other words, Condition 1 holds for n whenever there is a prime between n — pa and n.
The sequence of integers n for which there is no prime between n — pa and n can be found in the On-Line Encyclopedia of Integer Sequences (OEIS) [17] with the reference A290203 [3]. Its first terms are the following:
126, 210, 330, 630,1144,1360, 2520, 2574, 2992, 3432, 3960, 4199,... (2.1)
Banderier's conjecture [1] claims that if pn# denotes the n-th primorial, that is,
Pn# = PlP2 • • • Pn
where p1,...,pn are the first n primes, and q is the largest prime below pn#, then either
Pn# — q =1 orp„# — q is a prime.
Proposition 2.7. If Banderier's conjecture is true, then the sequence (2.1) contains all primorials pn# such that pn# — 1 is not a prime.
Proof. If pn# — 1 is not a prime, then pn# — q is a prime according to Banderier's conjecture. Since pn# — q does not divide pn#, we infer that pn# — q is bigger than pn, which is the largest prime power dividing pn #.	□
The first primorials pn# such that pn# — 1 is not a prime are
p4 # = 210, p7 # = 510510, p8# = 9699690, p9# = 223092870.
Inspecting this list could be a strategy to seek for a counterexample for Question 1.1. The complementary list of primorials can be found in OEIS with reference A057704 [11].
For any fixed value of d, the number fi in Theorem 2.5 is smallest when q is as close as possible to n/d. For this reason, we focus our attention on the largest prime qd below n/d for various values of d. This motivates the next definition.
Definition 2.8. For positive integers n and 1 < d < n/2, let qd be the largest prime smaller than n/d and let = n — dqd. For each integer c with 0 < c < (d +1)/2, we call [cqd, cqd + ,0d] a dangerous interval.
By Theorem 2.5, if we attempt to prove that Condition 1 holds with p and qd assuming that qd > n/(d + 1) —that is, assuming that the dangerous intervals are disjoint— we only need to care about values of k that lie in a dangerous interval and are multiples of the largest power of p dividing n.
In the case d =1, the only dangerous interval below n/2 is [0, n — q1]. When d =2, we have that [0, n — 2q2] and [q2, n — q2] are dangerous intervals. Since n — q2 > n/2, the second interval may be replaced by [q2, n/2] to carry our study further, as we do in the next section.
Example 2.9. The largest prime below n = p7# = 510510 is qi = 510481 and the largest prime dividing n is p = 17. Here n — q1 = 29 and therefore (k) is divisible by 17 or 510481 for all k except for k =17.
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On the other hand, the largest prime below n/2 = 255255 is q2 = 255253. Thus p2 = n — 2q2 =4 and therefore [0,4] and [255253, 255257] are dangerous intervals. The second interval contains a multiple of 17, namely n/2. However, since
510510 = 6 • 174 + 1 • 173 + 15 • 172 + 8 • 17, 255255 = 3 • 174 + 0 • 173 + 16 • 172 +4 • 17,
we infer from Lucas' Theorem that (250210) is divisible by 17. Consequently, (k) is divisible by 17 or 255253 for all k.
3 Using the nearest prime below n/2
Nagura showed in [14] that, if m > 25, then there is a prime between m and (1 + 1/5)m. Therefore, there is a prime q such that 5n/6 < q < n when n > 30. This implies that, if n > 30 and the largest prime-power divisor pa of n satisfies pa > n/6, then there is a prime q between n — pa and n and hence Condition 1 holds for n with p and q. The following result is sharper.
Proposition 3.1. If n > 2010882 and the largest prime-power divisor pa of n satisfies pa > n/16598, then n satisfies Condition 1 with p and the nearest prime q below n.
Proof. Schoenfeld proved in [15] that for m > 2010760 there is a prime between m and (1 + 1/16597)m. Hence, if n > 2010882 and the largest prime-power divisor pa of n satisfies pa > n/16598 then there is a prime between n — pa and n, and therefore Condition 1 holds for n by Corollary 2.6.	□
The following are consequences of Nagura's and Schoenfeld's bounds.
Lemma 3.2. Let qd be the largest prime below n/d for positive integers n and d.
(a)	If n > 120 and d < 5, then n/(d +1) < qd.
(b)	If n > 3.34 • 1010 and d < 16597, then n/(d +1) < qd.
Proof. By Nagura's bound [14], if n/d > 30, then 5n/6d < qd < n/d. Therefore, n — dqd < n/6. If d < 5, then 6d < 5(d + 1) and hence
5n(d + 1)
n < 6d < qd(d +1),
as claimed. The proof of part (b) is analogous using Schoenfeld's bound [15].	□
In order to apply Theorem 2.5 with d = 2 for a given n, we need that there is a prime q such that n/3 < q < n/2. If q2 denotes the nearest prime below n/2, then the inequality n/3 < q2 holds if n > 120 by Lemma 3.2. Since by (2.1) we have that n — q1 < pa if n < 126, we may assume that n/3 < q2 without any loss of generality.
Note that the inequality n/3 < q is equivalent to n — 2q < q, so the intervals [0, n — 2q] and [q, n — q] are disjoint.
Theorem 3.3. For an odd positive integer n and a prime power pa dividing n, suppose that there is a prime q with n/3 < q < n/2 and n — 2q < pa. Then n satisfies Condition 1 with p and q.
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Proof. By Theorem 2.5, in order to infer that (k) is divisible by p or q, the only cases that we need to discuss are those values of k that are multiples of p° with k e [0, n - 2q] or k e [q, n—q]. By assumption, there are no multiples ofp° in [0, n—2q]. Since n —q > n/2, we may focus on the interval [q, n/2]. Since n is odd, n/2 is not an integer; hence we are only left to prove that there is no multiple k of p° with q < k < n/2. We will prove this by contradiction.
Thus suppose that q < Ap° < n/2 for some integer A. The assumption that n — 2q < p° implies that n — p° < 2q and hence
n/2 — p°/2 < q < Ap°.
Consequently, Ap° < n/2 < (A + 1/2)p°. If we now write n = mpa, we obtain that 2A < m < 2A +1, which is impossible for an integer m.	□
The rest of this section is devoted to the case when n is even.
Lemma 3.4. Suppose that n is even and there is a prime q with q < n/2 and n — 2q < p°, where p° is the largest power of p dividing n. If there is a multiple k of p° in the interval [q, n/2], then p is odd and k = n/2.
Proof. Suppose first that p is odd. Then the integer n/2 is a multiple of p°, so we may write n/2 = Ap° for some integer A. If there is another multiple of p° in the interval [q, n/2], then q < (A — 1)p° < n/2, and this implies that
n/2 — p° = Ap° — p° = (A — 1)p° > q.
Hence n — 2q > 2p°, which is incompatible with our assumption that n — 2q < p°.
In the case p =2 (so that 2° is the largest power of 2 dividing n), we have that n/2 is divisible by 2°-1, and we may write n/2 = A2°-1 with A odd. If there is a multiple of 2° in the interval [q, n/2), then q < < n/2, so ^ < A/2 and ^ < (A — 1)/2 because A is odd. Therefore
n/2 — 2°-1 = (A — 1)2°-1 > > q. Hence, as above, n — 2q > 2°, which contradicts that n — 2q < 2°.	□
Theorem 3.5. For an even positive integer n, suppose that there is a prime q such that n/3 < q < n/2 and n — 2q < p°, where p° is the largest power ofp dividing n.
(a)	Ifp = 2, then n satisfies Condition 1 with 2 and q.
(b)	If p = 2, then n satisfies Condition 1 with p and q if and only if (n/2) is divisible by p.
Proof. By Theorem 2.5 and Lemma 3.4, the only case left is k = n/2 for p odd. Consequently, if (n/2) is divisible by p, then n satisfies Condition 1 with p and q. Moreover, (n/2) is not divisible by q, since the base q expansions of n and n/2 are, respectively,
2 • q + (n — 2q) and 1 • q + (n/2 — q). Hence the assumption that (n/2) be divisible by p is necessary.	□
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Our last remarks in this section correspond to the case when n is even, and they are only relevant if p = 2, by Theorem 3.5. Next we give sufficient conditions to infer that a prime p divides (nJ2). The greatest integer less than or equal to a real number x is denoted by |_xj, and we write vp(n) = a if pa is the maximum power of p such that pa divides n.
Recall from [12] that
>(n!) = E
k= 1
- sp(n) p — 1 '
(3.1)
where sp(n) denotes the sum of all the digits in the base p expansion of n.
Proposition 3.6. Suppose that n is even. A prime p divides (nJ2) if and only if at least one of the numbers |_n/pr J with r > 1 is odd.
Proof. By comparing vp(n!) and vp((n/2)!) we see that, for each r,
n	=2	n/2
pr		pr
if |_n/prJ is even. If |_n/prJ is even for all r, we conclude that vp(n!) = 2vp((n/2)!), and hence p does not divide (n„2). However, if |_n/pr J is odd, then
n	=2	n/2
pr		pr
+ 1
□
□
and consequently vp(n!) is greater than 2vp((n/2)!).
Corollary 3.7. If n is even and (n — sp(n))/(p — 1) is odd, then p divides (n„2). Proof. This follows from Proposition 3.6 and Legendre's formula (3.1). Corollary 3.8. Suppose that n is even.
(a)	If any of the digits in the base p expansion of n/2 is larger than |p/2j, then p
divides („n2).
(b)	If one of the digits in the base p expansion of n is odd, then p divides (n„2).
Proof. If a digit of n/2 in base p is larger than |p/2J, then when we add n/2 to itself in base p to obtain n there is at least one carry. Similarly, if n has an odd digit in base p, then there is a carry when adding n/2 and n/2 in base p. Hence, by Kummer's Theorem [10] with k = n/2, if there is at least one carry when adding n/2 to itself in base p, then p divides („„2).	□
Corollary 3.9. Let n be an even positive integer Suppose that there is a prime q such that n/3 < q < n/2 and n — 2q < pa, where pa denotes the largest power of p dividing n. If pLi°g „ / log pi > n/2, then p divides („„2) and therefore n satisfies Condition 1 with p and q.
Proof. The largest value of r such that pr < n < pr+1 is |log n/ log pj. Therefore, in Proposition 3.6, the exponent r is bounded by |_log n/ log pj. Also note that r > a, where a is the largest exponent of p such that pa divides n. If pLlog n / logpi > n/2, then |_n/prJ = 1. Because this is odd, p divides („„ J by Proposition 3.6.	□
p
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In those cases when the inequalities n - q1 < pa and n - 2q2 < pa both fail for the largest prime power pa dividing n, a possible strategy would be to analyze the inequality n - dqd < pa for bigger values of d, where qd is the largest prime below n/d.
Up to 1,000,000 there are 88 integers that do not satisfy n - 2q2 < pa, where pa is the largest prime power dividing n. The On-Line Encyclopedia of Integer Sequences has published these numbers with the reference A290290 [4]. Among these, there are 25 that do not satisfy the inequality n - 3q3 < pa; there are 7 that do not satisfy the inequality n - 4q4 < pa either; there are 5 for which the inequality n - 5q5 < pa also fails, and there is only one integer for which the inequality n - 6q6 < pa still fails (namely, n = 875160). However, the value of n - dqd need not decrease as d grows, and the number of dangerous intervals that one needs to inspect when n - dqd < pa increases linearly with d. Therefore this strategy is not conclusive, although it often works in practice.
Example 3.10. The largest prime power dividing n = p14# = 13082761331670030 is p = 43. In this case, n - q1 = 89 and n - 2q2 = 268. Thus, Condition 1 fails for p and q1 and it also fails for p and q2. Nevertheless, n - 3q3 = 27 works, as the dangerous interval [q3, n - 2q3] contains one multiple of 43, namely n/3, and (n/3) is divisible by 43. Therefore Condition 1 holds for p = 43 and q3 = 4360920443890001.
Example 3.11. For n = 210, the inequality n - q1 < 7 fails while n - 2q2 < 7 is true. However, is not divisible by 7. Hence we look for greater values of d and find that
n - 5q5 <7 with q5 = 41. Now 42 G [41,46] and 84 G [82, 87], yet S210) and S^0) are both divisible by 7. Hence Condition 1 is satisfied with p = 7 and q5 =41.
Example 3.12. For n = 875160, the inequality n - dqd < 17 is satisfied with d =11 but not with any smaller value of d. There are 6 dangerous intervals of length n - 11q11 = 11. Each of these intervals (except the first) contains one multiple of 17, and in each case the corresponding binomial coefficient (k) happens to be divisible by 17. Therefore Condition 1 is satisfied with p = 17 and q11 = 79559.
4 On the N-variation of Condition 1
Recall from Definition 1.2 that n satisfies the N-variation of Condition 1 if there are N primes p1,... ,pN such that if 1 < k < n - 1 then is divisible by at least one of
p1,... ,Pn .
Theorem 4.1. If an even positive integer n satisfies n - 2q < pa for a prime q with n/3 < q < n/2, where pa is the largest power of p dividing n andp = 2, then n satisfies the 3-variation of Condition 1 with p, q and any prime that divides (n/2).
Proof. According to the statement of part (b) of Theorem 3.5, the only binomial coefficient (k) with 1 < k < n - 1 that might fail to be divisible by p or q is (n/2). Hence it suffices to add an extra prime with this purpose.	□
Proposition 4.2. For a positive integer n, let q1 be the largest prime smaller than n, let pa1 be the largest prime-power divisor of n and let pa2 be the second largest prime-power divisor of n. Ifp^1 p^2 > n - q1, then n satisfies the 3-variation of Condition 1 with p1, p2 and q1 .
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Proof. By Lucas' Theorem, for any k such that 1 < k < p^1, the binomial coefficient (k) is divisible by pi, and for any k such that n - q1 < k < n/2 the binomial coefficient (k) is divisible by q1. Thus we need to add a prime that divides at least the binomial coefficients (k) with p^1 < k < n - q1 in which k is a multiple of p^1. For this, we pick p2 and therefore we only need to consider those values of k that are, in addition, multiples of p^2. The least k that is a multiple of both prime powers is p^1 p^2. Therefore, if p(L1 p^2 > n - q1, then all values of k lying in the interval p^1 < k < n - q1 are such that (k) is divisible by pi or p2.	□
In the statement of Proposition 4.2, the condition that p^1 p^2 > n - q1 holds by Nagura's bound [14] if we impose instead that p^1 p^2 > n/6.
For each n, we are interested in the minimum number N of primes such that n satisfies the N-variation of Condition 1. We next discuss upper bounds for N.
Proposition 4.3. For positive integers n and d, suppose that there is a prime q such that n/(d +1) < q < n/d and a prime-power divisor pa of n such that n - dq < pa. Then n satisfies the N-variation of Condition 1 with N = 2 + |_d/2_|.
Proof. By Theorem 2.5, the binomial coefficients (k) are divisible by q except possibly if k lies in a dangerous interval. In the dangerous intervals we only need to consider those integers that are multiples of pa, since otherwise (k) is divisible by p. Since we are assuming that n - dq < pa, we know that in each dangerous interval there is at most one multiple of pa. This means that the worst case is the one in which there is a multiple of pa in every dangerous interval [cq, cq + ft] with 1 < c < |d/2|. Hence we pick one extra prime for each such interval.	□
Corollary 4.4. If 1 < d < 5 and pa > qd + where pa divides n and qd is the largest prime below n/d, and = n - dqd, then n satisfies Condition 1 with p and qd.
Proof. By Lemma 3.2, we may assume that n/(d +1) < qd. If 1 < d < 5, then |_d/2j equals 1 or 2. If |_d/2j = 1, then the assumption that pa > qd + implies that no multiple of pa falls into any dangerous interval until n/2. If |d/2j =2, then we need to check that 2pa > 2qd + in order to ensure that 2pa does not fall into the third dangerous interval. The minimum value of pa such that our assumption pa > qd + holds is qd + + 1. The next multiple of qd + + 1 is 2qd + + 2, which is greater than 2qd + and therefore 2pa does not fall into the third dangerous interval.	□
In order to refine the conclusion of Proposition 4.3, we consider the Diophantine equation
pax - qdy = J,	(4.1)
for 0 < J < = n - dqd, where pa is a prime-power divisor of a given number n and qd is the largest prime below n/d with d > 1. We keep assuming, as above, that qd > n/(d +1). We will also assume that p = qd, which guarantees that (4.1) has infinitely many solutions for each value of J. Specifically, if (x1, y1) is a particular solution for some value of J, then the general solution for this J is
x = X1 + rqd, y = y1 + rpa,
where r is any integer. In the next theorem we denote by N(J) the number of solutions (x, y) of (4.1) with x > 0 and 0 < y < |d/2j for each value of J with 0 < J < ^ Thus N(J) = 0 precisely when (4.1) has no solution (x, y) subject to these conditions.
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Theorem 4.5. For positive integers n and d, suppose that the largest prime qd below n/d satisfies qd > n/(d + 1), and let ¡3d = n — dqd. Let pa be a prime power dividing n with p = qd. Then n satisfies the N-variation of Condition 1 with
3d
N = 2 + Y N(S),
S=0
where N (S) is the number of solutions (x, y) of pax — qdy = S with x > 0 and 0 < y < |_d/2j for each value of S with 0 < S < 3d.
Proof. The number N(S) counts how many times a multiple of pa falls into a dangerous interval [cqd, cqd + 3d] at a distance S from the origin of that interval. Thus we pick an extra prime for each such case, and add two to the sum in order to account for p and qd. □
Example 4.6. The largest prime-power divisor of n = 96135 is p = 29. For d = 4 we find that q4 = 24029 and 34 = 19. Since 24029 = 17 (mod 29), the only solution (x, y) of the Diophantine equation 29x — 24029y = S with x > 0 and 0 < y < 2 is (829,1) for S = 12. Thus, N(12) = 1 and N = 3 for d = 4. In other words, the only occurrence of a multiple of 29 in a dangerous interval for d =4 is 24041 G [24029, 24048]. This example shows that the bound 2 + |d/2j given in Proposition 4.3 can be lowered.
The number N given by Theorem 4.5 is not a sharp bound. For those multiples pax of pa falling into a dangerous interval [cqd, cqd + 3d], it often happens that the corresponding binomial coefficient (p"x) is divisible by p, as in Example 4.6 or in other examples given in the previous sections. It could also be divisible by qd if d > qd. When d < qd, we have that n satisfies Condition 1 with p and qd if and only if the binomial coefficient (pnx) is divisible by p for every solution (x, y) of (4.1) with x > 0 and 0 < y < [d/2J, since n = dqd + 3d and pax = yqd + S with S < 3d < qd and y < [d/2j < d, so ( ™x) is not divisible by qd by Lucas' Theorem. Note also, for practical purposes, that (pnx) = (n/£ ) (mod p) .
5 Every number has multiples for which Condition 1 holds
We next prove that every positive integer n has infinitely many multiples for which Condition 1 holds. We are indebted to R. Woodroofe for simplifying and improving our earlier statement of this result, which was based on prime gap conjectures.
It follows from the Prime Number Theorem [7] that, given any real number e > 0, there is a prime between m and m(1 + e) for sufficiently large m. This fact can be used to prove the following:
Theorem 5.1. For every positive integer n and every prime p, the number npk satisfies Condition 1 with p and another prime, for all sufficiently large values of k.
Proof. For any prime p and any k > 0, let m = npk — pk = pk (n — 1). Then
kk
np = m + p = m 1 +
n — 1
Therefore, by the Prime Number Theorem, there is a prime between m and npk for all sufficiently large values of k. Choose the largest prime q with this property. Thus,
npk — pk < q < npk,
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307
so npk - q < pk, from which it follows, according to Corollary 2.6, that npk satisfies Condition 1 with p and q.	□
Theorem 5.2. For every positive integer n there is a number M such that ifp is any prime with p > M then np satisfies Condition 1 with p and another prime.
Proof. Given n, let e = 1/(n - 1). Choose m0 such that there is a prime between m and m(1 + e) for all m > m0, and let M = em0. If p is any prime such that p > M, then for
m = p(n — 1) we have
np = m + p = m f1 + —^ = m ( 1+--) = m(1 + e).
V m/	\ n — 1/
Therefore, by our choice of m0, there is a prime between m and np. If q is the largest prime with this property, then np — p < q < np, and consequently np satisfies Condition 1 with p and q.	□
Prime gap conjectures provide information relevant to our problem. For example, if pj denotes the ¿-th prime, then Cramer's conjecture [6] claims that there exist constants M and N such that if pj > N then
pi+1 — pi < M(logpj)2.
Proposition 5.3. Let m be the number of distinct prime factors of n. If Cramer's conjecture is true and n grows sufficiently large keeping m fixed, then n satisfies Condition 1.
Proof. If n has m distinct prime factors, then ^n < pa, where pa is the largest prime-power divisor of n. Let M and N be the constants given by Cramer's conjecture. Pick n0 such that if n > n0 then M(log n)2 < m/n. For every n such that n > n0 and N < pi < n < pi+1 (where pi denotes the i-th prime), we have
n — pi < pi+i — pi < M(logpi)2 < M(logn)2 < /n < pa,
from which it follows that n satisfies Condition 1 with p and pi.	□
We note that the argument used in the proof of Proposition 5.3 yields an alternative proof of the fact that Condition 1 holds for a set of integers of asymptotic density 1 if Cramer's conjecture holds, a result first found in [16, § 5]:
Theorem 5.4 ([16]). If Cramer's conjecture is true, then the set of numbers in the sequence (2.1) has asymptotic density zero.
