ARS MATHEMATICA CONTEMPORANEA
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 7 (2014) 519-536
Chamfering operation on k-orbit maps
*
Maria del Rio Francos
Institute of Mathematics Physics and Mechanics, University of Ljubljana, Slovenia, Jadranska 19, Ljubljana 1000, Slovenia
Received 20 September 2013, accepted 25 September 2014, published online 6 October 2014
A map, as a 2-cell embedding of a graph on a closed surface, is called a k-orbit map if the group of automorphisms (or symmetries) of the map partitions its set of flags into k orbits. Orbanic, Pellicer and Weiss studied the effects of operations as medial and truncation on k-orbit maps. In this paper we study the possible symmetry types of maps that result from other maps after applying the chamfering operation and we give the number of possible flag-orbits that has the chamfering map of a k-orbit map, even if we repeat this operation t times.
Keywords: Map, flag graph, symmetry type graph, chamfering operation. Math. Subj. Class.: 52B15, 05C10, 57M15, 51M20, 52B10
1 Introduction
Topologically, a map M is a cellular embedding of a connected graph on a closed surface, with no boundary. While combinatorially, we define a map by an edge coloured cubic graph Gm, to which we refer as the flag graph of the map M, as Lins and Vince (1982-83) define it in [18] and [25], respectively. The vertex set of Gm is the set of flags of the map, and the edges define the connectivity between pairs of flags. Flags are a very important tool in describing combinatorially the structure of a map. They have been used not only for maps but also for hypermaps [9, 23], maps on the surfaces with boundary [1], abstract polytopes [22] ormaniplexes [28].
A map M is called a k-orbit map if its group of automorphisms, or symmetries, partitions the set of flags into exactly k orbits. The most symmetric maps are well known as regular (or reflexible) maps, those for which its automorphism group acts transitively on their set of flags, i.e. they have exactly one flag-orbit. Other highly symmetric type of maps
* This paper is a part of Bled'11 Special Issue.
E-mail address: maria.delrio@fmf.uni-lj.si (Maria del Rio Francos)
Abstract
©® This work is licensed under http://creativecommons.Org/licenses/by/3.0/
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are the so called chiral maps, which flags are partitioned into two orbits in such way that any two adjacent flags belong to different orbits, [12,13, 22]. In other words, the flag graph of a chiral map is a bipartite graph and each part is an orbit.
In [19] the question of possible symmetry types of maps resulting from other maps after applying various operations was raised. In particular, the medial and truncation operations on k-orbit maps were considered, for k < 4. In this paper we use the chamfering operation on k-orbit maps and determine, in terms of k, the number of possible flag-orbits that has the chamfering map of a k-orbit map. Table 1 depicts all possible cases.
The operation of chamfering an object is related to the idea of beveling ("to file down") the edges of a solid object. Given a map M, the chamfering operation replaces the edges of M with hexagonal faces while keeping the faces of M. This operation divides each flag of the map M into four different flags in the chamfering map. This operation is also used on the study of fullerenes (see [6]), for instance, which also leads to chemical applications as in [17]. Theorem 5.3 summarizes all the results presented in this paper.
To solve our problem, we define another graph to which we refer as the symmetry type graph of a map, this is, the quotient graph of the flag graph of a map under the action of its automorphism group. A strategy of how to generate symmetry type graphs is shown in [2]. Dress and Huson (1987) refer to such graphs as the Delaney-Dress symbol, [7]. Dress and Brinkmann (1996), as well as Balaban and Pisanski (2012), give applications to mathematical chemistry in [8] and [1], respectively.
In [20], Orbanic, Pellicer, Pisanski and Tucker (2011), show the 14 symmetry type graphs of edge-transitive maps. Later, in [4] and [5], the complete list of possible symmetry type graphs with at most 6 vertices is determined. In particular, in [4] are described some properties of the symmetry type graphs, and also, the advantages of symmetry type graphs were applied to completely solve the problem of symmetry types of medial maps. While, in [5] is given an extension of the results in [19] of all possible symmetry type graphs of a map and its truncated map might have, for up to 7 and 9 vertices.