Proof. Suppose that Cramer's conjecture holds with constants M and N, and denote by w(n) the number of distinct prime divisors of n. Thus n1/w(n) < pa, where pa is the largest prime-power divisor of n. According to [8, § 3.2], for every e > 0 the inequality
w(n) < (1 + e)loglogn	(5.1)
holds for all n except those of a set of asymptotic density zero. Since
n1/log log n
lim ~n-=
n^TO (log n)k
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for all k, there is an n0 such that n1/w(n) > M (log n)2 if n > n0. Now, if n is bigger than n0 and satisfies N < pi < n < pi+1, and moreover n is not in the set of integers for which (5.1) fails, then
n - pi < pi+1 - pi < M (log pi )2 < M (log n)2 < n1/w(n) < p°. Therefore, n satisfies Condition 1 with p and pi.	□
6 Multinomials
We also consider a generalization of Condition 1 to multinomials. We say that a positive integer n satisfies Condition 1 for multinomials of order m if there are primes p and q such that the multinomial coefficient
( n \	n!
\ki,k2,...,km/ ki!k2! ••• km!
is divisible by either p or q whenever k1 + • • • + km = n with 1 < ki < n — 1 for all i.
Proposition 6.1. If n satisfies Condition 1 with two primes p and q, then n satisfies Condition 1 for multinomials of any order m < n with p and q.
Proof. This follows from the equality
/ n \ = / n\/n — kA /n — k1 — kA \k1,k2,...,k^	\ k2 /V k3 y \kW '
and the fact that (kj is divisible by p or q by assumption.	□
Therefore, if Condition 1 is proven for binomial coefficients, then it automatically holds for multinomial coefficients.
ORCID iDs
Silvia Casacubert^ © https://orcid.org/0000-0001-5684-4585 References
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[2]	J. Bertrand, Memoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme, J. École Roy. Polytechnique 18 (1845), 123-140.
[3]	S. Casacuberta, Sequence A290203 in The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org.
[4]	S. Casacuberta, Sequence A290290 in The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org.
[5]	K. S. Davis and W. A. Webb, Lucas' theorem for prime powers, European J. Combin. 11 (1990), 229-233, doi:10.1016/s0195-6698(13)80122-9.
[6]	A. Granville, Harald Cramer and the distribution of prime numbers, Scand. Actuar. J. 1995 (1995), 12-28, doi:10.1080/03461238.1995.10413946.
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[7]	G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Math. 41 (1916), 119-196, doi:10.1007/ bf02422942.
[8]	G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n, Quart. J. Pure Appl. Math 48 (1917), 76-92.
[9]	A. E. Ingham, The Distribution of Prime Numbers, Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, Cambridge, 1932.
[10]	E. E. Kummer, Uber die Erganzungssatze zu den allgemeinen Reciprocitatsgesetzen, J. Reine Angew. Math 44 (1852), 93-146, doi:10.1515/crll.1852.44.93.
[11]	E. Labos, Sequence A057704 in The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org.
[12]	A.-M. Legendre, Théorie des nombres, Firmin Didot frères, Paris, 3rd edition, 1830.
[13]	E. Lucas, Theorie des fonctions numeriques simplement periodiques, Amer. J. Math. 1 (1878), 184-196, doi:10.2307/2369308.
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[15]	L. Schoenfeld, Sharper bounds for the Chebyshev functions 0(x) and ^(x). II, Math. Comp. 30 (1976), 337-360, doi:10.2307/2005976.
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ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 19 (2020) 311-324 https://doi.org/10.26493/1855-3974.2096.c9d
(Also available at http://amc-journal.eu)
Distance-balanced graphs and travelling salesman problems*
*
Matteo Cavaleri ©, Alfredo Donno f ©
Université degli Studi Niccold Cusano, Via Don Carlo Gnocchi, 3 00166 Roma, Italia
Received 26 August 2019, accepted 21 August 2020, published online 19 November 2020
For every probability p e [0,1] we define a distance-based graph property, the pTS-distance-balancedness, that in the case p = 0 coincides with the standard property of distance-balancedness, and in the case p = 1 is related to the Hamiltonian-connectedness. In analogy with the classical case, where the distance-balancedness of a graph is equivalent to the property of being self-median, we characterize the class of pTS-distance-balanced graphs in terms of their equity with respect to certain probabilistic centrality measures, inspired by the Travelling Salesman Problem. We prove that it is possible to detect this property looking at the classical distance-balancedness (and therefore looking at the classical centrality problems) of a suitable graph composition, namely the wreath product of graphs. More precisely, we characterize the distance-balancedness of a wreath product of two graphs in terms of the pTS-distance-balancedness of the factors.
Keywords: Distance-balanced graph, pTS-distance-balanced graph, total distance, wreath product of graphs.
Math. Subj. Class. (2020): 05C12, 05C38, 05C76, 90C27
1 Introduction
The investigation of distance-balanced graphs began in [13], though an explicit definition was provided later in [15, 18]. Such graphs generated a certain degree of interest also by virtue of their connection with centrality measures [2, 8] and with some well known
*We thank the anonymous referees for their valuable comments and suggestions which improved the exposition of the paper.
t Corresponding author. Partially supported by the Italian Ministry of Education, University and Research under PRIN grant No. 20154EHYW9.
E-mail addresses: matteo.cavaleri@unicusano.it (Matteo Cavaleri), alfredo.donno@unicusano.it (Alfredo Donno)
Abstract
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
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and largely studied distance-based invariants, such as the Wiener and the Szeged index [2, 14, 15, 16]. For instance, it was proven in [14] that, in the bipartite case, distance-balanced graphs maximize the Szeged index.
Throughout the paper we will denote by G = (VG, EG) a simple connected finite graph G with vertex set VG and edge set EG. We say that such a graph has order n if | VG | = n. For a pair of adjacent vertices u, v e VG (we say u ~ v in G) we define
where dG(u, v) denotes the geodesic distance in G. In other words, WG is the set of vertices of G which are closer to u than to v.
Definition 1.1. A graph G = (VG, EG) is distance-balanced if |WG | = | WG|, for every pair of adjacent vertices u, v e VG.
Cyclic graphs and complete graphs are simple examples of distance-balanced graphs. More generally, it is known that the class of distance-balanced graphs contains vertex-transitive graphs [18], which are graphs G = (VG, EG) whose group of automorphisms Aut(G) acts transitively on the vertex set. On the other hand, the Handa graph H24, introduced in [13], is an example of a non-vertex-transitive distance-balanced graph.
Recall that semisymmetric graphs are regular graphs which are edge-transitive (the group Aut(G) acts transitively on the edge set) but not vertex-transitive graphs. In particular, a semisymmetric graph is bipartite, and the two sets of the bipartition coincide with the orbits of Aut(G). As for such graphs there exists no automorphism switching two adjacent vertices, they appear as good candidates to be not distance-balanced. However, in [18] it is explicitly proven that there exist infinitely many semisymmetric graphs which are not distance-balanced, as well as infinitely many semisymmetric graphs which are distance-balanced.
In [15], the behaviour of the four classical graph compositions with respect to the distance-balanced property is investigated. More precisely, it is shown that the Cartesian product G □ H of two connected graphs is distance-balanced if and only if both G and H are distance-balanced; the lexicographic product G o H of two connected graphs is distance-balanced if and only if G is distance-balanced and H is regular; on the other hand, it is shown there, by explicit counterexamples, that the direct product G x H and the strong product G K H do not preserve the property of being distance-balanced. A generalization of the distance-balancedness, called ¿"-distance-balancedness, is studied in [19]. In [12], Cartesian and lexicographic graph products which are 2-distance-balanced are characterized.
In [3], in order to construct an algorithm that recognizes whether a given graph is distance-balanced or not, the authors establish a connection with some graph centrality measures; more precisely, they characterize the distance-balancedness of a graph in terms of its median vertices, and therefore in terms of their total distance (also known as transmission in the literature).
We denote the normalized total distance of a vertex u e VG as
which is the average of the distances of u from each vertex of G. A vertex u e VG is said to be median if dG(u) = minveVc dG(v). The graph G is said to be self-median if every vertex u e VG is median.
WUV = {z e Vg : dG(z,u) < dG(z,v)},
(1.1)
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Theorem 1.2 ([3, Theorem 3.1]). A graph G = (VG, EG) is distance-balanced if and only if it is self-median.
Using this characterization, Cabello and Luksic studied in [6] the complexity of the problem of finding the minimum number of edges that can be added to a given graph to obtain a distance-balanced graph.
According to Theorem 1.2, distance-balanced graphs are graphs where all vertices have the same relevance in some sense, but they are not necessarily indistinguishable (notice that there are even examples of distance-balanced graphs with a trivial automorphism group [17]). Therefore, distance-balanced graphs are of special interest in the study of social networks, as all people in such graphs are equal with respect to the total distance. A related measure of this equality is given by the opportunity index, which is defined as follows. Given a graph G = (VG, EG) with |VG| = 2n, and two subsets Vl and V2 of VG such that | Vl | = | V21 = n and V1 U V2 = VG (called a half-partition of G), the opportunity index of G is defined as opp(G) = max{|WVl (G) - WV2 (G)| : {Vl,V2} is a half-partition of G} where, for a given U C VG, Wv(G) denotes the sum of the distances between all pairs of vertices in U. In particular, distance-balanced graphs are characterized as those graphs whose opportunity index is zero [2].
In the present paper, aimed at generalizing distance-balancedness in a probabilistic direction, we start exactly from this point of view, and we interpret the set of median vertices of a graph, and the whole class of distance-balanced graphs itself, as solutions of particular facility location problems, very typical in graph centrality investigations. In order to deeper understand this correspondence, let us suppose that G = (VG, EG) represents a city; its vertex set is the set of the buildings/locations, the edges are connections between the buildings and then, for any u, v e VG, the geodesic distance dG(u, v) represents the distance between buildings u and v, or the cost of reaching the vertex v from the vertex u. In these terms, the quantity dG(u) is the average distance of the location u from all locations, and the median vertices are those vertices solving the following problem.
Problem 1.3. Find the location for a facility in order to minimize its average distance from all the buildings of the city.
Consequently, distance-balanced graphs are those graphs whose vertices are all equal with respect to Problem 1.3. That is, they solve this second problem.
Problem 1.4. Find a city where Problem 1.3 is solved by any location.
From another point of view, that we will develop in the last part of the paper, our work can be interpreted as the investigation of the distance-balancedness in a wreath product of graphs. In this sense, it is the natural continuation of [8]. The wreath product of graphs represents the graph analogue of the classical wreath product of groups, as it is true that the wreath product of the Cayley graphs of two finite groups is the Cayley graph of the wreath product of the groups. In [10], this correspondence is proven in the more general context of generalized wreath products of graphs, inspired by the construction introduced in [1] for permutation groups. Also, observe that in [9] the wreath product of matrices has been defined, in order to describe the adjacency matrix of the wreath product of two graphs: spectral computations using this matrix representation have been developed for some infinite families of wreath products in [5, 4, 11].
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The paper is organized as follows. In Section 2, we consider two optimization problems, namely Problem 2.5 and Problem 2.6, that are the analogue, respectively, of Problem 1.3 and Problem 1.4, where the centrality measure at the vertex u is not yet the normalized total distance, but the quantity dpG (u), that is, the expectation of the length of a shortest path starting from u that satisfies some random requirements depending on the probability p. In particular, these new problems collapse to the classical ones in the case p = 0.
Problems 2.5 and 2.6 are of some interest on their own, given their connection with the Travelling Salesman Problem, which is among the most studied optimization problems, largely investigated in literature also in its several probabilistic versions.
Then we extend the classical definition of distance-balanced graph by introducing the notion of pTS-distance-balanced graph in Definition 2.9, and we prove in Theorem 2.10 a pTS analogue of Theorem 1.2: pTS-distance-balanced graphs are exactly the graphs that solve Problem 2.6 (that is, the TS-version of Problem 1.4). We present examples and non-examples of pTS-distance-balanced graphs.
In Section 3, we recall the definition of the wreath product Gl H of two graphs G and H (Definition 3.1). It turns out that, when the order of H is m, the uniform probability distribution on the vertices of G l H is compatible, in a precise sense explained in Lemma 3.3, with the model introduced in Section 2 for G, when p = m-1.
It follows that the TS-problems considered on the graph G are equivalent to the classical problems on the wreath product G l H, for a suitable choice of the graph H. More precisely we characterize, in Theorem 3.4, the distance-balancedness of a wreath product in terms of pTS-distance-balancedness of its factors. Finally, we investigate the class of graphs that are pTS-distance-balanced for every p g [0,1], giving several equivalent characterizations in Theorem 3.11. We conclude the paper by asking if this class actually coincides with the class of vertex-transitive graphs (Question 4.1).
2 pTS centrality
As a natural generalization of Problem 1.3, suppose that every day each building (vertex) of the city (graph) G = (VG, EG), independently, with the same probability p g [0,1], requires a visit from the facility and with probability 1 - p does not. An example could be a postoffice with a postman delivering parcels. We want to find a location for the postoffice in order to minimize the expectation of the length of a shortest walk starting from the postoffice, visiting at least once each building waiting for a parcel, and finally arriving at the postman's house, that can be on each vertex with the same probability n (observe that the postoffice and the postman's house locations may coincide). This set-up is justified if, for example, we have to decide the postoffice location prior to be aware of the location of the postman's house, or for example if every day the postman can be different. We are going to formalize this model in what follows.
Definition 2.1. Let G = (VG, EG) be a graph and let A C VG. We define a map pA on VG x VG such that, for any pair of vertices u and v in VG, the number pA(u, v) is the length of a shortest walk joining u and v, visiting at least once all vertices in A.
Remark 2.2. Let G = (VG,EG) be a graph of order n, and let u,v,z g Vg and A C VG. We list some properties of the map pA; see [7] for more details.
•	p0(u, v) = dG(u, v).
•	pA(u, v) = pA(v,u). (Symmetry)
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•	PAuB(u,v) < pa(u,z)+ pb(z,v). (Triangle inequality)
•	B C A pB(u,v) < pA(u,v). (Monotonicity)
•	pa(u,v) < n2.
Combining the first with the third property we have
Ipa(u,z) — pa(v,z)l< dG(u,v).	(2.1)
Let G = (VG, Eg) be a graph of order n, and let u,v G VG. A Hamiltonian path from u to v in G is a path from u to v visiting each vertex of G exactly once. A Hamiltonian cycle is a Hamiltonian path between adjacent vertices u and v. A graph is Hamiltonian if it admits a Hamiltonian cycle, that is equivalent to say that pva (u, u) = n for some, or equivalently, for every u G VG. A graph G is Hamilton-connected if, for every pair u,v G VG, there exists a Hamiltonian path from u to v. It is easy to observe that
{n_1 if u == v
^^ G is Hamilton-connected. (2.2)
n if u = v
The computation of pVa for Hamilton-connected graphs is rather easy; however, to determine pA is in general very hard. This is not the case for the easiest example of Hamilton-connected graph, that is, the complete graph Kn.
Example 2.3. Let Kn = (VKn ,EKn) be the complete graph on n vertices. For every nonempty A C VKn and every u,v G VKn we have
|A| + 1	ifu,v G A
| A|	if u G A, v G A or viceversa
pA(u, v) = | A| — 1	if u,v G A,u = v
|A|	if u = v G A, |A| > 1
0	if u = v G A, |A| = 1.
The hypothesis that each vertex independently with probability p requires a visit implies that the probability for a given subset A C VG to be the random subset waiting for the parcels is
PA := pA(1 — p)n-IA.	(2.3)
Then we define the quantity dGG(u), that is, the expected length of a walk from u, visiting the random set A and arriving to the random vertex v (uniformly distributed on VG), as follows:
dG(u) := " E E PA pA(u,v).	(2.4)
n
veva acvg
Remark 2.4.
If p = 0 we have pA ^i1 if A ^ and dG(u) = dG (u).
0 otherwise G
If p = 1 we have pa = and d^u) = ^ Eveva Eacvg pa(u, v).
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If p =1 we have pa = <| J	and dpG (u) = i Y,veva PVg v).
We are now in position to formulate the pTS versions of Problem 1.3 and Problem 1.4, respectively.
Problem 2.5. Find a vertex u G VG such that dGG(u) = minveVa dGG(v).
Problem 2.6. Find a graph such that Problem 2.5 is solved by any vertex.
This leads us to introduce a notion of medianity in this setting, as a solution of the above mentioned problems.
Definition 2.7. In a graph G = (VG,EG) a vertex u G VG is pTS-median if it solves Problem 2.5. The graph G is self-pTS-median if it solves Problem 2.6.
Remark 2.8. Notice that, as a consequence of Remark 2.4, when p = 0 the Problem 1.3 and Problem 1.4 and their pTS versions, Problem 2.5 and Problem 2.6 respectively, are equivalent.
In analogy with Equation (1.1), for any subset A C VG and any pair of adjacent vertices u, v g VG, we define the vertex subsets
WUV := {z G Vg : pa(z,u) < pa(z,v)},
and the expectation of their cardinality is
<v := E PaWav |.	(2.5)
AC Vg
A natural generalization of the distance-balancedness is given in the following definition.
Definition 2.9. A graph G = (VG,EG) is pTS-distance-balanced if wppv = wppu, for every pair of adjacent vertices u,v G VG. A graph G is TS-distance-balanced if it is pTS-distance-balanced for each p g [0,1].
The following is the pTS-version of Theorem 1.2.
Theorem 2.10. A graph G = (VG, EG) is pTS-distance-balanced if and only if it is self-pTS-median.
Proof. Observe that, as the graph G = (VG, EG) is connected, the statement is proved if we show that, for every pair of adjacent vertices u,v g VG, one has:
dPG(u) - dG (v) = 1(wVu - wUvV ).	(2.6)
Now, by the definition of dGG(u) in Equation (2.4), we get
dPG(u) - dG(v) = - EE P a(P A(u, z) - PA(v, z))
n zeVa ACVq
= - E Pa E (Pa(U,Z) - Pa(V,Z)) = - E Pa(iWvaj-iWavi)
ACVq zeVa	ACVq
M. Cavaleri and A. Donno: Distance-balanced graphs and travelling salesman problems 317
since, being u and v adjacent, by virtue of Equation (2.1), we have
f1	ifz e WA
pA(u,z) - pA(v,z)= <| -1	if z e WUV
[ 0	otherwise.
Finally, by Equation (2.5), we have
dG(u) - dG(v) = n E pa(|w„au| - |wA|) = n«„ - wpv),
ACVG
that proves Equation (2.6).	□
Remark 2.11. As we have already observed,
G is distance-balanced ^^ G is oTS-distance-balanced.
Moreover, if, for two given vertices u, v e VG, there exists ^ e Aut(G) such that y(u) = v, one has d^(u) = d^(v), for every p e [0,1]. This implies
G is vertex-transitive G is TS-distance-balanced.
On the other hand, when p = 1, Hamilton-connected graphs satisfy d^(u) = n - 1 + n by Equation (2.2) and Remark 2.4 for every vertex u e VG, and then:
G is Hamilton-connected G is iTS-distance-balanced.
Notice that the converse of the last implication is not true, since there exist graphs which are vertex-transitive but not Hamilton-connected (e.g., cyclic graphs).
Example 2.12. The graph W7 depicted in Figure 1 is the Wheel graph on 7 vertices. Being Hamilton-connected, the graph W7 is iTS-distance-balanced. Clearly, it is not distance-balanced and then it is not oTS-distance-balanced.
By an explicit computation (brute-force) we computed dW?(u) = 727 • 3842, whether u is the central vertex or it belongs to the external cycle. As a consequence, the graph W7 is 1 TS-distance-balanced. We found quite surprising that this graph, that has a so recognizable central vertex, presents such an equality property.
Figure 1: The Wheel graph W7 on 7 vertices.
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Example 2.13. Let H9 be the graph on 9 vertices depicted in Figure 2. This graph has been introduced in [14] as the smallest example of a non-regular distance-balanced graph. In particular, it is oTS-distance-balanced, but an explicit computation gives djj2^) = • 26688 and dH(v2) = g^ • 26656. By virtue of Theorem 2.10, it is not 2TS-
distance-balanced.
Figure 2: The graph H9.
3 pTS-distance-balancedness and wreath product of graphs
We start this section by recalling the definition of the wreath product of two graphs.
Definition 3.1. Let G = (VG,EG) and H = (VH,EH) be two graphs. Let us fix an enumeration of the vertices of G so that VG = {x1,..., xn}. The wreath product G I H is the graph with vertex set V^0 x VG = {(yi,... ,y„)xj : xi e VG,y e Vh , j G {1,..., n}}, where two vertices u = (y1,..., yn)xi and v = (y',..., y'n)xk are connected by an edge if:
•	(edges of type I) either i = k and yj = yj for every j = i, and yi ~ y' in H;
•	(edges of type II) or yj = yj, for every j e {1,..., n}, and xi ~ xk in G.
It follows that if |VG| = n and |VH | = m, the graph G I H has nmn vertices. It is proved that G I H is connected, bipartite or vertex-transitive, if and only if both G and H are, respectively, connected, bipartite or vertex-transitive [7]. Moreover, if G is a regular graph of degree rG and H is a regular graph of degree rH, then the graph G I H is an (rG + rH)-regular graph.
There exists a practical and clarifying interpretation of the graph wreath product, given by the Lamplighter random walk [20]. Suppose that a lamplighter moves along G, so that each vertex of G represents a possible position of the lamplighter: at each vertex of G, there is a lamp. The vertices of the graph H represent the possible colors of each lamp. Therefore, a vertex (y1,..., yn)xi in G I H can be regarded as a configuration of colors (y1,..., yn) (each yj is from VH) together with a particular position xi (from VG) of the lamplighter: the lamp at the position xj e VG has the color yj e VH . Two vertices of G l H are adjacent if either they share the same configuration of colors and have adjacent positions for the lamplighter in G (such an edge of type II corresponds to the situation of the lamplighter moving to a neighbor vertex in G but leaving all lamps unchanged); or they share the same position but the configuration of colors differ, by two adjacent colors,
M. Cavaleri and A. Donno: Distance-balanced graphs and travelling salesman problems 319
exactly for the lamp associated at that position (such an edge of type I corresponds to the situation of the lamplighter changing the color of the lamp, to an adjacent color in H, in the position where he stays). For this reason, the wreath product G l H is also called the Lamplighter graph, with base graph G and color graph H.
The Lamplighter interpretation allows us to highlight the relationship between the wreath product of graphs and Problems 2.5 and 2.6. We explicit this connection in the two following lemmas, where the distance and the normalized total distance in G l H are expressed in terms of distance and normalized total distance in H, and of their TS-version in G.
Lemma 3.2 ([7]). Let G = (VG, EG) and H = (VH, EH) be two graphs of order n and m, respectively. For any vertices u = (yi,..., yn)x, v = (yj ,...,y^ )x' G VGiH , we have:
n
doiH (uv) = pa(„,v)(x,x/) + dH (yi,yi ^
i=i
where J(u, v) := {xi G VG : yi = y/}.
Observe that the sum J2n=i dH (yi, yi) can be interpreted as the geodesic distance in the n-th iterated Cartesian product of H with itself. Moreover, in the Lamplighter interpretation, the subset J(u, v) consists of those vertices xi of the base graph G where the color of the associated lamps yi and yi does not coincide. In other words, J(u, v) is the set of the vertices of G that the lamplighter has to visit in order to move from the lamp configuration of u to that of v.
Notice that fixing a vertex u in the graph G l H and considering a random vertex v = (y/,..., y/n)x/, with uniform probability "j in VGiH, induces a random subset J(u, v) of VG and a random vertex x/ in VG. More precisely, for a given subset A C VG, the probability that the random set J(u, v) is equal to A is exactly n(m - 1)|A|/nmn, because the Lamplighter may be in any position, and there are |A| vertices where the lamps may take m -1 distinct colors, whereas the colors of the lamps at the remaining n — | A| vertices are fixed. This probability is exactly pA of Equation (2.3) for p = m-1. This means that the model considered in Section 2 can be simply derived by assigning a uniform probability distribution to the vertices of the wreath product. This is formally proved in Lemma 3.3.
Lemma 3.3. Let G = (VG, EG) and H = (VH, EH) be two graphs of order n and m, respectively. Let u = (y1,... yn)x G VG/H andp = m-1. Then:
n
dGH (u) = dG(x) + dH (yi).
i=1
Proof. From Theorem 17 in [8], where the focus is on the non-normalized total distance, we have:
n
nmndc)H(u) = nm"'^ mdH(yi) + (m — 1)|A| ^^ pa(x,x/).
i=1	ACVa	x'eVa
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Taking into account that p = m-1, we obtain:
n	1	(m - 1)IAI
dom(u) = Y} dH (yi) + - Y y2 -1-PA(x,x0
z—'	n z—' z—' mn
i=1	ACVc x'eVG
n1
= Xdm(yi) + n X ]T p|A|(1 -p)n-|A|pA(x,x/). i=1	ACVg x'eVa
The claim follows by using Equations (2.3) and (2.4).	□
In the following theorem we give necessary and sufficient conditions for a vertex of a wreath product to be median and for the wreath product itself to be distance-balanced.
Theorem 3.4. Let G = (VO,EO) and H = (VH, EH) be two graphs with |Vo| = n, |VH| = m. Let u = (y1,..., yn)x G VO)H, andp = m-1. Then u is median in G i H if and only if y1,..., yn are median in H and x is pTS-median in G. In particular:
G is pTS-distance-balanced	„ TT . ,. , , ,
„ . ,. , , , -<=>■ G l H is distance-balanced . H is distance-balanced
Proof. Suppose that yi* is not median in H, so that there exists y G VH such that dH (y) < dm(yi*). Denoting by u/ the vertex (yi,..., yi*_1, y, yi» + i,..., yn)x G VG^h, by Lemma 3.3 we have do>H(u/) < do>H(u), and then u is not median in Gi H.
Similarly, suppose that x is not pTS-median in G, so that there exists X G Vo such that dO(x) < dO(x). Denoting by u// the vertex (y1,..., yn)x G VGiH, by Lemma 3.3 again we have dO>H (u//) < dO>H(u), and then u is not median in Gi H.
Viceversa, suppose that u is not median in GiH and then dO>H (u) is not minimal, then one among {dH(yi)}i=1,...,n or dO(x) cannot be minimal, and the statement follows. □
Corollary 3.5. If H and H/ are two distance-balanced graphs of the same order, then Gi H is distance-balanced if and only if Gi H/ is distance-balanced.
Remark 3.6. Another consequence of Theorem 3.4 is the equivalence of the TS-problems for G with the classical problems for G i H, where H is any distance-balanced graph. More precisely, let H = (VH, EH) be a distance-balanced graph of order m, and p = m-1. Then we have:
x G Vo	^^	(y1,...,yn)x G Vo;h
is solution of Problem 2.5	is solution of Problem 1.3
G	G i H
is solution of Problem 2.6	is solution of Problem 1.4.
Example 3.7. We know from Example 2.12 that the graph W7 is 2 TS-distance-balanced. By virtue of Theorem 3.4, the graph W7 i K2 is distance-balanced. Moreover, the graph W7 i K2 has order 896, it is non-regular (since W7 is non-regular), and it is not bipartite (since W7 is not bipartite).
Example 3.8. We know from Example 2.13 that the distance-balanced graph H9 is not 2TS-distance-balanced. As a consequence, the graph H9 i K2 is not distance-balanced.
M. Cavaleri and A. Donno: Distance-balanced graphs and travelling salesman problems 321
In the light of Example 2.12 and Example 3.7, we deduce that the distance-balancedness of the wreath product Gl H does not imply the distance-balancedness of the first factor G. Moreover, in the light of Example 2.13 and Example 3.8, we deduce that the distance-balancedness of the graphs G and H does not imply the distance-balancedness of their wreath product.
We conclude this section by investigating the class of graphs G such that G l H is distance-balanced whenever H is distance-balanced. By virtue of Theorem 3.4, this class must contain the class of TS-distance-balanced graphs. The two classes actually coincide, as we will prove in Theorem 3.11. We need a preliminary definition and lemma.
Definition 3.9. The total distance vector of the vertex u G VG is the (n + 1)-vector
Wp(u, G) = (Wpo (u, G), WP1 (u, G),..., WPn (u, G)), where, for each k G {0,1,..., n}, we set
Wpfc(u,G):= 53 53 PA(u,v).
ACVa,\A\ = k veVa
In particular, observe that Wp0 (u, G) is the non-normalized total distance of the vertex u in G.	0
Lemma 3.10. For every u G VG and for every p G [0,1], we have:
1 n
dG(u) = n Epk(1 - p)n-kWpk(u, G).	(3.1)
k=0
Proof. The claim follows by combining Equation (2.4) with Definition 3.9, since the expression of pA in Equation (2.3) only depends on the cardinality of A.	□
Theorem 3.11. Let G = (VG, EG) be a graph of order n. The following are equivalent.
(1)	G is TS-distance-balanced;
(2)	G l H is distance-balanced for every distance-balanced graph H;
(3)	G l Kn32n is distance-balanced;
(4)	G is pTS-distance-balancedfor more than n distinct values ofp G [0,1];
(5)	the total distance vector Wp(u, G) does not depend on the particular vertex u G VG.
Proof. (1) (2): It is a consequence of Theorem 3.4.
(2) (4): If G l H is distance-balanced for every distance-balanced graph H, in particular G l Km is distance-balanced for m = 2,..., n + 2, and then, by Theorem 3.4, the graph G is pTS-distance-balanced for eachp g {2, |,..., n+i}.
(4) (5): For a given vertex u G VG, we define the following polynomial of degree n in the variable x
n
Pu(x) := ^xk(1 - x)n-kWpfc(u, G).
k=0
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By Lemma 3.10, we have Pu(p) = ndG(u). Combining with hypothesis (4), for any pair u, v e VG, the polynomials Pu and Pv share more than n evaluations, and so Pu = Pv. It is an easy exercise to prove that this implies that WPk (u, G) = Wpk (v, G), for each k e {0,1,..., n}, and so Wp(u, G) = Wp(v, G).
(5) (1): It is a consequence of Lemma 3.10 and Theorem 2.10.
(2)	(3): It is true since the graph Kn32n is distance-balanced.
(3)	(5): As we already observed in Remark 2.2, for every A C VG, for every u, v e VG we have pA(u, v) < n2. Since the number of subsets of VG having cardinality k is clearly less than 2n, this implies
0 < Wpfc (u, G) = £ £ pA(u, v) < 2nn3.	(3.2)
ACVg, |A| = kv£VG
We set m := 2nn3 and p := m-1. Since, by hypothesis, G I Km is distance-balanced, it follows that, for every u, v e VG:
dG(u)nmn = dG(v)nmn,
and then, by Lemma 3.10:
n	n
£(m - 1)kWpfc(u, G) = ^(m - 1)kWpfc (v, G).	(3.3)
k=0 k=0
By virtue of Equation (3.2) we can regard the quantities Wpk (u, G) (resp. Wpk (v, G)) as the digits of dG(u)nmn (resp. dG(v)nmn) in base (m - 1); therefore, Equation (3.3) implies that Wpk (u, G) = Wpk (v, G), for each k e {0,1,..., n}, and so Wp(u, G) = Wp(v, G).	□
Example 3.12. Lemma 3.10 and the characterization (5) in Theorem 3.11 make us able to investigate distance-balancedness (at least in those cases for which the total distance vector is known) simply by studying roots of polynomials. For the graph H9 from Example 2.13, we first computed the total distance vectors, which are given by
Wp(vi, H9) = (14, 252,1345, 3711, 6279, 6941, 5065, 2363, 641, 77) Wp(v2, H9) = (14, 252,1360, 3762, 6333, 6933, 5001, 2307, 620, 74).
By using Equation (3.1) we were able to determine dG(v1) and dpG(v2) for a general p. In particular, the graph H9 is pTS-distance-balanced exactly for all values of p e [0,1] satisfying the equation dG(v1) = dG(v2). It turns out that these values are p = 0 and p « 0.48219, which is the unique real root of the polynomial 2x5 — 13x4 + 38x3 — 63x2 + 54x - 15.
As we already observed in Remark 2.11, vertex-transitive graphs satisfy the equivalent properties of Theorem 3.11. Moreover, we recall that regularity, vertex-transitivity and bipartiteness are all properties preserved by the wreath product. This yields the following infinite families of examples.
Example 3.13. Let H9 be the non-regular non-bipartite distance-balanced graph from Example 2.13. Let H24 be the Handa graph which is non-regular, bipartite and distance-balanced [13]. Consider the Generalized Petersen Graph P(7, 3), that is regular, distance-balanced but not vertex-transitive [15]. Then:
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•	{Kn i H9}neN is a family of non-regular, non-bipartite, distance-balanced graphs;
•	{Kn i P (T, 3)}neN is a family of regular, non-vertex transitive, distance-balanced graphs;
•	{Kn,n i H24}neN is a family of non-regular, bipartite, distance-balanced graphs.
It is clear that, in order to obtain other infinite families with the same properties, one can replace Kn n or Kn with any (bipartite or not) vertex-transitive graph, and the second factor with any distance-balanced graph sharing the same properties of regularity, vertex-transitivity, bipartiteness.
4 Conclusions
Vertex-transitive graphs are TS-distance-balanced. More generally, if u and v are vertices of a graph G = (VG, EG) for which there exists p G Aut(G) such that p(u) = v, one has Wp(u, G) = Wp(v, G). In other words, the total distance vector is constant on the orbits of VG under the action of Aut(G). This suggests that it is possible to use it as an invariant in order to distinguish vertices: it is a finer invariant than the standard total distance (see Example 3.12). Therefore, it is natural to ask whether this invariant is complete on the orbit partition of vertices. The following question is a total-distance-analogue of Question 1 in [7] about the Wiener vector and the isomorphism problem.
Question 4.1. Does there exist a graph G = (VG, EG) with two vertices u, v G VG for which there exists no automorphism p such that p(u) = v, but Wp(u, G) = Wp(v, G)?
A negative answer would imply that the equivalent conditions of Theorem 3.11 are also equivalent to the vertex-transitivity property.
A last remark is that, regardless of the answer to our Question 4.1, the wreath product construction produces new infinite families of distance-balanced graphs, which cannot be obtained via the classical graph products. Moreover, the graphs in these families possibly inherit good properties from their factors (see Example 3.13). We believe that this new approach may provide different examples and counterexamples in the field of distance-balancedness and its generalizations, and for this reason it deserves to be further investigated and exploited.
ORCID iDs
Matteo Cavaleri © https://orcid.org/0000-0002-6711-8316 Alfredo Donno © https://orcid.org/0000-0002-8523-1881
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ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 19 (2020) 325-335 https://doi.org/10.26493/1855-3974.2122.1e2
(Also available at http://amc-journal.eu)
On generalized truncations of complete graphs*
XueWang©, Fu-Gang Yin®, Jin-Xin Zhou f©
Mathematics, Beijing Jiaotong University, Beijing, 100044, P.R. China Received 21 September 2019, accepted 23 August 2020, published online 20 November 2020
For a k-regular graph r and a graph Y of order k, a generalized truncation of r by Y is constructed by replacing each vertex of r with a copy of Y. E. Eiben, R. Jajcay and P. Sparl introduced a method for constructing vertex-transitive generalized truncations. For convenience, we call a graph obtained by using Eiben et al.'s method a special generalized truncation. In their paper, Eiben et al. proposed a problem to classify special generalized truncations of a complete graph Kn by a cycle of length n — 1. In this paper, we completely solve this problem by demonstrating that with the exception of n = 6, every special generalized truncation of a complete graph Kn by a cycle of length n — 1 is a Cayley graph of AGL(1, n) where n is a prime power. Moreover, the full automorphism groups of all these graphs and the isomorphisms among them are determined.
Keywords: Truncation, vertex-transitive, Cayley graph, automorphism group. Math. Subj. Class. (2020): 05C25, 20B25
1 Introduction
In [6], the symmetry properties of graphs constructed by using the generalized truncations was investigated. In particular, a method for constructing vertex-transitive generalized truncations was proposed (see [6, Construction 4.1 and Theorem 5.1]), and this method was used to construct vertex-transitive generalized truncations of a complete graph Kn by a cycle of length n — 1 for some small values of n. The vertex-transitive generalized truncations of a complete graph Kn by a graph Y in context of [6, Theorem 5.1] can be defined as follows.
Let Kn be a complete graph of order n with n > 4, and let V (Kn) = [v\, v2,..., vn}. Let G be an arc-transitive group of automorphisms of Kn. Then G acts 2-transitively
* The authors also thank the anonymous referee for the valuable comments and suggestions. This work was partially supported by the National Natural Science Foundation of China (grant numbers 11671030 and 12071023).
1 Corresponding author.
E-mail addresses: xuewang@bjtu.edu.cn (XueWang), 18118010@bjtu.edu.cn (Fu-Gang Yin), jxzhou@bjtu.edu.cn (Jin-Xin Zhou)
Abstract
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
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on V(Kn). Let v = vi, and let be a union of orbits of the stabilizer Gv acting on
{{x, y} | x = y, x, y € V(Kn) \ {v}}. Let T be the graph with vertex set {v2, v3,..., vn} and edge set . For each u € V(Kn), let Vu = {(u,w) | w € V(Kn) \ {u}}. The special generalized truncation of Kn by T, denoted by T(Kn, G, T), is the graph with the vertex set |JueVVu, and the adjacency relation in which a vertex (u, w) is adjacent to the vertex (w, u) and to all the vertices (u, w') for which there exists a g € G with the property ug = v and {w, w'}g € .
Based on the analysis of special generalized truncations of a complete graph Kn by a cycle of length n - 1 for some small values of n, the authors of [6] proposed the following problem.
Problem 1.1 ([6, Problem 5.4]). Classify the special generalized truncations of Kn (n > 4) by a cycle of length n - 1 .
The main purpose of this paper is to give a solution of this problem. Before stating the main result of this paper, we first set some notation. For a positive integer n, we denote by Zn the cyclic group of order n, and by D2n the dihedral group of order 2n. Let Z^ be the multiplicative group of units mod n (Z£ consists of all positive integers less than n and coprime to n). Also we use An and Sn respectively to denote the alternating and symmetric groups of degree n. For two groups M and N, N xi M denotes a semidirect product of N by M. For a group G, the automorphism group of G and the socle of G will be represented by Aut(G) and soc(G), respectively. For a graph r we denote by V(r), E(r), A(r) and Aut(r) the vertex set, edge set, arc set and full automorphism group of r, respectively. A graph r is said to be vertex-transitive (resp. arc-transitive (or symmetric)) if Aut(r) acts transitively on V(r) (resp. A(r)). Cayley graphs form an important class of vertex-transitive graphs. Given a finite group G and an inverse closed subset S C G \ {1}, the Cayley graph Cay(G, S) on G with respect to S is the graph with vertex set G and edge set {{g, sg} | g € G, s € S}. Finally, we use Kn and Cn respectively to denote the complete graph and cycle with n vertices.
Let p be a prime and e a positive integer. Let GF(pe) be the Galois field of order pe and let x be a primitive root of GF(pe). Then
AGL(1,pe) = {aXi: z ^ zx! + z', Vz € GF(pe) | i € Zpe_1,z' € GF(pe)},
and AGL(1,pe) is a 2-transitive permutation group on GF(pe). Let
H = {a1jZ/: z ^ z + z', Vz € GF(pe) | z' € GF(pe)}, K = 0 : z ^ zx®, Vz € GF(pe) | i € Zpe_1}.
Then H is regular on GF(pe) and the point stabilizer AGL(1,pe)0 of the zero element 0 of GF(pe) is K. So AGL(1,pe) = H x K.
Construction 1.2. Let z' be a non-zero element of GF(pe). For each i € Zpe_1 with i < £e_1, let
Kpe = Cay(AGL(1,pe), {a_1,z', «^,0, «x-,o}) (p > 2), K2e = Cay(AGL(1, 2e), {aM,, aai,o, aa-i,o}) (p = 2).
X. Wang, F.-G. Yin and J.-X. Zhou: On generalized truncations of complete graphs 327
Remark 1.3. Let z', z" be two non-zero elements of GF(pe). There exists xj e GF(pe) \ {0} such that z'xj = z''. So
[a~i,z>,axi,o, ax-i= {a_i,z", «x\0, ax-i,0} (p > 2), {«1,z' ,«xi,0,«x-i ,0}"xj,° = {«1,z" ,«xi,0,«x-i,0} (p = 2).
It follows that
Cay(AGL(1,pe), {a_M',«x>,0, ^-¿,0}) =
Cay(AGL(1,pe), {a_i,z",0^,0, ax-,0}) (p > 2), Cay(AGL(1, 2e), {«i,z-, «x\0, ^-¿,0}) =
Cay(AGL(1, 2e), {aM„, «x-,0, «^-,0}) (p = 2).
In view of this fact, up to graph isomorphism, Kp- is independent of the choice of z'.
The following is the main result of this paper.
Theorem 1.4. Let Kn be a special generalized truncation of Kn (n > 4) by Cn_1. Then IK n is isomorphic to either T (K6, A5, C5) (see Figure 1), or one of the graphs Kp- (i e Zpe_1, i < p-_1). Conversely, each of the above graphs is indeed a special generalized truncation of Kn (n > 4) by a cycle of length n — 1, where n = 6 or a prime power.
Furthermore, for any distinct i,i' e Zp-_1 with i,i' < p-_1, Kp- = Kp- if and only if i' = ipj or —ip (mod pe — 1) for some 1 < j < e. Moreover, the following hold:
(i)	Aut(T(Ke,A5, C5)) = A5;
(ii)	Aut(K4) = S4;
(iii)	Aut(K7) = D42 x Z3;
(iv)	Aut(K?1) = PGL2(11);
(v)	Aut(K23) = PGL2(23);
(vi)	if Kp- is not isomorphic to one of the graphs: K4, K£, Kf1 and K^g, then Aut(Kp-) = AGL(1,pe).
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2	Preliminaries
All groups considered in this paper are finite and all graphs are finite, connected, simple and undirected. For the group-theoretic and graph-theoretic terminology not defined here we refer the reader to [3, 12].
Let r = Cay(G, S) be a Cayley graph on a group G relative to a subset S of G. It is easy to prove that r is connected if and only if S is a generating subset of G. For any g G G, R(g) is the permutation of G defined by R(g) : x ^ xg for x G G. Set R(G) = {R(g) | g G G}. It is well-known that R(G) is a subgroup of Aut(r). For briefness, we shall identify R(G) with G in the following. In 1981, Godsil [7] proved that the normalizer of G in Aut(r) is G x Aut(G, S), where Aut(G, S) is the group of automorphisms of G fixing the set S set-wise. Clearly, Aut(G, S) is a subgroup of the stabilizer Aut(r)i of the identity 1 of G in Aut(r). We say that the Cayley graph Cay(G, S) is normal if G is normal in Aut(Cay(G, S)) (see [13]). If r = Cay(G, S) is a normal Cayley graph on G, then we have Aut(G, S) = Aut(r)1, and if, in addition, r is also arc-transitive, then Aut(G, S) is transitive on S. From this we can easily obtain the following lemma.