The paper is organized in the following way. In Section 2, we formally define a map and its flag graph. In Section 3, we define the symmetry type graph of a k-orbit map and give some of its properties, also studied in [4]. In Section 4, we define the chamfering map and find some conditions for the original map as for its chamfering map in manner to determine whether the chamfering map of a k-orbit map has 4k flag-orbits or not. Finally, we conclude with Theorems 5.1 and 5.3 where we obtain the number of flag-orbits that the chamfering map has if we repeat this operation t times on the same map.
2 Maps
A map M is defined as a cellular embedding of a connected graph on a surface. Let BS be the barycentric subdivision of M and let $ be a triangle in BS. Label the vertices of $ by $o, $i and according to whether they represent a vertex, an edge or a face (mutually incident) in the map M. Note that every triangle of BS is adjacent to other three triangles, see Figure 1. If two triangles $ and ^ of BS are adjacent by the edge with vertices $j and $k, with j, k G {0,1,2} and j = k, then we say that $ and ^ are ¿-adjacent, for i G {0,1, 2} and i = j, k. In this case we shall denote ^ by $® (likewise $ by and note that for every triangle $ and i G {0,1,2}, ($i)i = $.
Combinatorially, a map can be seen as a set F (M) of flags, and the relation between pairs of elements in F(M) in the following way. To each flag in F(M), we assign a
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triangle $ in BS described by the ordered triple ($0, $1, $2), represented by the vertices of $ in BS, and denote by $i the corresponding ¿-adjacent flag of $ in M, with i G {0,1,2}. Note that as it happens for the degenerated cases: as a map with a single vertex and edge but two faces, or a map with two vertices, one edge and a face; two adjacent flags can be represented by the same triple, however they are assigned to different triangles. We shall say that a map M is a non-degenerated map if the triples ($0, $1; $2) are in one to one correspondence to the flags of M.
Let s0, s1 and s2 be the three permutations in the symmetric group Sym(F(M)) such that, for every flag $,
$si = $ • si = $\
with i = 0,1,2. Note that s0, s1, s2 and s0s2 are fixed point free involutions. Furthermore, by the connectivity of the map the action of the subgroup of Sym(F(M)) generated by these three distinguished involutions, denoted by Mon(M) := (s0, s1; s2}, is transitive on the set of flags F(M). The group Mon(M) is known as the monodromy (or connection) group of the map M, [10].
An automorphism of the map M is a bijection between vertices, edges and faces preserving their adjacency on the map. Thus, an automorphism of M induces a permutation of the flags in F(M) such that its action commutes with the elements of Mon(M). In other words, for every automorphism a of M, every flag $ G F(M) and every i G {0,1, 2} it follows that
$si a = ($a)Si,
[14]. The connectivity of the map implies that the only automorphism that fixes a flag is the identity one. That is, the action of the automorphism group Aut(M) over the set F(M) is semi-regular, and hence divides F(M) into k orbits of the same size; in such case M is called a k-orbit map. If the action of Aut(M) over the set F(M) is transitive we say that the map is regular (or reflexible). The 2-orbit maps were widely studied and classified (in different contexts) in [9] and [15]. The most studied and understood type of 2-orbit maps is the chiral one, which has two orbits on its flags where any two adjacent flags belong to different orbits.
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2.1 Flag graph
Given a map M, we can construct a graph Gm in the following way. The set of flags F(M) of the map M corresponds to the vertex set of the graph Gm, and two vertices $ and ^ in V(Gm) are adjacent by an edge of colour i = 0,1, 2 if and only if the corresponding flags are i-adjacent in M (see Figure 2). We shall refer to the graph Gm as the flag graph of the map M.
$
$		
		
$2
Figure 2: Local representation of the flag graph Gm of a map M.
Observe that the distinguished generators so, s1 and s2 of the monodromy group Mon(M) of the map M label the coloured edges of its flag graph Gm in a natural way. Hence, for each flag $ G F(M), a word w = sio sil ■ ■ ■ sin G Mon(M) describes a path along the edges in Gm, coloured by i0, i1,..., in, starting at the vertex $ and ending at the vertex $w, with
$w = ($io) ■ si1 ■ •• sin =: $io-il'--in.