Lemma 2.1. There does not exist an arc-transitive normal Cayley graph of odd valency at least three on a cyclic group.
A Cayley graph Cay(G, S) on a group G relative to a subset S of G is called a CI-graph of G, if for any Cayley graph Cay(G, T), whenever Cay(G, S) = Cay(G, T) we have T = Sa for some a G Aut(G). The following proposition is a criterion for a Cayley graph to be a CI-graph.
Proposition 2.2 ([1, Lemma 3.1]). Let r be a Cayley graph on a finite group G. Then r is a CI-graph of G if and only if all regular subgroups of Aut(r) isomorphic to G are conjugate.
Let r be a connected vertex-transitive graph, and let G < Aut(r) be vertex-transitive on r. For a G-invariant partition B of V(r), the quotient graph rB is defined as the graph with vertex set B such that, for any two different vertices B, C G B, B is adjacent to C if and only if there exist u G B and v G C which are adjacent in r. Let N be a normal subgroup of G. Then the set B of orbits of N in V(r) is a G-invariant partition of V(r). In this case, the symbol rB will be replaced by rN.
In view of [11, Theorem 9], we have the following proposition.
Proposition 2.3. Suppose that r is a connected trivalent graph with an arc-transitive group G of automorphisms. If N < G has more than two orbits in V(r), then N is semiregular on V(r), and rN is a trivalent symmetric graph with G/N as an arc-transitive group of automorphisms.
3	Proof of Theorem 1.4
3.1 Special generalized truncations of Kn by Cn-1
In this subsection, we shall prove the first part of Theorem 1.4 by determining all special generalized truncations of Kn (n > 4) by Cn-1. Throughout this subsection, we shall use the following assumptions and notations.
X. Wang, F.-G. Yin and J.-X. Zhou: On generalized truncations of complete graphs 329
Assumption 3.1.
(!) Let Kn be a complete graph of order n with n > 4, and let V (Kn) = {vi, v2,..., Vn}.
(2)	Let G < Aut(Kn) be an arc-transitive group of automorphisms.
(3)	Let v = vi, and let Ov be a union of orbits of the stabilizer Gv acting on {{x, y} | x = y, x,y G V(Kn) \ {v}}. Let T be the graph with vertex set {v2,v3,..., vn} and edge set Ov.
(4)	For each u G V(Kn), let Vu = {(u,w) | w G V(Kn) \ {u}}.
(5)	Let Kn = T(Kn, G, T) be the graph with the vertex set [jueV(Kri) Vu, and the adjacency relation in which a vertex (u, w) is adjacent to the vertex (w, u) and to all the vertices (u, w') for which there exists a g G G with the property ug = v and
{w,w'}g G Ov.
In view of [6, Theorem 5.1], we have the following proposition.
Proposition 3.2. Use the notations in Assumption 3.!. Then Aut(Kn) has a vertex-transitive subgroup G such that P = {Vu | u G V(Kn)} is an imprimitivity block system for G. Furthermore, the following hold.
(!) The quotient graph of Kn relative to P is isomorphic to Kn.
(2)	G = G.
(3)	G acts faithfully on P.
For the two groups G, G in the above proposition, we shall follow [6] to say that G is the lift of G. The next lemma shows that if T = Cn-i then G is a 2-transitive permutation group on P and the point stabilizer GVu is either cyclic or dihedral.
Lemma 3.3. Use the notations in Assumption 3.!. Let T = Cn-i and let G be the lift of G. Then for each u G V(Kn), the subgraph of Kn induced by Vu is a cycle of length n — 1, and the subgroup GVu of G fixing Vu set-wise acts faithfully and transitively on Vu. In particular, G acts faithfully and 2-transitively on P, and GVu = Zn-i, or Dn-i (if n is odd), or D2(n-i).
Proof. By Assumption 3.1 (3) and (5), the subgraph of Kn induced by Vv is isomorphic to T. By Proposition 3.2, P = {Vu | u G V(Kn)} is an imprimitivity block system for G, and so for each u g V(Kn), the subgraph of Kn induced by Vu is a cycle of length n — 1.
For any two vertices u, w of Kn, by Assumption 3.1 (5), {(u, w), (w, u)} is the unique edge of Kn connecting Vu and Vw. This implies that the subgroup K of GVu fixing Vu point-wise will fix every block in P. It then follows from Proposition 3.2 (3) that K =1, and so GVu acts faithfully on Vu. Since G is transitive on V(Kn), GVu is transitive on Vu. Since the subgraph of Kn induced by Vu is a cycle of length n — 1, one has GVu = Zn-i, or Dn-i (if n is odd), or D2(n-i).
Again since {(u, w), (w, u)} is the unique edge of Kn connecting Vu and Vw, it follows that GVu also acts transitively on P \ {Vu}. This implies that G acts 2-transitively on P. By Proposition 3.2 (3), G acts faithfully on P.	□
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The above lemma enables us to determine the structure of G in the case when Y =
Cn-1.
Lemma 3.4. Use the notations in Assumption 3.1. Let Y = Cn-1 and let G be the lift of G. Then one of the following holds:
(1)	n = 6 and soc(G) = A5;
(2)	n = 4 and G ^ AGL(1, 22) or ArL(1, 22);
(3)	n = pe = 4 and G = AGL(1, pe), where p is a prime and e is a positive integer.
Proof. By Lemma 3.3, G can be viewed as a 2-transitive permutation group on P with point stabilizer isomorphic to Zn-1, or Dn-1 (if n is odd), or D2(n-1). By [5, Propositon 5.2], soc(G) is either elementary abelian or non-abelian simple, and furthermore, if soc(G) is non-abelian simple, then by checking the list of the simple groups which can occur as socles of 2-transitive groups in [5, p. 8], we have soc(G) = A5. In order to complete the proof of this lemma, it remains to deal with the case when soc(G) is elementary abelian.
In what follows, assume that soc(G) = Zp for some prime p and positive integer e. View soc(G) as an e-dimensional vector space over a field of order p, and let 0 denote the zero vector of soc(G). Recall that Go = Zpe_1, Dpe-1 (p odd), or D2(pe-1). By checking Hering's theorem on classification of 2-transitive affine permutation^groups [8] (see also [10, Appendix 1]), we have G < ArL(1,pe) with point-stabilizer G0 < TL(1,pe). As G = soc(G) x G0, to determine G, we only need to determine all possible subgroups of rL(1,pe) which are isomorphic to Zpe_1, Dpe_1 (p odd), or D2(pe_1), and transitive on soc(G) \ {0}.
Note that TL(1,pe) can be constructed in the following way. Let GF(pe) be the Galois field of order pe, and view soc(G) as the additive group of GF(pe). It is well-known that the multiplicative group GF(pe) * of GF(pe) is cyclic, and let x be a generator of GF(pe) *. Then GL(1,pe) = (x). Let y be the Frobenius automorphism of GF(pe) such that y maps every g G GF(pe) to gp. Then we have
rL(1,pe) = (x, y | xp -1 = ye = 1, y-1xy = xp).
In the following, we shall first determine all possible cyclic subgroups of TL(1,pe) of order either pe -1 or (p odd) (Claim 1), and then this is used to determine all possible subgroups of rL(1,pe) which are isomorphic to Zpe_1, Dpe_1 (p odd), or D2(pe_1), and transitive on soc(G) \ {0}.
Claim 1. Let T be a cyclic subgroup of TL(1,pe) of order p-_ with either r = 1 or r = 2
andp is odd. Then either T = (xr), or pe = 32, T = Zp°-i and T = (xy) or (x3y).
2
Proof of Claim 1. Let I = pe -1 or (p odd). Since T is acyclic subgroup of TL(1,pe) of order we may let T = (xjyk) with 0 < j < pe — 2 and 0 < k < e — 1. If k = 0, then T < (x) and so T = (xr) with either r =1 or r = 2 and p is odd.
Assume now that 0 < k < e — 1. Then yk = 1. Since y_1xy = xp, one has
yxpy_1 = x, and hence (yxy_1)p = x. Clearly, pe = 1 (mod pe — 1), so yxy_1 = xp
X. Wang, F.-G. Yin and J.-X. Zhou: On generalized truncations of complete graphs 331
It follows that yk xj y k = xjp e , and so yk xj = xjp e yk .By this equality, we have for any positive integer m,
mk (e — 1)_j
(xjyk)m = xj(1+pfc(e—1)+P2k(e—1)k(e—1))ymfc = pk(=-i)-i ymk. (3.1)
. pek(e—1) — 1
From Equation (3.1) it follows that (xjyk)e = xj pk(e—1)— 1 . Since pe — 1 | pfce(e-1) — 1, one has
(xj yk )e(pk(e—1)-1) = xj(pek(e—1)-1) = 1.
This implies that the order of xjyk divides e(pk(e-1) — 1), namely, ^ | e(pk(e-1) — 1). Since I = pe — 1 or (p odd), we have pe — 1 | 2e(pk(e_1) — 1).
Suppose that e > 3. If (p, e) = (2,6), then I = pe — 1 = 63. However, it is easy to check that 63 \ 6(25k — 1) for any k < 5, contrary to I | e(pk(e-1) — 1). Thus, (p, e) = (2,6). Then by a result of Zsigmondy [14], there exists at least one prime q such that q divides pe — 1 but does not divide p4 — 1 for any positive integer t < e. Clearly, p = q, so p is an element of Z* = Zq-1 of order e. In particular, we have q > e. Since q | pe — 1 and pe — 1 | 2e(pk(e_1) — 1), we have q | pk(e-1) — 1, implying k(e — 1) > e. Since k < e — 1, we may let k(e — 1) = me +1 for some positive integers m and t < e, and sincepme(p4 — 1) = (pk(e-1) — 1) — (pme — 1), we have q | p4 — 1. However, this is impossible because it is assumed that q \ p4 — 1 for any t < e.
Thus, e < 3. Since 0 < k < e — 1, one has e = 2 and k =1, and thenp2 — 1 | 4(p — 1). It follows that p +1 | 4 and hence p = 3. Then (xjy)2 = x4j has order at most 2 since (x) = Z8, and then xjy has order dividing 4. This implies that I = = 4 and T = (xy) or (x3y). This completes the proof of Claim 1.	□
By now, we have shown that Claim 1 is true. Recall that G0 < TL(1,pe), G0 = Zpe_1, D pe_1 (p odd), or D2(pe_1) and G0 is transitive on soc(G) \ {0}. We shall finish the proof by considering the following three cases.
Case 1. G0 = Zpe_1.
In this case, by Claim 1, we must have G0 = (x) = GL(1, pe) and so G = AGL(1, pe).
Case 2. G0 = Dpe_1 (p odd).
In this case, by Claim 1, either x2 G G0, or pe = 9 and G0 contains xy or x3y. For the former, we have G0 = (x2,f), where f is an involution of TL(1,pe) such that
0
2 — (2 — 1 imnliAC that a ic pi/pn anH Z- — _
/x2/ = x 2 and / G (x). Note that G0 is transitive on soc(G) \ {0}. We may let / = xyk and 0 < k < e - 1. By Equation (3.1), /2 = (xyk)2 = 1 implies that e is even and k
e(e-l)
e(e-l)
and furthermore, xp + 1 = 1. It follows that pe — 1 | p 2 + 1. However, since pe(pI + 1) = (pe<e2 + 1) + (pe— 1), we would have pe — 1 | pe + 1, forcing that p = 2, a contradiction. _
For the latter, we have G0 = D8. However, it is easy to check that in rL(1, 9) = (x, y | x8 = y2 = 1, y-1xy = x3} there does not exist an involution inverting xy or x3y, a contradiction.
Case 3. G0= D2(pe_1).
By Claim 1, we must have G0 = (x} x (ye} with y IxyI = x-1. On the other hand, since y-1xy = xp, we have y Ixye = xp 2 and hence xp 2 = x-1. It follows that
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p- = — 1 (mod pe — 1) and hence pe — 1 divides p- + 1. Consequently, we have pe = 4, Go = (x,y) = rL(1,4), and (5 = ArL(1,4) = S4.	□
Now we are ready to determine all possible special generalized truncations of Kn by
Cn-1.
Lemma 3.5. Use the notations in Assumption 3.1. Let T = Cn-1 and let G be the lift of G. Then Kn = T(Kn, G, T) is isomorphic to either T(K6, A5, C5) (see Figure 1), or one of the graphs Kp- (i G Z*e_1, i < p-_-) (see Construction 1.2 for the definition of these graphs).
Proof. If soc(G) = A5, then by [6, Example 5.3], we have G = A5 and up to graph isomorphism, there exists a unique graph, and so we may denote this graph by T(K6, A5, C5) (see Figure 1).	_	_
In what follows, we assume that soc(G) ^ A5. Then from Lemma 3.4 we see that G has a subgroup, say T such that T = AGL(1,pe) and T acts regularly on V(Kn), where p is a prime and e is a positive integer such that pe > 4. It follows that Kn is a Cayley graph on T (= AGL(1,pe)) and n = pe. For each u G V(Kn), by Lemma 3.3, the subgraph of Kn induced by Vu is a cycle of length n — 1, and the subgroup GVu of G fixing Vu set-wise acts faithfully and transitively on Vu. Furthermore, G acts faithfully and 2-transitively on P. For convenience, we may identify P with GF(pe), identify Vu with the zero element 0 of GF(pe) and identify T with AGL(1,pe). We shall use the notations for T = AGL(1,pe) as well as its elements and subgroups H and K introduced in the paragraph before Construction 1.2. Then TVu = K = Zp-_1.
Take (u, w) G Vu, and assume that (u, w1) and (u, w2) are two vertices in Vu adjacent to (u, w). Since TTVu = K = Zp-_1 is transitive on Vu, there exists a unique axi 0 G TTVu such that (u, w)°xi,° = (u, w1) and (u, w)°x-i,° = (u, w2), and since the subgraph of Kn induced by Vu is a cycle of length n — 1, i is coprime to pe — 1 (n = pe). So we may let
Kn = Cay(AGL(1,pe), {«x^.o, "x-,0, «xj}),
where axjis an involution. Since Kn is connected, if p is odd, then we have axj =
a p--i = a- 1z' and z' = 0, and if p = 2, then we have axj z/ = a1 z/ and z' = 0, and
x 2 ,z'	'	'	'
correspondingly, we obtain the two graphs Kp- (p > 2) and K2- (see Construction 1.2).
□
From Figure 1 it is easy to see that T(K6, A5, C5) is a special generalized truncation of K6 by a cycle of length 5. The following lemma shows that each of the Cayley graphs Kp-(i G Zp-_1, i < p-_1) is also indeed a special generalized truncation of Kp- by a cycle of length pe — 1.
Lemma 3.6. Each of the graphs Kp- (i G Zp-_1, i < p-_1) (see Construction 1.2) is a special generalized truncation of Kp- by a cycle of length pe — 1.
Proof. Recall that each Kp- (i G Z*-_1, i < p-2z1) is a trivalent Cayley graph on AGL(1,pe) defined as follows:
Kp- = Cay(AGL(1,pe), {a_1,z', a^ ,o, ax-.,o}) (z' = 0,p > 2), K2- = Cay(AGL(1, 2e), {a1,z', ax.,o, ax-.,o}) (z' = 0).
X. Wang, F.-G. Yin and J.-X. Zhou: On generalized truncations of complete graphs 333
(Keep in mind we use the notations for AGL(1,pe) as well as its elements and subgroups H and K introduced in the paragraph before Construction 1.2.) Note that AGL(1,pe ) = H x K, where
H = {a1tZ„ : z ^ z + z", Vz G GF(pe) | z" G GF(pe)}, K = {axj,0 : z ^ zxj, Vz G GF(pe) | j G Zpe-1}.
Moreover, K is maximal in AGL(1,pe) since AGL(1,pe) is 2-transitive on GF (pe). As i G Z*e_1, one has K = (axi,0) and then the maximality of K implies that (a_1jZ/, aXi,0) = AGL(1,pe) for p > 2 and (aM',axi,0) = AGL(1,2e). Thus, every Kp-(i G Zpe_1, i < p-_) is connected.
It is easy to see that Cay(K, {axi,0, ax-i,0}) = Cp-_1 is a subgraph of Kp- (i G Zp-_1, i < p-_1 ). Since AGL(1,pe) acts on V(Kp- ) by right multiplication, the subgraph of Kp- induced by Kg for any g G AGL(1, pe ) is a cycle of length pe - 1. As AGL(1, pe ) acts 2-transitively on B = {Kg | g G AGL(1,pe)}, the quotient graph of Kp- relative to B is a complete graph Kp-. So we have Kp- = T(Kp-, AGL(1,pe), Tj), where Y is the subgraph with vertex set B - {K} and edge set {{KYg, KYaxi,0g} | g G K} where Y = a_1jZ/ for p> 2 and 7 = a1jZ/ for p = 2.	□
3.2 Automorphisms and isomorphisms
In this subsection, we shall determine the automorphism groups and isomorphisms of special generalized truncations of Kn by Cn_ 1, and thus prove the second part of Theorem 1.4. By checking [6, Table 1], we have the following lemma.
Lemma 3.7. Aut(T(Ke^Gs)) = A5.
In the following two lemmas, we shall determine the automorphisms and isomorphisms of the graphs Kp- (i G Zp-_1, i < p-_1 ). We keep using the notations for AGL(1,pe) as well as its elements and subgroups H and K introduced in the paragraph before Construction 1.2.
Lemma 3.8. Let r be one of the graphs Kp- (i G Zp-_1, i < p-_1 ) (see Construction 1.2). Then Theorem 1.4 (ii)-(vi) hold.
Proof. Recall that r is a connected trivalent Cayley graph on X = AGL(1,pe). Let A = Aut(r). For convenience of the statement, we view X as a regular subgroup of A.
Suppose first that r is arc-transitive. Let N = p|geA Xg. If N =1, then by [9, Theorem 1.1], we have Aut(r) = PGL2(pe) withpe = 11 or 23. If pe = 11, then since i G Zf0 and i < 5, we have i = 3 and hence r = K31. If pe = 23, then i = 3,5,7 or 9 as i G Z22 and i < 11, and by Magma [4], Aut(K23) == PGL2(23) if and only if i = 7, and hence r = K23. If N > 1, then N < A, and in piirticular, N < X. Since soc(X) = Zp is the unique minimal normal subgroup of X = AGL(1,pe), one has soc(X) < N. Clearly, soc(X ) is a Sylow p-subgroup of N since N < X .So soc(X ) is characteristic in N and hence normal in A. Consider the quotient graph S of r relative to soc(X). Clearly, S has pe - 1 vertices. Since pe - 1 > 2, by Proposition 2.3, S would be a trivalent arc-transitive Cayley graph on X/ soc(X) = Zp-_1. Furthermore, by [2, Corollary 1.3], either S = K3,3, or S is a trivalent normal arc-transitive Cayley graph on X/ soc(X) = Zp-_1. However, the latter case cannot happen by Lemma 2.1. For the former, we have pe - 1 = 6
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and so p = 7 and e = 1. In this case, we have i = 1 and r = K7. By Magma [4], we have Aut(K7) = D42 x Z3.
Suppose now that r is not arc-transitive. If A > X, then the vertex-stabilizer Aa is a 2-group for any a e V(r). Then Aa fixes one and only one neighbor of a. Assume that the neighbor of a fixed by Aa is b. Then B = {{a, b}g | g e A} is a system of blocks of imprimitivity of A on V(r). It follows that r - B is a union of several cycles with equal lengths, and the set of vertex-sets of these cycles forms an A-invariant partition of V(r). Let C be the cycle of r containing the identity 1 of X. Since r is a Cayley graph on X, X acts on V(r) = X by right multiplication, and since V(C) is a block of imprimitivity of A acting on V(r), C is actually a subgroup of X. From the definition of r = Kp-, one may see that V(C) = K = {axi,o: z ^ zx®, Vz e GF(pe) | i e Zp-s1}, and the vertex set of every cycle of r - B is just a right coset of K. Let B = {Kg | g e X}. Then B is an A-invariant partition of r. Clearly, X acts 2-transitively and faithfully on B, so the quotient graph of r relative B is Kp-. Now it is easy to see that r = T(Kp-, A, Y®), where Y® is the subgraph with vertex set B - K and edge set {{KYg, K7axi,og} | g e AK} where 7 = a_i,z/ for p > 2 and 7 = a^z' for p = 2. Clearly, Y® = Cp-s1. From Lemma 3.4 it follows that either pe =4 and A = ArL(1,4) = S4, or A = X = AGL(1,pe).	□
Lemma 3.9. For any distinct i, i' e Z*-s1 with i, i' < p-_1, Kp- = Kp- if and only if there exists 1 < j < e such that i' = ipj or —ip (mod pe — 1).
Proof. If pe = 4 or 7, then we must have i =1, and so we have only one graph for each of these two cases. If pe = 11 or 23, then by MAGMA [4], for any distinct i, i' e Z*-s1 with i, i' < p-_1, one may check that Kp- = Kp'- ifandonlyif i' = ipj or —ipj (mod pe — 1).
Suppose that Kp- is not isomorphic to one of the graphs: K4, K1, Kf1 and K23. By Lemma 3.8, Aut(Kp-) = AGL(1,pe) and by Proposition 2.2, Kp- is a CI-graph. Recall that
Kp- = Cay(AGL(1,pe), {a_M', «x-,o}) (p > 2),
K2- = Cay(AGL(1, 2e), {aM', a^,o, aa-i,o}) (p = 2).
Since Kp- is a CI-graph, for any distinct i, i' e Zp- s1 with i, i' < p-_1, Kp- = Kp- if and only if there exists 7 e Aut(AGL(1,pe)) such that {axi,o, ax-i,o}Y = {axi' o, ax-i' o} and either a! 1 z = as1,z' for p > 2 or aY z' = a1,z' for p = 2.
Note that Aut(AGL(1,pe)) = ArL(1,pe) = AGL(1,pe) x (n), where n is induced by the Frobenius automorphism of GF(pe) such that = aaP,bP for any aa,b e AGL(1,pe). Suppose first that i' = ipj or —ip (mod pe — 1) for some 1 < j < e.
Then one may check that a±1 Zz'' ^ = a±1,z' and {axi,o, ax-i,o}n a(z')-pjz',» =
{axi' o,ax-i' o}. So Kp- = Kp-. Conversely, if Kp- = Kp-, then there exists 7 e Aut(AGL(1,pe)) such that {ax»,o, ax-i,o}Y = {axi' o, ax-i' o} and either al 1 z' = as1,z' for p > 2 or aYz = a1,z' for p = 2. Since K = (axi,o), 7 normalizes K, and since NArL(1,p-)(K) = K x (n), one has 7 = axk,onj, for some k e Z*-s1 and 1 < j < e. Then
axi,o = axi,o = axi,o = axipj ,o e {axi' ,o,ax-i' ,o}. It follows that i' = ipj or —ipj (mod pe — 1).	□
X. Wang, F.-G. Yin and J.-X. Zhou: On generalized truncations of complete graphs 335
3.3 Proof of Theorem 1.4
From Lemmas 3.5 and 3.6 we can obtain the proof of the first part of Theorem 1.4, and
from Lemmas 3.8 and 3.9, we obtain the proof of the second part of Theorem 1.4.
ORCID iDs
Xue Wang © https://orcid.org/0000-0001-5131-6353 Fu-Gang Yin © https://orcid.org/0000-0001-8328-0690
Jin-Xin Zhou © https://orcid.org/0000-0002-8353-896X
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[7]	C. D. Godsil, On the full automorphism group of a graph, Combinatorica 1 (1981), 243-256, doi:10.1007/bf02579330.
[8]	C. Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order, II, J. Algebra 93 (1985), 151-164, doi:10.1016/0021-8693(85)90179-6.
[9]	J. J. Li and Z. P. Lu, Cubic s-arc transitive Cayley graphs, Discrete Math. 309 (2009), 60146025, doi:10.1016/j.disc.2009.05.002.
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ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 19 (2020) 337-349 https://doi.org/10.26493/1855-3974.2022.44a
(Also available at http://amc-journal.eu)
Properties of double Roman domination on cardinal products of graphs*
Antoaneta Klobucar
Faculty of Economics, University of Osijek, 31000 Osijek, Croatia
Ana Klobucar ©
Faculty ofMechanical Engineering and Naval Architecture, University ofZagreb,
10000 Zagreb, Croatia
Received 14 June 2019, accepted 25 August 2020, published online 20 November 2020
Double Roman domination is a stronger version of Roman domination that doubles the protection. The areas now have 0,1, 2 or 3 legions. Every attacked area needs 2 legions for its defence, either their own, or borrowed from 1 or 2 neighbouring areas, which still have to keep at least 1 legion to themselves. The minimal number of legions in all areas together is equal to the double Roman domination number.
In this paper we determine an upper bound and a lower bound for double Roman domination numbers on cardinal product of any two graphs. Also we determine the exact values of double Roman domination numbers on P2 x G (for many types of graph G). Also, the double Roman domination number is found for P2 x Pn, P3 x Pn, P4 x Pn, while upper and lower bounds are given for P5 x Pn and P6 x Pn.
Finally, we will give a case study to determine the efficiency of double protection. We will compare double Roman domination versus Roman domination by running a simulation of a battle.
Keywords: Roman domination, double Roman domination, cardinal products of graphs, paths, cycles. Math. Subj. Class. (2020): 05C69, 68U20
*This work has been fully supported by the Croatian Science Foundation under the project IP-2018-01-5591. E-mail addresses: aneta@efos.hr (Antoaneta Klobucar), aklobucar@fsb.hr (Ana Klobucar)
Abstract
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
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Ars Math. Contemp. 19 (2020) 173-187
1 Introduction
In the 4th century AD Constantine I (274 - 337 AD) ruled the Roman Empire. To defend the Empire against barbarians, he had to arrange Roman legions in a way that all strategically important places were protected with as low costs as possible.
If at least one Roman legion was stationed at a certain location, that location was considered to be secured. Unsecured locations, on the other hand, had no legions, but they had to be adjacent to at least one secured location. If an unsecured location was under attack, one could send a legion from some neighbouring secured location. But to avoid making that secured location unsecure, it had to have at least two legions itself. Maintaining of an army was expensive, so Constantine had to secure the Empire with as few legions as possible.
This historical background motivated Ian Stewart (1999) to suggest the new variant of graph domination known as Roman domination (RD). If we represent locations of the Empire as graph vertices and roads of the Empire as graph edges, the problem of defending the Roman Empire becomes a problem of graph domination. Double Roman domination (DRD) is stronger version in which we double protection.
There are many works dealing with Roman domination [8, 9, 13, 14], but only few about double Roman dominations. Foundations of DRD are set in [4]. In [3, 15, 16] we can find bounds on the DRD and the most recent work is [2]. For more details on Roman domination and double Roman domination and their variants see [5, 6, 7].
In this paper we determine exact values or upper and lower bounds for double Roman domination numbers on cardinal products of some graphs.
Apart from this introduction, the work is organized in the following way. In Section 2 we define dominating function on a graph G, Roman dominating function on G, double Roman dominating function on G and on a cardinal product of graphs. Domination number, Roman domination number and double Roman domination number are defined and some basic relations among them are given.
In Section 3 we determine one upper and one lower bound for double Roman domination numbers on cardinal product of any two graphs. Then we determined the exact values of double Roman domination numbers of P2 x G for many types of graph G. Finally, the double Roman domination number is found for P2 x Pn, P3 x Pn, P4 x Pn, while upper and lower bounds are given for P5 x Pn and P6 x Pn.
In Section 4 we give a case study to determine the efficiency of double protection. We will simulate a battle between Romans and barbarians in the cases of double Roman domination and standard Roman domination.
2 Definitions
Dominating function (DF) on G = (V, E) is a function f: V ^ {0,1} satisfying the condition that every vertex u for which f (u) =0 is adjacent to at least one vertex v for which f (v) = 1. Depending on values of f, we get the ordered partition (V0, V) of V where each vertex in V0 is adjacent to at least one vertex in V1. The set V1 is called a
dominating set.
We have bijection between the set of all functions f: V ^ {0,1} and the set of all ordered partitions (V0, Vi). Thus we are allowed to write f = (V0, Vi). The weight of f equals w(f) = EveV f (v) =0 • |Vo | + 1 • |Vi| = |Vi|. Of course, we will look for dominating functions with the minimum weight. This weight 7(G) is called the domination
A. Klobučar and A. Klobučar: Properties of double Roman domination on cardinal products ... 339 number of G.
Further, Roman dominating function (RDF) on G = (V, E) is a function f: V ^ {0,1,2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. Since this function also induces the ordered partition of V with V = {v € V : f (v) = i}, i G {0,1, 2}, we are allowed to write f = (V0, Vi, V2). The set VlU V2 is called a Roman dominating set. The weight of an RDF f equals w(f) = f (v) = 0 • |V0| + 1 • + 2 • |Vs| = |Vi| + 2|V2|. The minimum such weight yr(G) is called the Roman domination number on G.
In Roman domination, one legion is required to defend any attacked area. Double Roman domination is a stronger version of Roman domination that doubles the protection by ensuring that any attack can be defended by at least two legions.
Finally, double Roman dominating function (DRDF) on G = (V, E) is a function f: V ^ {0,1, 2,3} if it satisfies the following conditions:
1.	If f (v) = 0, then the vertex v has at least two neighbours in V2 or one neighbour in V3.
2.	If f (v) = 1, then the vertex v has at least one neighbour in V2 U V3,
where by V we denote the set of vertices assigned with i by the function f. The set Vi U V2 U V3 is called a double Roman dominating set. The weight of a DRDF equals w(f) = Evev f(v) = |Vi| + 2|V2| + 3|V3|.
Double Roman domination number YdR(G) equals the minimum weight among all double Roman dominating functions on G. A double Roman dominating function on G with weight YdR(G) is called a YdR-function of G.
In Roman domination at most two Roman legions are deployed at any location. But as we will see in what follows, the ability to deploy three legions at a given location provides a level of defense that is both stronger and more flexible. Also, the additional security we get is usually greater than the additional costs.
Here we can see a real benefit of double Roman domination. In the example of the star graph K1,n_1 (see Figure 1), it is obvious that YdR(K1,n-1) = 3. Note that this is only one more than YR(K1in-1) = 2. So we doubled the defense of a graph (at least two legions against each attack) with an added cost of no more than 50% of the cost of defending against each attack with one legion.
Figure 1: Double Roman domination on star graph.
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In [4], it is observed that YdR(G) < 2|Vi| + 3|V2| for any RDF f = (Vo, Vi, V2). It is also proved a relation between domination and double Roman domination number of any graph G, i.e.
2Y(G) < YdR(G) < 37(G),
and a relation between Roman domination and double Roman domination number of any nontrivial connected graph G
Yr(G) < YdR(G) < 2YR(G).
Graphs where YdR(G) = 3y(G) are called double Roman graphs. There is an open problem to characterize such graphs. Double Roman trees are characterized in [1]. For more domination parameters and for the terminology see [10, 11, 12].
In this paper we will consider double Roman domination number of cardinal product of graphs. For arbitrary graphs G and H, the cardinal product of G and H is the graph G x H which satisfies the following:
1.	Its vertex set is V(G x H) = V(G) x V(H).
2.	Two vertices (g, h), (g', h') G V(G x H) are adjacent if and only if g is adjacent to g' in G and h is adjacent to h' in H.
If H c V(G) then G[H] is the subgraph induced with H. The cardinal product of two paths Pm x Pn has two connected components. If the vertices of Pm and Pn are denoted by {1, 2,3,..., m} and {1,2, 3,..., n} respectively, then the component of Pm x Pn containing the vertex (1,1) will be denoted by K1 and the other component by K2. If at least one of the parameters m or n is even, the components K1 and K2 are isomorphic (see Figure 2). Otherwise, the component K1 has one vertex more than the component K2.
12
P2 O-O
1	2	3
P3 o—o—o
(1,1) (1,2) (1,3)
(2,1) (2, 2) (2, 3)
P2 x P3
Figure2: Ki = P2 x P3 [{(1,1), (2, 2), (1, 3)}] and K2 = P2 x P3 [{(2,1), (1, 2), (2,3)}].
3 Specific values of double Roman domination numbers for cardinal products of graphs
As for introduction, we will show here some basic results and bounds for double Roman domination.
A. Klobučar and A. Klobučar: Properties of double Roman domination on cardinal products .
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Remark 3.1. In [2] it is proved that
YdR(Pn) =
and
YdR (Cn) =
J3[n 1, n = 0, 2 (mod 3) n 1- 1, n = 1 (mod 3)
J n, n = 0 (mod 3) [n +1, n = 1, 2 (mod 3)
fn, n = 0, 2, 3,4 (mod 6) In + 1, n = 1, 5 (mod 6).
Observartion 3.2. For any graphs G and H of order n and m YdR(G x H) > r 3mn
A(G)A(H) + 1 rr,	2 + 1n((1 + S(G)S(H ))/2)
where by A(G) (S(G)) we denote the maximum (minimum) degree of all vertices on G. Proof. In [16] it is proved that for any graph G of order n with maximum degree A(G) > 1
3n
A(G) + 1
YdR (G) >
Further, it holds
A(G x H) = A(G) • A(H).
Combining two previous statements we get the lower bound. Next, in [14] it is proved that for cardinal product any graphs G and H of order n and m
YR(G x H) < mn2 + ^(Hi'+'H))/2'.
Then the statement follows from YdR(G) < 2yr(G).	□
Now we will calculate the exact values of double Roman domination numbers for cardinal products of some graphs.
Theorem 3.3. For any tree T and any graph G without cycles of odd length we have
YdR(P2 x T) = 2YdR(T) < YdR(P2) • YdR(T), YdR(P2 x G) = 2YdR(G) < YdR (P2) • YdR(G).
Proof. The proof is trivial, since P2 x T and P2 x G consist of two disjoint copies of T and G, respectively and YdR(P2) = 3.	□
Theorem 3.4. For the path P2 and any odd cycle G2n+i, n > 1,
YdR(P2 x C2„+i) = 4n + 2.
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Proof. Note that the cardinal product of P2 and C2n+1 is a cycle C4n+2. Namely, if we denote the vertices of P2 with a and b, and the vertices of C2n+1 with 1,2,..., 2n+1, then the vertices of the product P2 x C2n+1 are adjacent in this order: (a, 1), (b, 2), (a, 3), (b, 4),..., (a, 2n +1), (b, 1), (a, 2),..., (b, 2n +1) and the last vertex (b, 2n +1) is adjacent to (a, 1), which makes a cycle of length 2(2n +1) = 4n + 2. Remark 3.1 implies that
YdR(C4n+2) = 4n + 2.
□
Definition 3.5. For a fixed m, 1 < m < n, the set (Pk )m = {(i, m) : i = 1,..., k} is called a column of Pk x Pn. Similary, for a fixed l, 1 < l < k, the set (Pn); = {(l, j) : j = 1,..., n} is called a row of Pk x Pn.
Theorem 3.6. Let n > 2. Then
YdR (P3 x P„)
7,	n = 3
2n + 2, otherwise.
Proof. It is easy to see that YdR(P3 x P3) = 7. Hence we assume n > 4. First we show that YdR(P3 x Pn) < 2n + 2. Define f: V(P3 x Pn) ^ {0,1,2, 3} by f ((2,2)) = f ((2, n - 1)) = 3, f ((2, j)) = 2 for j G {1,..., n} - {2, n - 1} and f (x) = 0 otherwise. Clearly f is a double Roman dominating function on P3 x Pn of weight 2n + 2 and so YdR(P3 x Pn) < 2n + 2. To prove inverse inequality, let f = (V0, 0, V2,V3) bea YdR(P3 x Pn)-function. Since the vertices (2, 2) and (2, n - 1) are strong support vertices, we have (2,2), (2, n - 1) G V3. On the other hand, since V2 U V3 is a dominating set of P3 x Pn, we have |V2UV3I > y(P3 xPn) = n (see [11]). Thus we have y^r(P3 xP„) = 2|V2|+3|V3| = 2(|V2| + |V3|) + |V3| > 2n + 2. Thus YdR(P3 x Pn) = 2n + 2 for n > 4 and the proof is complete.	□
Theorem 3.7. Let n> 2. Then
YdR(P4 x Pn) =
3n,	n	= 0	(mod	4)
3n + 3,	n	= 1	(mod	4)
3n + 2,	n	= 2	(mod	4)
3n + 1,	n	= 3	(mod	4)
Proof. First we show that
YdR(P4 x Pn) <
3n,	n	= 0	(mod	4)
3n + 3,	n	= 1	(mod	4)
3n + 2,	n	= 2	(mod	4)
3n + 1,	n	= 3	(mod	4)
Since P4 x Pn consists of two isomorphic components, we consider only K1 and we multiply the result by 2.
Case 1: n = 0 (mod 4). Define f: V(K1) ^ {0,1, 2,3} by f ((2,4j + 2)) = f ((3,4j + 3)) = 3, j = 0,1,..., [f J - 1, and f (x) = 0 otherwise. Clearly f is a double Roman dominating function of weight 3- on K1 and so YdR (P4 x Pn) < 3n, for n = 0 (mod 4).
A. Klobučar and A. Klobučar: Properties of double Roman domination on cardinal products .
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Case 2: n = 1 (mod 4). Define f: V(Ki) ^ {0,1, 2, 3} by f ((2,4j + 2)) = f ((3,4j + 3)) = 3, j =0,1,..., [ f J -1, f ((2, n— 1)) = 3 and f (x) = 0 otherwise. It can easily be seen that f is a double Roman dominating function of weight 6 (f-1) + 3 = 3f±3 on K1 and so YdR(P4 x Pn) < 3n + 3, for n = 1 (mod 4).
Case 3: n = 2 (mod 4). Define f: V(Ki) ^ {0,1, 2,3} by f ((2,4j + 2)) = 3, j = 0,1,..., [ f J — 1, f ((3,4j + 3)) = 3, j = 0,1,..., [ f J — 2, f ((3, n — 1)) = 3, f ((1, n — 1)) = f ((4, n — 4)) = 2 and f (x) = 0 otherwise. Hence f is a double Roman dominating function of weight 6( f-2) + 4 = 3f±2 on K1 and so YdR(P4 x Pn) < 3n + 2, for n = 2 (mod 4).
Case 4: n = 3 (mod 4). Define f: V(K1) ^ {0,1, 2, 3} by f ((2,4j + 2)) = f((3,4j + 3)) = 3, j = 0,1,..., [fJ — 1, f((2,n — 1)) = 3, f((4,n — 1)) = 2 and f (x) = 0 otherwise. Therefore f is a double Roman dominating function of weight
6(f-3e + 5 = 3fti on K1 and so ydR(P4 x Pn) < 3n +1, for n = 3 (mod 4).
Proof of the minimality: In [16] is proved that if G is a graph of order n with maximum degree A(G) > 1
YdR (G) > Ia^^TT
The order of P4 x Pn is 4n and A(P4 x Pn) = 4. Therefore
-12n"
"5"
Let n = 0 (mod 4). From the fact that YdR(P4 x Pn) < 3n and (3.1), it follows
YdR(P4 X P„) >
= 3n.	(3.1)
YdR (P4 x Pn) = 3n, n = 0 (mod 4).
In more details, for this case each vertex from V0 is double Roman dominated by only one vertex from V3. Next, V2 = 0, and V3 is dominating set (see [11]). Also, on the last n-th column from P4 x Pn all vertices are from V0 (see Figure 3).
Figure 3: The function f (V (K1)) on P4 x P8.
Let n = 1 (mod 4). Then from (3.1) on the first n — 1 columns on P4 x Pn double Roman function f has a weight at least 3(n — 1). Further, if the function f has the exactly
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weight 3n - 3, then the vertex (2, n - 1) G V0 (on Ki). But (2, n - 1) is strong suport vertex, so must be in V3. The same situation is on K2. It follows that YdR(P4 x Pn) > 3n - 3 + 6 = 3n + 3. Hence,
YdR(P4 x P„) = 3n + 3, n = 1 (mod 4).
Let n = 2 (mod 4). It is easy to see that YdR(P4 x P6) = 20 (on each component 10) and that the last 4 x 4 block and nth and (n - 1)th columns make a 4 x 6 block. It follows that on the last 6 columns we need at least weight 20, and on the first n - 6 columns 3(n - 6). Then YdR(P4 x Pn) > 3n - 18 + 20 = 3n + 2. So,
YdR(P4 x Pn) = 3n + 2, n = 2 (mod 4).
Let n = 3 (mod 4). Then on the first n-3 columns on P4 x Pn double Roman function f has a weight at least 3(n - 3). On the last 3 columns we need at least weight 5 on one, or 10 on both components giving YdR(P4 x P3) = 10. It follows that YdR(P4 x Pn) > 3n - 9 + 10 = 3n + 1. Therefore,
YdR(P4 x P„) = 3n +1, n = 3 (mod 4). For P5 x Pn and P6 x Pn from the formula
2y(g) < YdR(G) < 3y(G), and [11] we have the following bounds:
□
2
n + 2,	n=	2, 3, 4	
11,	n=	7	
4n+6 3 , 4n+4 3 , 4n+8 3 ,	n = n = n =	0,	3 2, 5 1,	4	(mod 6) (mod 6) (mod 6),
4 A case study
< YdR(P5 X Pn),
Ydfl(P X Pn) < 3 <
n + 2,
11,
4n+6 3 , 4n+4 3 , 4n+8 3 •
^n - |JJ) < YdR(P6 X Pn) < 6(n - [nj)
n = 2, 3, 4 n = 7
n = 0, 3 (mod 6) n = 2, 5 (mod 6) n = 1,4 (mod 6), n > 7, n 5-
In this section we simulate a battle between Romans and barbarians to test efficiency of the double protection versus the ordinary protection (standard Roman domination). Cardinal product P4 x Pn is used to model the battlefield, more precisely component Ki. We could use any other cardinal product of graphs, but we use P4 x Pn because of its convenience: it is large enough to have multiple outcomes, but not too large for visualization.
Instead of Romans and barbarians, we could have ambulances and patients or firefighters and fires. Ambulances would respond to medical emergencies and firefighters would
A. Klobučar and A. Klobučar: Properties of double Roman domination on cardinal products .
345
extinguish fires in their local area. Position of hospitals and fire stations would correspond base vertices, respectively. We are still speaking about Romans and barbarians in order to conform with the usual terminology dealing with dominations. But as we can see, the whole situation has also some modern interpretations, which are more practical and more useful.
First, we will give some basic rules and restrictions to avoid exceptions. The following rules could be easily adapted to ambulances and patients, and to firefighters and fires.
•	The simulation is organized in turns. The first turn is played by the barbarians.
•	The simulation stops if all cities are destroyed by the barbarians or if all barbarians are defeated by legions and legions are returned to their base cities.
•	The legions and the barbarian groups move only by one edge in each turn.
•	The barbarian groups destroy an unprotected city if Romans do not send enough help in the next turn.
•	If the barbarian group attacks a city with a legion, they fight immediately (no waiting for the next turn).
•	The destroyed city stays destroyed, but both the legions and barbarian groups can pass through it.
•	If the legions are outnumbered, they all die and no barbarian group dies. An analogous rule holds if the barbarians are outnumbered.
•	If there is an equal number of legions and barbarian groups, Romans always win.
•	Base cities defend only their direct neighbours.
•	A base city does not send help to a neighbour if it cannot send enough help.
•	At least one legion must stay in its base city.
•	If a direct neighbour is secured, the legion returns to its base city.
•	If a city is destroyed, the barbarian group moves to the closest undestroyed city. If there is more then one, then it moves randomly.
•	Different barbarian groups move independently.
In the case of double Roman dominations, the initial number of Roman legions and their positions on the graph will be defined as for minimum double Roman domination sets in Theorem 3.7. In the case of standard Roman dominations, the layout of Roman legions will be defined as for minimal Roman domination sets [14], i.e. for P4 x Pn, n = 0 (mod 6) and K1 the minimal Roman domination set is:
Vi = {(1,6j + 5), (4, 6j + 2): j = 0,1,..., [6J - l} and V = {(2, 6j + 2), (3, 6j + 5) : j =0,1,..., [nJ - l}.
Vertices with initial legions are called the base vertices or base cities. The initial number and placement of barbarian groups is arbitrary, but we will put at most 2 barbarian groups into one city. We do not want to destroy all cities in the very beginning.
We consider the placement of the barbarian groups as the first move done by barbarians. The next turn is on the legions. In each turn we have to check the state of each city i.e. the
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number of barbarian groups and legions and determine their next move. The number of legions and barbarian groups is fixed (it can only decay by turns).
Because on P4 x Pn we have two symmetrical components we will consider only K1. On this component from Theorem 3.7, in the case of double Roman dominations, on the graph P4 x P12 we have 6 base cities and 3 legions in each of them. So 18 legions defend 24 cities. In case of standard Roman dominations, we have 8 base cities with total of 12 legions.
We have noticed that for P4 x P12 there exists a second layout for base vertices. It has also total sum of 12 legions, but they are placed differently while still satisfying Roman domination.
In Figure 4, Figure 5 and Figure 6, we see an initial placement for the standard and double protections.
1
Figure 4: First version of initial placement of legions for P4 x P12 according to Roman dominating set with the lowest weight [14].
Figure 5: Second version of initial placement of legions for P4 x P12 according to Roman dominating set with the lowest weight.
Now we will test the both cases simultaneously. For the standard case we will take both layouts into consideration. Further, for a fixed number of barbarians, we will reproduce 30 random possibilities of attack for each case and measure number of destroyed cities and legions. Numbers of destroyed cities and legions will be represented with their arithmetic means.
First, we compare Roman dominating set with the first layout and double Roman dominating set. As shown in Table 1, Roman dominating set of 12 legions can survive at most
A. Klobučar and A. Klobučar: Properties of double Roman domination on cardinal products .
347
Figure 6: Initial placement of legions for P4 x P12 according to double Roman dominating set with the lowest weight.
25 barbarian groups according to our simulation, while double Roman dominating set of 18 legions can survive maximum 45 barbarion groups. So 50% more legions can survive 80% more barbarian groups on P4 x P12 which is efficiency increase of 30%. Also, when all cities and legions are destroyed in case of Roman dominations, only 27% of cities and 9% of legions are destroyed for double Roman dominations.
Table 1: Average number of destroyed cities and legions at the end of the simulations.
	RD 1. layout		RD 2. layout		DRD	
Barb. legions	Destroyed cities	Destroyed legions	Destroyed cities	Destroyed legions	Destroyed cities	Destroyed legions
10	6.67	1.46	3.70	0.50	0.90	0.10
20	20.66	9.13	15.87	6.13	4.53	0.73
25	23.63	11.63	20.97	9.33	6.56	1.66
29	24	12	23.53	11.53	9.80	3.67
30	24	12	24	12	10.53	4.40
40	24	12	24	12	21.53	15.2
45	24	12	24	12	23.83	17.76
46	24	12	24	12	24	18
Second, we compare Roman dominating set with the second layout and double Roman dominating set. Now Roman dominating set of 12 legions can survive at most 29 barbarian groups. So 50% more legions can survive 55% more barbarion groups. The increase in efficiency is considerably less than for the first layout.
What is common to the second layout of Roman dominating set and double Roman dominating set is that base cities are closer and bigger. It means that it is better to have few base cities with higher number of legions than more base cities with smaller number.
5 Conclusion
In this paper bounds for double Roman domination numbers for the cardinal product of any two graphs are given. Also, the exact values are given for the cardinal product of P2 with any graph, for P3 x Pn and for P4 x Pn. Furthermore, upper and lower bounds for double Roman domination numbers of P5 x Pn and P6 x Pn are given.
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We have also created a case study in wich we have compared Roman domination and double Roman domination on a cardinal product of graphs. The case study has confirmed that double Roman domination is more efficient because for small cost we can multiple protection.
Double Roman domination can be useful even today, not only in military sense. For example, in unsecure parts of a town, where calls for police are common, there should be at least three teams ready to go out after a call. So, when two teams are gone, the thmining team can react to some new call. Such services already exist in emergency medical stations.
ORCID iDs
Ana Klobucar © https://orcid.org/0000-0002-0260-5439 References
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[2]	H. Abdollahzadeh Ahangar, M. Chellali and S. M. Sheikholeslami, On the double Roman domination in graphs, Discrete Appl. Math. 232 (2017), 1-7, doi:10.1016/j.dam.2017.06.014.
[3]	J. Amjadi, S. Nazari-Moghaddam, S. M. Sheikholeslami and L. Volkmann, An upper bound on the double Roman domination number, J. Comb. Optim. 36 (2018), 81-89, doi:10.1007/ s10878-018-0286-6.
[4]	R. A. Beeler, T. W. Haynes and S. T. Hedetniemi, Double Roman domination, Discrete Appl. Math. 211 (2016), 23-29, doi:10.1016/j.dam.2016.03.017.
[5]	M. Chellali, N. Jafari Rad, S. M. Sheikholeslami and L. Volkmann, Roman domination in graphs, in: T. W. Haynes, S. T. Hedetniemi and M. A. Henning (eds.), Topics in Domination in Graphs, Springer, Cham, volume 64 of Developments in Mathematics, pp. 365-409, 2020, doi:10.1007/978-3-030-51117-3_11.
[6]	M. Chellali, N. Jafari Rad, S. M. Sheikholeslami and L. Volkmann, Varieties of Roman domination, in: T. W. Haynes, S. T. Hedetniemi and M. A. Henning (eds.), Structures of Domination in Graphs, Springer, Cham, volume 66 of Developments in Mathematics, 2020.
[7]	M. Chellali, N. Jafari Rad, S. M. Sheikholeslami and L. Volkmann, Varieties of Roman domination II, AKCEInt. J. Graphs Comb. (2020), doi:10.1016/j.akcej.2019.12.001.
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[15]	D. A. Mojdeh, A. Parsain and I. Masoumi, Bounds on double Roman domination number of graphs, in: Proceedings of the 2nd International Conference on Combinatorics, Cryptography and Computation (I4C2017), 2017 pp. 429-434, http://i4c.iust.ac.ir/2017/ UPL/Paper17/i4c17-10 61.pdf.
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ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 19 (2020) 351-362
https://doi.org/10.26493/1855-3974.2262.9b8
(Also available at http://amc-journal.eu)
On the Smith normal form of the Varchenko matrix*
*
Tommy Wuxing Cai ©
Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2 Canada
Yue Chen ©
School of Sciences, South China University of Technology, Guangzhou 510640, PR China
Lili Mu t ©
School of Mathematics,Liaoning Normal University, Dalian 116029, PR China Received 24 February 2020, accepted 1 September 2020, published online 21 November 2020
Let A be a hyperplane arrangement in A isomorphic to Rn. Let Vq be the q-Varchenko matrix for the arrangement A with all hyperplane parameters equal to q. In this paper, we consider three interesting cases of q-Varchenko matrices associated to hyperplane arrangements. We show that they have a Smith normal form over Z[q].
Keywords: Hyperplane arrangement, Smith normal form, Varchenko matrix. Math. Subj. Class. (2020): 15A21, 52C35
1 Introduction
Let M be an n x n matrix over a commutative unital ring R. We say that M has a Smith normal form (SNF for short) over R if there are matrices P, Q G Rnxn such that det(P) and det(Q) are units in R and PMQ is a diagonal matrix diag(di, d2,..., dn) where d divides dj in R for all i < j.
* The first author and the third author would like to thank M.I.T. for their hospitality and the China Scholarship Council for their support. Part of the work was done when they were visiting the M.I.T. Department of Mathematics during the 2013-2014 academic year. They thank Richard Stanley for his comprehensive help on this work. The first author and the second author thank Naihuan Jing for his help. The authors thank the anonymous referee for his/her careful reading and helpful comments.
t Corresponding author. The author was supported by the National Natural Science Foundation of China (No. 11701249) and the Natural Science Foundation of Liaoning Province (Grant No. 2019-BS-152).
E-mail addresses: caiwx@scut.edu.cn (Tommy Wuxing Cai), 353043301@qq.com (Yue Chen), lly-mu@hotmail.com (Lili Mu)
Abstract
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
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Recently, there is an interest in SNF in combinatorics. A survey of this topic was given by Stanley in [11]. The SNF of a matrix of a differential operator was considered by Stanley and the first author in [2], where they proved a special case of a conjecture given by Miller and Reiner [7]. In [13], interesting results concerning the SNF of random integer matrix were found.
It is well known that M has an SNF if R is a principal ideal domain (PID), but not much is known for general rings. In this paper we are interested in the integer polynomial ring Z[q]. Some matrices in Z[q]nxn do not have an SNF over R. For example, it is not hard to show that [2 0] does not have an SNF over Z[q]. However, lots of matrices in Z[q]nxn do have sNf over Z[q]. For example, it is asked whether every matrix of the form A = (qaij), where a^ are nonnegative integers, has an SNF over Z[q]. There is not a general solution to this question. But we could give a positive answer which arises from some special cases of geometrical structures. The matrices we are interested in are called Varchenko matrices (see [12]). These matrices are associated to a hyperplane arrangement (see Definition 1.2). The Varchenko matrix was studied in the papers of Varchenko [12], Schechtman and Varchenko [8], and Brylawski and Varchenko [1]. These matrices describe the analogue of Serre's relations for quantum Kac-Moody Lie algebras and are relevant to the study of hypergeometric functions and the representation theory of quantum groups [6]. Entries appearing in the diagonal of a Smith normal form of a matrix are called invariant factors. Applications of invariant factors of a q-matrix can be found in [3, 4, 9]. We are going to prove that Varchenko matrices associated to some hyperplane arrangements do have an SNF.
We use the notation and terminology on hyperplane arrangements in [10]. A finite (real) hyperplane arrangement A is a finite set of affine hyperplanes in some affine space A isomorphic to Rn.
For a hyperplane H in A, let
Ah = {H n H' : H' e A such that H' n H = 0 and H' = H}.
This is a hyperplane arrangement in the affine space H. We also write A - {H} for the arrangement from A with H removed.
Let A be a hyperplane arrangement in A. Then A is divided into some regions by these hyperplanes. Explicitly, a region is a connected component of A - |JHeA H. We let R(A) denote the set of regions of A.
Example 1.1. In the following picture, arrangement Ap is an example of the so-called peelable arrangement, which is treated in Section 2. Here we see straight lines a, b, c form a hyperplane arrangement in the plane R2. There are 7 regions of Ap which we denote by 1', 2', 3', 1, 2, 3,4. (We write it in this way for the example in Section 2.) The hyperplane arrangement A contains two affine hyperplanes A = b n a, B = b n c (two points in b).
Arrangement Ap is also an example of the regular n-gon arrangement Gn, which is treated in Section 4. It is a regular triangle arrangement. (Although in Figure 1 the central triangle is not so much like a equilateral triangle. This does not matter, because the Varchenko matrix that we are concerned with is a topological invariant.) As another example for the n-gon arrangement, a picture of the pentagon arrangement Gs is given in Section 4.
Arrangement C4 in Figure 1 is an example of arrangement Cn, which is treated in Section 3.
T. W. Cai, Y. Chen and L. Mu: On the Smith normal form of the Varchenko matrix	353
d x c
(a) Arrangement Ap
(b) Arrangement C4
Figure 1: Arrangements Ap and C4.
Definition 1.2. Let A be a finite hyperplane arrangement and R( A) its set of regions, and let aH for H e A be indeterminates. The Varchenko matrix V = V (A) is indexed by R(A) with the entries given by
Vrr =	aH,
HeSepA(R,R')
(1.1)
where Sep^(R, R') is the set of hyperplanes in A which separate R and R'. We write Vq = Vq (A) for V(A) when we set each aH = q, an indeterminate, and call Vq the q-Varchenko matrix of A.
Thus (Vq)RR = q#Sep(R,R ). Also note that V(A) and Vq(A) are symmetric matrices with 1's on the main diagonal.
We are interested mostly in the q-Varchenko matrix Vq. We are going to prove that Vq(A) has an SNF over the ring Z[q] for the peelable arrangements (in Section 2), arrangement Cn (in Section 3) and regular n-gon arrangement Gn (in Section 4). (Since this ring is not a PID, an SNF does not a priori exist.) In Section 5, we compute the SNF of the Varchenko matrices for two arrangements which are not included in the previous sections.
2 Peelable hyperplane arrangements
Example 2.1. Let us look at the arrangement Ap in Example 1.1. Its Varchenko matrix
Vq = Vq (Ap) is
Vq
1	q	q2	q	q2	q3	q2 "
q	1	q	q2	q	q2	q3
q2	q	1	q3	q2	q	q2
q	q2	q3	1	q	q2	q
q2	q	q2	q	1	q	q2
q3	q2	q	q2	q	1	q
q2	q3	q2	q	q2	q	1
where the columns are indexed by the regions in the order 1', 2', 3', 1,2,3,4, and so are the rows. We will briefly show that this matrix has an SNF. We write Vq as a block matrix the
a
b
b
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way it is partitioned:
Vq
A1	B1	C1
A2	B2	C2
As	Bs	Cs
Notice that (Bi, Ci) = q(B2, C2) and A2 = qA^ (This is not a coincidence. We see that [B1, C1] is the submatrix indexed by 1', 2', 3' (rows) and 1, 2, 3,4 (columns), while [B2,C2] is the submatrix indexed by 1,2,3 (rows) and 1,2,3,4 (columns). There is one more line, line b, to separate regions i' and j' than regions i and j.) We can multiply by the following matrix on the left to cancel B1:
P
Is -qls 0 0 Is 0 0 0 1
We have
As Vq is a symmetric matrix, so is PVPl. We thus have
	A1 - qA2	0	0
PVq =	A2	B2	C2
	As	Bs	Cs
	A1 - qA2	0	0
PVq P1 =	0	B2	C2
	0	Bs	Cs
(1 - q2)Ai 0
0
M1
where we write
M1 =
B2 Bs
C2 Cs
and we use that A2 = qA1. The matrix A1 is the q-Varchenko matrix of A£. (See Example 1.1 for the notation A£). The matrix M1 is the q-Varchenko matrix of Ap - {b}. We can use induction to transform PVqP1 into an SNF.
This example motivates us to define a peelable hyperplane in an arrangement.
Definition 2.2. Let A be a finite hyperplane arrangement and H be a hyperplane in A. We say that H is peelable (from A) if there is one side Hf of H such that if R is a region of A and R is in Hf, then R n H is the closure of a region of AH.
For example, the hyperplane b is peelable from Ap in Example 1.1. Let us see why this is. On the side above b there are three regions 1', 2' and 3'. For each one of these regions, the intersection of its closure with b is actually a closure of a region of A£. For instance, the closure of region 2' intersects b at a line section AB, and this line section is actually a closure of a region of A£. (In fact Ab has 3 regions: the part to the left of A, the part between A and B, and the part to the right of B.)
Theorem 2.3. Assume that H is peelable from A. Then there is a matrix P with entries in Z[q] such that det(P) = 1 and
PVq (A)P' :
(1 - q2)Vq (AH ) 0
0
Vq (A-{H })
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Remark 2.4. Under the same assumption, a similar result can be given for the Varchenko matrix V(A), and the proof is almost the same. Using this result, we can prove that the Varchenko matrix V(A) associated to a peelable hyperplane arrangement (as defined below) has a "diagonal form" in Z[aH : H € A], that is, we can find matrices P, Q whose determinants are units and PV (A)Q is a diagonal matrix. Let us mention that, subsequent to our work, Gao and Zhang [5] gave a necessary and sufficient condition on an arrangment A for V (A) to have a diagonal form.
The main idea of the proof of this theorem is in the previous example. We will give a rigorous proof in a while, in order to make sure there is no gap that might have occurred when we move from the more visualizable two-dimensional example.
Iteratively using this result, the Varchenko matrices of a special type of hyperplane arrangement can be shown to have an SNF.
Definition 2.5. Let A = {Hi, H2,..., Hm} be a finite hyperplane arrangement. We inductively define A to be peelable as follows.
1.	If m = 1 then A = {H1} is peelable.
2.	If there is one peelable hyperplane H in A such that both A - {H} and AH are peelable, then we say that A is peelable.
Now it is easy to see that we have the following result.
Corollary 2.6. The q-Varchenko matrix Vq (A) of a peelable hyperplane arrangement A has an SNF over Z[q]. Moreover, its SNF is of the form
diag((1 - q2)"1, (1 - q2)"2,..., (1 - q2)"^),
where 0 < n1 < n2 < ■ ■ ■ < nr is a sequence of nonnegative integers and r is the number of regions of A.
We will need the following two results, which are not hard to prove.
Lemma 2.7. Let H be a hyperplane in A. Assume that R is a region such that R n H contains a point which is an interior point of some region R1 in AH. Then R = R n H.
Lemma 2.8. Let H be a hyperplane in A. Assume that R is a region such that R n H is the closure of some region of AH. Then there is a unique region R' on the other side of H such that R' n H = R n H.
To simplify the wording of the proof of Theorem 2.3, we introduce a new notation.
Definition 2.9. Let A be a hyperplane arrangement. Let R1, R2 be two subsets of R(A). We denote by Vq (R1, R2) the submatrix of Vq (A) with rows indexed by R1 and column indexed by R2.
Now let us prove Theorem 2.3. Assume that H is peelable from A and Hf is a side of H with the properties as in Definition 2.2. Let R1,R2,... ,Rs be the set of the regions in Hf. Let H, A, Hf be as in the Definition 2.2. Let R = {1', 2',..., r'} denote the set of regions in Hf, and let R1 = {1,2,..., r} denote the corresponding regions on the other side of Hf as given by the previous lemma. Let R2 = {r +1,..., r + s} be the set of other
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regions. Let R' = {1, 2,..., r +1}, i.e., R' is the union of Ri and R2. It is not difficult to prove the following facts:
Vq (Ri, Ri) Vq (R', R') Vq (Ri, R') Vq (Ri, Ri)
Vq (AH ) Vq (A-{H }) qVq (Ri, R') qVq(Ri, Ri).
The q-Varchenko matrix V = V( A) has the following block matrix form: Vq (A)
Now an argument similar to Example 2.1 can be applied to prove the theorem.
Vq (Ri, Ri) Vq (Ri, Ri) Vq (R'i, R2) Vq (Ri, Ri) Vq (Ri, Ri) Vq (Ri, R2) Vq (R2, Ri) Vq (R2, Ri) Vq (R2, R2 )
3 The case that all lines go through the same point
From now on, we consider hyperplane arrangements in R2. Define Cn to be the arrangement consisting of n lines intersecting in a common point in R2. We prove that the q-Varchenko matrix V(n) associated to Cn has a Smith normal form (over Z[q], as usual). This matrix has the form
V(n)
1 q q2 q3
2
qq
2 3 4
q q2 q3 q4
q
„n-1
q
i-1
qn—i qn-2
q
q2
1
Remark 3.1. This matrix is an example of circulant matrices C(ci, c2,. defined by
C(ci, C2, . . . , Cn)
ci	c2	c3 . .	. cn—i	cn
cn	ci	c2 . .	. cn —2	cn—i
cn—i	cn	ci . .	. cn—3	cn—2
c2
c3 c4
Ci
.., cn) which is
(3.1)
We see that V(n) is circulant because the regions of Cn are in a circular mode. Similar but more complicated situations occur in the regular n-gon arrangement, which is considered in the next section.
Proposition 3.2. Let n be a positive integer. Then the Varchenko matrix V(n) has the following Smith normal form over Z[q]:
diag(1, 1 - q2, . . . , 1 - q2, (1 - q2)2, (1 - q2)(1 - q2n), . . . , (1 - q2)(1 - q2n)). (3.2)
n	n —2
Proof. First successively apply the row operations r - qri—i (i = n, n - 1,..., 2), rn+i - qrn+i+i (i = 1, 2,..., n - 1), r2n - qri. This transforms V(n) into the block
1
n
q
q
c
n
T. W. Cai, Y. Chen and L. Mu: On the Smith normal form of the Varchenko matrix	357
matrix
1 a O M
q O
0 ¡3 1 - q2
where M is a 2(n — 1) x 2(n — 1) matrix, a, 3 are row vectors, O is a zero column vector and ¡'s components are all multiples of 1 — q2. It's easy to see that we only need to find the Smith normal form of M. Factoring 1 — q2 out of M, one finds that
where
M = (1 — q2)
%-2
A = ^ qk Tk, B = J2
A B Bt A1
2
k=0
=q
k=0
■t-1-kfrrit\ k
(T l)A
and T = (tj) with tijj = 6i+i,j. Note that A is a unitriangular matrix; in particular, it is invertible in Z[q]. Multiplying M on the left by
we transform M into
P
(1 — q2)
I
—BlA
O
-i
A
B
OA1- B lA-iB
We see that we only need to find the Smith normal form of A1 — BlA iB, but it can be seen from the following lemma that its SNF is
diag(1 — q2,1 — q2n,..., 1 — q2n).
n-2
Now the SNF of V(n) follows.
Lemma 3.3. Let m x m matrix T = (tj) with ti,j = . Let
(3.3)
□
i
i
A =E qkTk, B = ^ qm-k(T')k.
k=0
k=0
Then the matrix
C = (Im — qT' )(A' — B' A B)
is equal to a matrix with first row
(1 — q2, q3 — q2m+i, q4 — q2m, q5 — q2m-i, . . . , qm+i — qm+3)
the other diagonal entries all equal to 1 — q2m+2, and all other entries zero. Proof. First A-i = Im — qT, so
BlA-i = qmIm +y(q
m-i
m + E (qm-k — qm+2-k)Tk. k=i
I
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Ars Math. Contemp. 19 (2020) 173-187
Then one computes BlA 1B and finds it is equal to
M =
q""
qm+l

,2m+l
qm+l
qm+2
q5 - q
q2m 2m+l
„6
qm+2 qm+3
m+l
m+3
qm+2 _ qm+4 qm+3 - qm+5
q2m-1 _ q2m+1
2m
Now let N = (Im _ qTl)M. We find that the first row of N is the same as that of M, the other diagonal entries of N are all equal to q2m+2 and all other entries are zero. Now we see that C is as claimed in the lemma since
C = (Im _ qTl)Al _ (Im _ qTl)M = Im _ (Im _ qTl)M = Im _ N.
□
4 The case of regular n-gon arrangement Gn
Let Gn be the arrangement in R2 obtained by extending the sides of a regular n-gon. Let Vq(Gn) be the Varchenko matrix associated to Gn. We are going to prove the following
Theorem 4.1. Let Vq(Gn) be the Varchenko matrix associated to the regular n-gon arrangement Gn. A Smith normal form of Vq (Gn) over Z[q] is
diag(1,1 _ q2,..., 1 _ q2,(1 _ q2)2, . . . , (1 _ q2)2)
(4.1)
(p-l)n
where p is the integer part of (n + 1)/2.
The above result can be proved by using some results and tools in [3, 12]. But we want to prove it directly. First, it is easy to calculate the number of regions of the arrangement Gn. For instance, one uses the formula that the number of regions is one more than the sum of the number of the lines and the number of intersection points.
Lemma 4.2. The number of the regions associated to the regular n-gon arrangement Pn is np + 1, where p is the integer part of (n + 1) /2.
The main idea of the proof of Theorem 4.1 is to group the regions by their shapes. We then write the Varchenko matrix as a block matrix. The columns of each block are labeled by regions of a same shape and so are the rows of a block. For regions of the same shape, we order them clockwise. The key property of this treatment is that each block is a circulant matrix. Once we write the block matrix down, it will be relatively easy to do cancelations and turn it into an SNF, although it takes some space to write the process down. To show how to write the block matrix, we consider the example of G5. Then in the proof we write the block matrix for general n and then do the cancelation.
Example 4.3. We mark the regions of G5 (see Figure 2) as in the following.
They are regions A(j) (i = 1,2,..., 5; j = 1,2,3) together with a unmarked central region. Note that we mark the regions according to their shape. Precisely, for each j, the shape of the A(j) for i = 1, 2,..., 5 are the same. Let us call them the regions of type
2
3
4
q
q
q
3
q
4
q
T. W. Cai, Y. Chen and L. Mu: On the Smith normal form of the Varchenko matrix	359
v(3) 4
v(3)
A.
(3) 2
(3)
Figure 2: Arrangement G5.
(j) A(j)
, A
(j)
We
j. For regions of the same type, we label them clockwise as AY', A call AY' the leading region of the type j regions. The union of the three leading regions A^1', Af, a13' is the region inside an exterior angle of the pentagon. (So is the union of three region Ai1', Ai2', Ai3'.) We obtain the Varchenko matrix
Vq (Gs) =
1	Qi	Q2	Qs'
Qi	Eii Ei2 Eis
Q2	E2i E22 E2S
Qs Esi Es2 Ess
where the first (block) column is indexed by the central non-marked region. For j = 2, 3,4, the jth block column is indexed by the type j regions. The block rows are indexed in the same way. For example, the rows of the matrix Ei2 are indexed by the type 1 regions and the columns of it are indexed by type 2 regions. Because regions of the same type are ordered in a circular mode, the blocks Ej should all be circulant matrices (see Remark 3.1). In fact, it can be checked that the blocks are as follows
Qk = (qk,qk,qk, qk, qk) Eii = C (1, q2, q2, q2, q2 ) E22 = C (1,q2,q4,q4,q2 )
for k = 1, 2, 3,
Ei2 = C(q, qS, qS, qS, q)
E23 = C(q, qS,q5,qS,q)
Eis = C (q2,q4,q4,q2,q2 ), Ess = C(1,q2,q4,q4, q2).
We then use Gaussian elimination (in blocks) to turn the matrix into an SNF. For instance, at the beginning, we subtract the q times of the third block row (Q2 E21 E22 E23) from the fourth block row (Q3 E31 E32 E33).
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Proof. We write the Varchenko matrix of Gn in the following form of block matrix:
Vq (Gn) =
1	Qi	Q2 . .	. Qp
Q1	En	E12 .	. Ei p
Q2	E21	E22 .	. E2 p
Qpp Epi EP2 .
Ep
where Ekl = E/k, Qk is the row vector
Qk = (qk ,qk ,...,q
(4.2)
and Ej (i < j) is a circulant matrix:
Ei.
cfqj-i, qj-i+2, qj-i+4, . . . , qj-i+2(i-i), qi+j, . . .
i+j
qj+i-2 q>+i-4
q_,
n+1-i-j
.,qj+i-2(i-l),qj-i,...,qj-i
v-v-'
Now we apply Gaussian elimination to Vq(Gn) and transform it into the desired diagonal form. We do this in blocks and we will use the multiplication of elementary block matrices to realize the elimination. We proceed in four steps.
Step 1: We first apply some row eliminations. Let
R
i=
"qInXl In
and Rk
-qIn In
for k > 2, where Inxl is a column of n 1's and Rk comes from the (block) identity matrix by adding the —q times of its (k - 1)th block row to it's kth block row. Now compute the matrix Mi = R1R2 • • • RpVq(Pn).
Step 2: We apply some column eliminations. Let
Si =
i -qllXn In
In-
and Sk
In -qIn
for k > 2, where Iixn is a row of n 1's and Sk comes from the (block) identity matrix by adding the —q times of its (k — 1)th block column to its kth block column. (So Sk = Tkl.) Now compute the matrix M2 = MiSp • • • S2Si.
k
i
I
i

I
I

I

i
I

I

I
T. W. Cai, Y. Chen and L. Mu: On the Smith normal form of the Varchenko matrix	361
Step 3: We apply some more row eliminations. Let
Tk
-qJ In
with J
0	1	0 .	.0	0
0	0	1 .	.0	0
0	0	0 .	.0	1
1	0	0 •	• 0	0
where Tk come from the (block) identify matrix by adding the —q J times the kth block row to the (k + 1)th block row. Now compute M3 = Ti • • • Tp-iM2. We find the Varchenko matrix is now transformed to
1
0	0	0 .	.0	0
D	N12	N13 .	. Nip-1	Nip
0	D'	N23 .	. N2p-i	N2p
0	0	0 .	. D'	Np-ip
0	0	0 .	.0	D'
M3 =
where D = (1 — q2)In, D' = (1 — q2)2In and all non-diagonal entries are the multiple of the diagonal entry on the same row. This ensures that we can do the following:
Step 4: We apply more column eliminations to cancel the non-diagonal entries. This does not change the diagonal entries of M3. We finish the proof as the diagonal of M3 is the same as that of (4.1).	□
i
I
I

5 Two more examples
We now simply say that a hyperplane arrangement A has SNF if its Varchenko matrix Vq (A) has an SNF over Z[q]. We can use Theorem 2.3 to give more examples of hyperplane arrangements who have SNF. For example, starting from an arrangement which has SNF, for instance Cn, we can keep adding straight lines to it. As long as every time the line added does not separate the set of intersection points of the previous arrangement, the new arrangement will have SNF. This helps us to construct lots of examples of hyperplane arrangements having SNF. We now give two examples which can not be constructed this way. We found that they both have SNF.
1.	The Shi arrangement S3 with hyperplanes x® — Xj =0,1 for 1 < i < j < 3. We write the multiplicity of a diagonal element in brackets following that entry. For instance, 1 — q2 [3] indicates that 1 — q2 occurs three times as a diagonal element of the SNF. The diagonal elements of the SNF of Vq(S3) are 1 [1], 1 — q2 [6], (1 — q2)2 [6], and (1 — q2)(1 — q6) [3].
2.	Define a hyperplane arrangement A in R3 by the equations x = 0, y = 0, z = 0,
x — y — z = 0. We verified that its q-Varchenko matrix has an SNF over Z[q], with
diagonal entries 1 [1], 1—q2 [4], (1 — q2)2 [6], (1 — q2)3 [2], and (1—q2)2(1 —q8) [1].
Based on the previous examples, it is natural to consider the following problem.
Problem 5.1. Do all hyperplane arrangements have SNF?
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Ars Math. Contemp. 19 (2020) 173-187
ORCID iDs
Tommy Wuxing Cai © https://orcid.org/0000-0001-9816-4115
Yue Chen © https://orcid.org/0000-0002-8173-0894
Lili Mu © https://orcid.org/0000-0003-0911-519X
References
[1]	T. Brylawski and A. Varchenko, The determinant formula for a matroid bilinear form, Adv. Math. 129 (1997), 1-24, doi:10.1006/aima.1996.1530.
[2]	T. W. Cai and R. P. Stanley, The Smith normal form of a matrix associated with Young's lattice, Proc. Amer. Math. Soc. 143 (2015), 4695-4703, doi:10.1090/proc/12642.
[3]	G. Denham and P. Hanlon, On the Smith normal form of the Varchenko bilinear form of a hyperplane arrangement, Pacific J. Math. 181 (1997), 123-146, doi:10.2140/pjm.1997.181. 123.
[4]	G. Denham and P. Hanlon, Some algebraic properties of the Schechtman-Varchenko bilinear forms, in: L. J. Billera, A. Bjorner, C. Greene, R. E. Simion and R. P. Stanley (eds.), New Perspectives in Algebraic Combinatorics, Cambridge University Press, Cambridge, volume 38 of Mathematical Sciences Research Institute Publications, pp. 149-176, 1999, papers from the MSRI Program on Combinatorics held in Berkeley, CA, 1996 - 1997.
[5]	Y. Gao and Y. Zhang, Diagonal form of the Varchenko matrices, J. Algebraic Combin. 48 (2018), 351-368, doi:10.1007/s10801-018-0813-7.
[6]	P. Hanlon and R. P. Stanley, A q-deformation of a trivial symmetric group action, Trans. Amer. Math. Soc. 350 (1998), 4445-4459, doi:10.1090/s0002-9947-98-01880-7.
[7]	A. Miller and V. Reiner, Differential posets and Smith normal forms, Order 26 (2009), 197228, doi:10.1007/s11083-009-9114-z.
[8]	V. V. Schechtman and A. N. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139-194, doi:10.1007/bf01243909.
[9]	W. C. Shiu, Invariant factors of graphs associated with hyperplane arrangements, Discrete Math. 288 (2004), 135-148, doi:10.1016/j.disc.2004.07.009.
[10]	R. P. Stanley, An introduction to hyperplane arrangements, in: E. Miller, V. Reiner and B. Sturmfels (eds.), Geometric Combinatorics, American Mathematical Society, Providence, Rhode Island, volume 13 of IAS/Park City Mathematics Series, pp. 389-496, 2007, doi: 10.1090/pcms/013/08, lectures from the Graduate Summer School held in Park City, UT, 2004.
[11]	R. P. Stanley, Smith normal form in combinatorics, J. Comb. Theory Ser. A 144 (2016), 476495, doi:10.1016/j.jcta.2016.06.013.
[12]	A. Varchenko, Bilinear form of real configuration of hyperplanes, Adv. Math. 97 (1993), 110144, doi:10.1006/aima.1993.1003.
[13]	Y. Wang and R. P. Stanley, The Smith normal form distribution of a random integer matrix, SIAMJ. Discrete Math. 31 (2017), 2247-2268, doi:10.1137/16m1098140.
ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 19 (2020) 363-374 https://doi.org/10.26493/1855-3974.1851.b44
(Also available at http://amc-journal.eu)
Frobenius groups which are the automorphism groups of orientably-regular maps*
Hai-Peng Qu ©
School of Mathematics and Computer Science, Shan Xi Normal University,
ShanXi, P.R.C.
School of Mathematics and Information Science, Yan Tai University, Yan Tai, P.R.C. Received 12 November 2018, accepted 21 September 2020, published online 22 November 2020
In this paper, we prove that a Frobenius group (except for those which are dihedral groups) can only be the automorphism group of an orientably-regular chiral map. The necessary and sufficient conditions for a Frobenius group to be the automorphism group of an orientably-regular chiral map are given. Furthermore, these orientably-regular chiral maps with Frobenius automorphisms are proved to be normal Cayley maps. Frobenius groups conforming to these conditions are also constructed.
Keywords: Frobenius group, (orientably) regular map, automorphism group. Math. Subj. Class. (2020): 05C25, 05C30
1 Introduction
Maps are 2-cell embeddings of graphs in compact, connected surfaces. A flag of a map is a topological triangle whose corners are a vertex, the midpoint of an edge incident with the vertex, and the midpoint of a face incident to both the vertex and the edge. Thus, the supporting surface of any map can be decomposed into flags (considered as closed discs bounded by the triangles).
*The authors want to thank Professor Gareth A. Jones of University of Southampton for his valuable suggestions and encouragement. We also thank the referees for their helpful comments.
tCorresponding author. Supported by NSFC (No. 11771258, 11471198, 11671347, 61771019, 11671276), NSFS (No. ZR2017MA022, ZR201910270119) and J16LI02.
E-mail addresses: orcawhale@163.com (Hai-Peng Qu), wang_yan@pku.org.cn (Yan Wang), pktide@163.com (Kai Yuan)
Yan Wang t ©, Kai Yuan©
Abstract
©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/
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It is well known that the automorphism group of a map acts semi-regularly on its flags. If the automorphism group of a map is regular on the flags, then the map is called regular. Regular maps have the largest automorphism groups, acting regularly on flags of the map. Similarly, orientably-regular maps are maps in orientable surfaces that have the largest orientation preserving automorphism groups acting regularly on darts (edges with direction).
Regular and orientably-regular maps constitute the most meaningful generalization of the Platonic solids. Early recognition of the importance of regular maps in modern mathematics goes back to Kepler [12]; more recent development of the theory of maps was closely related to the theory of map colorings, with the topic of highly symmetric maps always at the center of interest. The study of regular maps is nowadays considered one of the 'classical' areas of mathematics (e.g., Heffter [7], Klein [13], Dyck [5], or Burnside [3]).
A group G acting on a set X is said to act regularly, if for any pair of elements x, y G X there exists a unique element g G G mapping x to y, xg = y. In such a case, X can be identified with the elements of G, and consequently, any mathematical structure with an automorphism group acting regularly on its base set can be identified with the group itself, the building blocks of the structure being identified with cosets of stabilizers of some blocks. This identification has been used in the theory of regular and orientably-regular maps as well and we just sum up the basics, referring for details to [11] and [2] for the theory of maps on orientable and on general surfaces, and to [16] for a recent survey of the theory of regular and orientably-regular maps. In all the forthcoming group presentations we will assume that the listed exponents are the true orders of the corresponding elements.
A finite regular map M can in this way be identified with a (partial) three-generator presentation of a finite group G, isomorphic to the automorphism group Aut(M) of M, of the form
G = (x, y, z | x2, y2, z2, (xy)2, (yz)£, (zx)m, .. .)	(1.1)
where the dots indicate possible presence of additional relators (at least one if the carrier surface of the map is not simply connected). In particular, all vertices of M have degree I and all the face boundary walks in M have length m; we will often refer just to face length m. The pair m) is the type of the regular map M. In such a representation of M, its flags are elements of G, the darts are (say) right cosets of the subgroup (x), while edges, vertices and faces are right cosets of the dihedral subgroups (x, y), (y, z) and (z, x) of order 4, 2^ and 2m, respectively. The three generators x, y, z correspond to involutory automorphisms of M taking a fixed flag onto its three neighboring flags, and the three dihedral subgroups correspond to the edge-, vertex- and face-stabilizers of M.
We will write M = Map(G; x, y, z) to formally identify a regular map M with a group presentation as in (1.1). The algebraic situation with finite orientably-regular maps is similar. Each such map M can be identified with a partial two-generator presentation of a group H, isomorphic to the group Aut+(M) of orientation-preserving automorphisms of M, of the form
H = (p, A | /,A2, (pA)m,...) .	(1.2)
Here, elements of H represent darts of M; right cosets of the cyclic groups (A), (p) and (pA) represent edges, vertices and faces of M. The generators A and p, stabilizing an edge e and a vertex v incident to e, represent a half-turn of M about the center of e and a turn of M about v in accord with a chosen orientation of the carrier surface of the map. Again, the pair (¿, m) is the type of the map, and we will use the notation M = Map(H; p, A) in this case.
H.-P. Qu, Y. Wang and K. Yuan: Frobenius groups which are the automorphism groups of.
365
If a regular map M = Map(G; x, y, z) is orientable (meaning that its carrier surface is orientable), M is also orientably-regular, with Aut+(M) = (p, A) for A = xy and p = yz. In fact, a regular map Map(G; x, y, z) is orientable if and only if the subgroup (xy, yz) has index 2 in G. Reversing this line of thought, an orientably-regular map M = Map(H; p, A) may also be regular. It happens if and only if the map admits an orientation-reversing automorphism, which (see e.g. [16]) is equivalent to the existence of an automorphism of H that fixes A and inverts p. In such a case we call the orientably-regular map M reflexible; otherwise, that is, when H = Aut+(M) = Aut(M), the map is called chiral.
A Cayley graph Cay(G, X) is a graph whose vertex set can be identified with the elements of a group G generated by a set X closed under taking inverses and not containing the identity 1G, with the pairs of adjacent vertices consisting of all pairs g, gx with g G G and x G X .A graph r is isomorphic to a Cayley graph Cay(G, X) if and only if Aut r contains a subgroup G acting regularly on the vertices of r [15]. A Cayley map is an orientable map M that admits a group of orientation preserving automorphisms G acting regularly on its set of vertices. Therefore, the underlying graphs of Cayley maps are Cayley graphs. It turns out that many of the orientably-regular maps obtained in the forthcoming sections fall in the class of Cayley maps the theory of which (without regularity assumptions) was initiated in [14] and further developed e.g. in [8] and [4].
An orientably-regular Cayley map can therefore be distinguished by M = Map(H; p, A), where H = J(p) for some subgroup J < H such that J n (p) = 1, vertices of M are right cosets of (p) in H, and the underlying graph of M is a Cayley graph Cay( J, S) for some unit-free inverse-closed generating set S of J. In the even more special instance when J is normal in H, i.e., when H is a semi-direct product J x (p), we speak about a normal (orientably-regular) Cayley map. In this case, conjugation by p induces an automorphism p of J and its restriction n = np to S is a cyclic permutation of S. It turns out that either all elements in S are involutions, or none of them is and then s-1 = sp£/2 = p-i/2spi/2 for every s G S, where I is the order of p (necessarily even in this case). Moreover, since we also know that J(p) = (A, p), the involution A can be taken to be equal to an arbitrary element of S in the all-involutions case, or to sp£/2 for an arbitrary s G S if no element in S is involutory.
In our paper we address the natural question whether for a given finite Frobenius group G there exists some orientably-regular or even regular map whose automorphism group is G. In Section 2, we list some properties of Frobenius groups which we will refer to in Section 3. In Section 3, necessary and sufficient conditions (Theorems 3.3, 3.5 and 3.6) for a Frobenius group to be the automorphism group of an orientably-regular chiral map are given. The Frobenius groups conforming to these conditions are also constructed.
2 Frobenius groups
A Frobenius group is a transitive permutation group G on a set Q which is not regular on Q, but has the property that the only element of G which fixes more than one point is the identity element. It was shown by Thompson [17, 18] that a finite Frobenius group G has a nilpotent normal subgroup K, called the Frobenius kernel, which acts regularly on Q. Thus, K is the direct product of its Sylow subgroups and G is the semidirect product K x H, where H is the stabilizer of a point of Q. Because of the vertex transitivity of the action, any two point stabilizers are conjugate. As a result, every point stabilizer has the form (hk)-1H(hk) = k-1Hk = Hk for some h G H and k G K. Each point stabilizer is
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called a Frobenius complement of K in G, so the choice of Frobenius complement is not unique. Because of the regularity of K acting on Q, one may identify Q with K such that K acts on itself by multiplication. Moreover, Gorenstein [6, pp. 38, 339] showed that every element of H \ {1} induces an automorphism of K by conjugation which fixes only the identity element of K. Combining all these results we give a lemma to express the relation between a Frobenius group and its Frobenius kernel as well as its Frobenius complements.
Lemma 2.1. Let G = K x H be a Frobenius group, where K is the Frobenius kernel and H is a Frobenius complement. Then, G can be divided in the following two ways.
(1)	G = UkeKHk, where Hki n Hk2 = 0 for any two different elements ki,k2 G K;
(2)	G = (UkeKHk) U K, where Hk = k-iHk denotes the conjugation of H by k, and Hkl n Hk2 = Hk n K = {1} for any elements ki, k2, k in K and ki = k2.
Given a(several) Frobenius group(s), one can get new Frobenius groups. In the following Lemmas 2.3 and 2.4, we give two methods to get new Frobenius groups from original ones.
Lemma 2.2 ([19, Lemma 3.8, p. 13]). Assume A, B are two groups and B acts on A. If A has a subgroup P which is invariant under the action of B, (|B|, \P|) = 1 and (Pa)b = Pa for some a G A and each b G B, then there is an element x G Pa such that xb = x for every b G B.
Lemma 2.3. Assume G = K x H, 1 < N < K and N < G. Then G is a Frobenius group with H as a Frobenius complement if and only if both N x H and K/N x H are Frobenius groups with H as a Frobenius complement.
Proof. Assume G = K x H is a Frobenius group with H as a Frobenius complement. It is obvious that N x H is a Frobenius group with H a Frobenius complement. So we only need to show that K/N x H is a Frobenius group. If not, then there is an h G H such that h fixes some non-identity element of K/N. That is, there is an element k G K but k G N such that Nk is fixed by h. Consider K x (h). From Lemma 2.2, there is an element x G Nk which is fixed by h. It is obvious that x = 1, and so h fixes at least two elements in K. This contradicts the assumption of G being a Frobenius group.
Conversely, assume both N x H and K/N x H are Frobenius groups with H as a Frobenius complement. If G is not a Frobenius group, then there exists 1 = k G K and 1 = h G H such that kh = k. Since N x H is a Frobenius group, k G N. Thus Nk = 1 in K/N. Clearly, (Nk)h = Nkh = Nk. This contradicts the assumption of K/N x H being a Frobenius group.	□
Lemma 2.4. Let Ki x H and K2 x H be two Frobenius groups. Then, (Ki x K2) x H is a Frobenius group, where H acts on Ki x K2 by (k\k2)h = khk!h, for any elements k1 K1, k2 K2 and h H.
Proof. Note that each non-identity element h H fixes exactly the identity element of
Ki x K2.	□
Lemma 2.5. Let G = K x H be a Frobenius group. For each g G G \ K, it satisfies the following two relations:
(1) (g) n K = {1};
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(2) As an element in the quotient group G/K, Kg has order o(g), where o(g) denotes the order of g in group G.
Proof. According to Lemma 2.1, there is an element k G K such that g G Hk .So (g) n
K < Hk n K = {1}. As a result, o(Kg) = |K(g)/K| = |(g)|/(g) n K| = |(g)| = o(g).	□
Corollary 2.6. Let G = K x H be a Frobenius group. For each h G H, h =1 and for each k G K, the orders of h, kh and hk are equal, that is o(h) = o(kh) = o(hk).
Proof. Note that Kh = K(kh) = K(hk), so o(h) = o(kh) = o(hk) according to Lemma 2.5(2).	□
Remark 2.7. If a group G = N x P is not a Frobenius group, then it may not satisfy the results in Lemma 2.5. For example, take G = SL2 (3) = Q8 x Z3. Let x = (-1 1). There
x G G \ Q8, o(x) = 6. But (x) n Q8 = (x3) = 1 and o(Q8x) = 3 = o(x).
3 Maps having Frobenius groups as automorphism groups
The following Lemma 3.1 will be referred to several times in this paper. The result is known and one can prove it very quickly. But for easy reference, we give a short proof.
Lemma 3.1. Let G be a finite group. If there is an involution t g Aut(G) such that t only fixes the identity element of G, then t maps each element in G to its inverse and G is an abelian group of odd order.
Proof. According to the property of t, one can check that G = {g-1gT | g G G}. Clearly (g-1gT)T = (g-1gT)-1. It follows that t maps each element in G to its inverse. So, for any two elements a, b G G, one can get (ab)T = b-1a-1 = aTbT = a-1b-1. That is to say, G is an abelian group. Since t only fixes the identity element of G, the group G does not have involutions. Thus, G is of odd order.	□
If a G L, then we use (a)L to denote the group generated by the elements x-1ax for
x G L.
Theorem 3.2. Other than dihedral groups of order 2n for any odd integer n, Frobenius groups cannot be the automorphism groups of regular maps.
Proof. Let G = K x H be a Frobenius group. If G can be the automorphism group of a regular map, then G has the following generating relations:
G = (x, y, z | x2, y2, z2, (xy)2, (yz)k, (zx)m,.. .),
where 2, k, m are the true orders of xy, yz and zx, respectively. As G/K = (Kx, Ky, Kz) = H, |H| is even. By Lemma 3.1, K is an abelian group of odd order and H has a unique involution. Consequently, H = Z2. Moreover, H is a Sylow-2 subgroup of G.
It is easy to see that (x, y) is a 2-group. So |(x, y)| < |H| = 2. Thus (x, y) = (xy). It follows that x = 1 or y = 1. In either case, G is a dihedral group of order 2n for some odd integer n. In this situation, the map is an embedding of a semi-star of valency n in the sphere or the disc, or the dual of the latter, an embedding of a circuit of length n in the boundary of the disc. It is obvious that these two infinite families of maps are reflexible with
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their full automorphism groups being the dihedral groups of order 2n. Apart from these two infinite families of maps, the only other possibilities for Frobenius automorphism groups are orientably-regular chiral maps.	□
According to Theorem 3.2, we only need to concentrate on Frobenius groups which can be automorphism groups of orientably-regular chiral maps. There are several well-known infinite families of examples of these, such as the embeddings of complete graphs Kn [9], Paley graphs, and generalized Paley graphs [10]. In the following Theorem 3.3, we will give the necessary conditions that a Frobenius group G = K x H should satisfy to be the automorphism group of an orientably-regular chiral map.
Theorem 3.3. Let G = K x H be a Frobenius group. If G = (p, A | pk, A2, (pA)m,.. .), k,m > 3, is the automorphism group of an orientably-regular chiral map M = Map(G; p, A), then one of the following two cases will happen.
(1) H is a cyclic group of even order and K is an abelian group of odd order. There are two subcases corresponding to the parity of k.
(1.1) If k is even, then H = Zk and
2, if k = 2 (mod 4), k, if k = 0 (mod 4).
k = 2 (mod 4), M has ^ vertices, ^ edges, ^^ faces and the genus of the corresponding orientable surface is 1 - 1 G 1 (46.-fc) ; when k = 0 (mod 4),
Moreover, the map M is an orientably-regular normal Cayley map of K. When
I G | .. | G | , 2 | G
^ vertices, ^ edges, -jt
K6z
4k
M has Jf vertices, ^ edges, ^ faces and the genus of the corresponding orientable surface is 1 — 1 G 1 44k-k). (1.2) If k is odd, then H = Z2k and m = 2k. The map M is an orientably-regular normal Cayley map of a group isomorphic to K x Z2. In this situation, M has ^kp vertices, ^ edges, J2kJ faces, so the genus of the corresponding orientable surface is 1 — 1 G 1 43k-k).
(2) H is a cyclic group of odd order and H = Zk, K is a 2-group and m = k. In this situation, M is an orientably-regular normal Cayley map of K. The map M has f
vertices, ^ edges, ^ faces and the genus of the corresponding orientable surface
. 1 _ I G I(4-k)
is 1	4k .
Proof. (1): If |H| is even, then there is an involution in Aut(K) which only fixes the identity element. So, K is an abelian group of odd order by Lemma 3.1. In this case, A / K and so o(KA) = 2 in the quotient group G/K. Note that p ^ K. Otherwise, by Corollary 2.6 one can get o(pA) = o(A) = 2, that is m = 2. So, o(Kp) = o(p) = k by Lemma 2.5. According to Lemma 3.1, there is only one involution in H = G/K = (Kp, KA), so KA belongs to the center of G/K which is therefore abelian.
(1.1): If k is even, then KA e (Kp). So, (Kp, KA) = (Kp) and H = Zk. According
k
to Lemma 2.1, one can assume H = (p) and A = ap2 for some non-identity element a e K without loss of generality. The vertices of M can be looked as the cosets of H.
m
H.-P. Qu, Y. Wang and K. Yuan: Frobenius groups which are the automorphism groups of.
369
Therefore, K acts regularly on the vertices of M which implies that M is an orientably-regular Cayley map of K. Now, we know that Aut(M) = K x (p). So, M is normal and from the construction method of M from G, one can get the corresponding Cayley subset
{a, ap, ap ,..., ap }. In this case, K = (a)H.
Since KA = Kpk, KpKA = Kpk+1. If k = 2 (mod 4), then m = o(pA) = o(KpKA) = o(Kpk+1) = k. The type of M is (k, k). Moreover, M has vertices,
edges, ^k^ faces and the genus of the corresponding orientable surface is 1 - |G|46fc-k). If k = 0 (mod 4), then m = k and consequently the type of M is (k, k). And M has
IGI	IGI	IGI
G vertices, hf edges, G faces and the genus of the corresponding orientable surface is
, _ | G |(4-fc) 1	4k '
(1.2): If k is odd, then KA belongs to the center of G/K which is therefore abelian. So, (Kp, KA) = (KpKA) and as a result H = Z2k. Because p G K, according to Lemma 2.1, we may assume p G H and H = (p) with p = p2. As a result, A = apk for some non-identity element a G K. Because Ap = apk+2, it follows that m = o(Ap) = o(pk+2) = 2k according to Lemma 2.6. The type of M in this subcase is therefore (k, 2k).
Let H = (p) be the index two subgroup of H and K = K x (pk) = K x Z2. It is clear that G = K x H. Now, A g K and so we have the relations G = (p, A) < (A)<p>(p) < KH = G. Therefore, K = (A)H.
The vertices of M can be looked as the cosets of H. Therefore, K acts regularly on the vertices of M which implies that M is an orientably-regular normal Cayley map of K with corresponding Cayley subset {A, Ap, Ap2,..., Apk 1}. In this case, M has ^ vertices, ^ edges, faces, so the genus of the corresponding orientable surface is 1 - |G|(43fc-fc).
(2): If |H| is odd, then A G K and so H = G/K = (Kp) is cyclic. Similar to (1.1), we can assume H = (p) and M is an orientalby-regular normal Cayley map of K with the corresponding Cayley subset {A, Ap,..., Ap }. Also in this case K = (A)H. It is known that K is nilpotent, so the Sylow-2 subgroup P of K is a characteristic subgroup of G. Note that G = (A, p) = (A)G(p) < P x (p) < K x (p). So, K = (A)G = P is a 2-group. According to Corollary 2.6, o(pA) = o(p) = k and so the type of M is (k, k). The map M
|G|	IGI	IGI
has ikL vertices, -G edges, -G faces and the genus of the corresponding orientable surface
is i IGI (4—k)	n
is 1 - 4k .	u
In the proof of Theorem 3.3, for a Frobenius group G = K x H that can be the automorphism group of an orientably-regular chiral map, we have described the relations between K and H. To be more clear, we rewrite these relations in Corollary 3.4.
Corollary 3.4. Let G = K x H be a Frobenius group. If G = (p, A | pk, A2, (pA)m,...) is the automorphism group of an orientably-regular chiral map Map(G; p, A), then one of the following three cases will happen:
(1)	k is even, H = (p) = Zk, K is an abelian group and K = (Apk )H;
(2)	k is odd, H = Z2k, K is abelian and G = K x H, where K = K x Z2, H = (p) is the index two subgroup of H and K = (A)H;
(3)	k is odd, H = (p) = Zk and K = (A)H is a 2-group.
In the following Theorems 3.5 and 3.6, we will show that a Frobenius group whose Frobenius kernel and Frobenius complement conforming to the conditions in Corollary 3.4
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Ars Math. Contemp. 19 (2020) 173-187
can be the automorphism group of an orientably-regular normal Cayley map which implies that the conditions are also sufficient.
Theorem 3.5. Let G = K x H be a Frobenius group, where K is abelian and K = (x)H for some x G K, H = (y) is cyclic of order 2n, n > 2. Then, there is an orientably-regular normal Cayley map M such that G = Aut(M) and the type of M is
((2n, n) or (n, 2n), if n is odd, (k, m) = \
l(2n, 2n),	if n is even.
Proof. Let p = y, A = xyn. Then G = (p, A). It is clear that o(p) = 2n, o(A) = 2, o(pA) = n if n is odd and o(pA) = 2n if n is even. So, G is the automorphism group of an orientably-regular map M of type (2n, n) or (2n, 2n) depending on whether n is odd or even. Because H = (y) = (p), it follows that the vertex set consists of the cosets of H in G. So, K acts regularly on the vertex set of M and as a result M is an orientably-regular normal Cayley map of K.
When n is odd, if we set p = y2, A = xyn, then o(p) = n, o(A) = 2 and o(pA) = 2n. We claim that G = (p, A). Set Q = (p, A), then Q = (y2, xy) because n is odd. From the requirement of n > 2, we have y2 = 1 and so CK (y2) = 1 in the Frobenius group G. A calculation shows that the commutator [y2, x-1] = (xy)y (xy)-1 G Q. Also, [y2, x-1] belongs to K. Note that K is abelian. We have
Q > [y2,x-1]^ = [y2,x-1]K^> = [y2,x-1]^>
= [y2,x-1]<y> = ([y2, (x-1)g] | g G (y)).
Define a function a: K ^ K such that ba = [y2, b] for each b G K. Now,
(6162)" = [y2,b1b2] = [y2,b2][y2,b1]b2 = [y2,b1][y2,b2] = b? b^. From CK(y2) = 1, one can get a G Aut(K). Therefore,
([y2, (x-1)g] | g G (y)) = (((x-1)g)" | g G (y)) = ((x-1)<y>)" = K" = K.
So, K < Q and (xy)K< Q. Consequently, Q = G. Let KK = K x (yn) and # = (p). Then, if = K x Z2, Hi is the index two subgroup of H and G = if x H. Therefore, G is the automorphism group of an orientably-regular normal Cayley map of KT of type
(n, 2n).	□
Theorem 3.6. Let G = K x H be a Frobenius group, where K is a 2-group and K = (x)H for some involution x G K, H = (y) is cyclic of order n for some odd integer n. Then, there is an orientably-regular normal Cayley map M such that G = Aut(M) and the type of M is (n, n).
Proof. Let p = y, A = x. Then G = (p, A). It is clear that o(p) = n, o(A) = 2, o(pA) = n. So, G is the automorphism group of an orientably-regular map M of type (n, n). Because H = (y) = (p), it follows that the vertex set consists of the cosets of H in G. So, K acts regularly on the vertex set of M and as a result M is an orientably-regular normal Cayley map of K.	□
H.-P. Qu, Y. Wang and K. Yuan: Frobenius groups which are the automorphism groups of.
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Corollary 3.7. Let Ki x H and K2 x H be Frobenius groups, where Ki = (x^H, K2 = (x2)H are both abelian groups whose orders are coprime with each other, H = (y) and o(y) = 2n for some integer n > 2. Then, the following two results follow from Lemma 2.4 and Theorem 3.5.
•	(Ki x K2) x H is a Frobenius group, Ki x K2 = (xix2)H and for each ai e Ki, a2 e K2, b e H the element b acts on aia2 in the way (aia2)b = aia2,
•	(Ki x K2) x H is the automorphism group of an orientably-regular normal Cayley map.
According to Theorem 3.5 and Corollary 3.7, one may concentrate on Frobenius groups whose Frobenius kernels are p-groups and satisfy the conditions in Theorem 3.5. Now, we want to give an example of Frobenius groups satisfying the conditions in Theorem 3.5.
In a finite group G, for each element g e G we use CG(g) to denote the centralizer of g in G, that is CG(g) = {h e G | hg = gh}.
Example 3.8. Let K = (ai) x (a2) x • • • x (ad), where o(a4) = pei, p is an odd prime number and these positive integers ei; 1 < i < d, satisfy ei > e2 > • • • > ed. Let H = Zk = (b) for some positive even integer k satisfying k | p — 1. Assume d < 4>(k), where ^ is the Euler's totient function, ti is a positive integer such that ti + peiZ is an element in Z*pSi of order k and ti + pZ = tj + pZ for any 1 < i = j < d. Set G = K x H,
where ab = ati, then G is a Frobenius group. Take a = f]d=i ai, then
K = (a)H = (a,

Proof. To show that G is a Frobenius group, we only need to show that for each element
y e H \ {1}, the equality CK(y) = 1 holds. Suppose x = nd=i x e CK(y), where
G (ai). It is obvious that f]^
n?
From the defining relation ab

for each i. That is xi G C^ai^(y).
then (ai) is an H-invariant subgroup, and so xi While y is a power of b and the action of b on (ai) has only one fixed point, that is the identity of (ai), so C{a.} (y) = 1.
For each 1 < i < d — 1, ab = d=i at-. If we look at the determinant
1
ti
i
1
t2
id-1
t
d-1
in the finite field Fp, then from the choices of ti this is a non-zero Vandermonde determinant. As a result,
a + $(K), ab + $(K),..., abd-1 + $(K)
is a basis of the linear space K/$(K), where $(K) is the Frattini subgroup of K. From the Burnside basis theorem, the result K = (a, ab,..., ab ) follows.	□
Corollary 3.9. Let K and H be groups in Example 3.8. Then, the Frobenius group G = K x H is the automorphism group of an orientably-regular normal Cayley map described in Theorem 3.5.
b
a
a
ti
xy

=a
1
1
2
d
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Ars Math. Contemp. 19 (2020) 173-187
Lemma 3.10. Let A be a group, B be a subgroup of A of index 3, and each a G A\B, a3 = 1. Then, [b, ba] = 1 for any b G B, a G A \ B.
Proof. Note that if b G B and a G A \ B, then ba G A \ B. So, (ba)3 = 1 and bab = a-1b-1a-1. The commutator [b, ba] = b-1a-1b-1aba-1ba = b-1(a-1b-1a-1) • (a-1ba-1)ba = b-1(bab)(b-1ab-1)ba = a3 = 1.	□
Corollary 3.11. Let G = K x H be a Frobenius group. If G satisfies the following two conditions:
(1)	G can be generated by two elements,
(2)	H = Z3,
then K is abelian. Moreover, if G is the automorphism group of an orientably-regular map, then K is isomorphic to the Klein group K4 and G is isomorphic to the alternating group A4.
Proof. Assume G = (a, b) and a G K .By Lemma 2.1 and Corollary 2.6, a3 = 1 and so G = K U Ka U Ka2. As a result, one of the three elements b, ba-1, ba-2 must belong to K. Suppose b G K, then G = (a, b) = (a)(b)G < (a)K = G. Because (a)nK = 1, it follows that K = (b)G. While (b)G = (b, ba, ba ), so K is abelian according to Lemma 3.10.
If G is the automorphism group of an orientably-regular map, then without loss of generality we can assume H = (a). So, K = (b,ba,ba ) is a 2-group according to Theorem 3.3. The fact of K being abelian implies that the rank d(K) of K satisfies d(K) < 3. Therefore, K = Zd(K). Moreover, from 3 | |K| — 1, one can get d(K) = 2 and K is isomorphic to K4 and G = A4.	□
Remark 3.12. In Corollary 3.11, the condition K = (b)H for some element b G K is necessary. In fact, one may check the list of small groups to find SmallGroup(192,1023) in Magma [1] to get a Frobenius group satisfying K = (a, b)H for two different elements a and b of K, H = Z3, but K is not abelian.
According to Theorem 3.3, if the Frobenius group G = K x H is the automorphism group of an orientably-regular map and |H| is odd, then K is a 2-group. By Corollary 3.11, in order to find a non-abelian 2-group as the Frobenius kernel, the smallest order of the Frobenius complement is 5.
Theorem 3.13. Let G = K x H be a Frobenius group. If K is a non-abelian 2-group, H is a cyclic group of odd order and G is the automorphism group of an orientably-regular map, then the group G of the smallest order is SmallGroup(1280,1116310) in Magma.
Proof. Since K is a non-abelian 2-group, its commutator subgroup K' is non-trivial and is a proper subgroup of K. Set |H | = n. Because K/K' x H and K' x H are both Frobenius groups, n | (|K/K'| — 1) = 2n — 1 and n | (|K'| — 1) = 2n — 1 for some integers n and n2. According to Corollary 3.11, n is an odd integer but n = 3.
If n = 5, then the smallest choices of n1, n2 are 4 and in this case |G| = 28 x 5. The Frobenius group satisfying these conditions really exists. It is SmallGroup(1280,1116310) in the list of groups in Magma.
We claim that no Frobenius groups of order less than 28 x 5 with non-abelian 2-groups as Frobenius kernels, cyclic groups of odd orders as Frobenius complements, exist that can
H.-P. Qu, Y. Wang and K. Yuan: Frobenius groups which are the automorphism groups of.
373
be automorphism groups of orientably-regular maps. Otherwise, suppose a group G =
(p, A | pn, A2,...} satisfies these conditions. Then, n = 7, n1 = n2 =3 and |G| =
26 x 7 = 448. It is SmallGroup(448,1394) in the list of groups of Magma. But, its
Frobenius kernel is abelian which is a contradiction.	□
ORCID iDs
Haipeng Qu © https://orcid.org/0000-0002-3858-5767
Yan Wang © https://orcid.org/0000-0002-0148-2932 Kai Yuan© https://orcid.org/0000-0003-1858-3083
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