Since in general the action of Mon(M) is not semi-regular on F(M), this implies that one can have differently "coloured" walks in Gm going from $ to another flag ^ that induce different words of Mon(M) that act on the flag $ in the same way.
Note that in Gm the edges of a given colour form a perfect matching (an independent set of edges containing all the vertices of the graph), and the union of two sets of edges of different colour is a subgraph whose components are even cycles; such subgraph is known as a 2-factor of Gm. In particular, note that since (s0s2)2 = 1 and s0s2 is fixed-point free, the cycles with edges of alternating colours 0 and 2 are all of length four.
Note that the connected components of the 2-factor of colours 0 and 2 in Gm, define the set of edges of M. In other words, the edges of M can be identified with the orbits of F(M) under the action of the subgroup generated by the involutions s0 and s2; that is, E(M) = {$<s°>s2> | $ g F(M)}. Similarly, we find that the vertices and faces of M are identified with the respective orbits of the subgroups (s1, s2) and (s0, s1) on F(M). That is, V(M) = {$<si's2> | $ g F(M)} and F(M) = {$<s°'si> | $ g F(M)}. Thus, the group (s0, s1; s2) acts transitively on the sets of vertices, edges and faces of M.
The automorphism group Aut(M) of M induces a bijection between the flags of M preserving their adjacencies, and an edge-coloured preserving automorphism of the graph Gm is a bijection between the vertices of Gm, preserving the adjacencies on the elements of F(M). Consequently, Aut(M) is isomorphic to the edge-coloured preserving automorphism group of the flag graph Gm.
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Recall that Aut(M) partitions the set F(M) into k orbits of the same size. Let Orb(M) := |O$|$ G F(M)} be the set of all flag-orbits of M. By the connectivity of Gm we have the following lemma.
Lemma 2.1. Let Oi, O2 G Orb(M), $ G F (M), and w G Mon(M). If $ G Oi and $w G O2, then ^ G Oi if and only if G O2, for any ^ G F (M).
3 Symmetry type graph of a map
We define a graph T(M) (fairly, a pre-graph as in [24]), that we call the symmetry type graph of M, as the quotient of the flag graph Gm, under the action of the automorphism group Aut(M) of the map. Hence, the vertices of T(M) correspond to the elements in Orb(M), where two vertices O$, O^ G Orb(M) are adjacent by an edge of colour i = 0,1, 2 if and only if there are flags $' G O$ and G O^ that are ¿-adjacent in Gm (if the two i-adjacent flags $' and belong to the same flag-orbit, then the edge of colour i is projected into a semi-edge in T(M)). By Lemma 2.1 the symmetry type graph T(M) is a 3-valent (pre-)graph of chromatic index 3.
It can be seen that the action of Mon(M) on the set Orb(M) is defined as O$ • w = O$w, for any w G Mon(M) and $ G F(M). This action is transitive, as is the action of Mon(M) on F(M). Since Gm is a connected graph, then its corresponding symmetry type graph T(M) is connected as well.
The symmetry type graph of regular maps is a graph with a single vertex and three semi-edges of colours 0, 1 and 2. Moreover, the symmetry type graph of chiral maps is a graph with two vertices and three parallel edges coloured by 0 ,1 and 2, connecting both vertices. In fact, chiral maps are commonly said to be of symmetry type 2.
The number of symmetry types of k-orbit maps is bounded by the number of connected cubic graphs with k vertices, properly three edge-coloured, where the colours 0 and 2 are as in the Figure 3. The reader can refer to [4] and [5] for all possible symmetry type graphs with at most 6 vertices.
D ^ 0 ^ <
Figure 3: Possible quotients of 0-2 coloured 4-cycles of Gm.
4 Chamfering map
The chamfering map Cham(M) of any (non-degenerated) map M is produced, as its name says: by chamfering the edges in M. More precisely, the edges of a map M are replaced by hexagonal faces, surrounding the faces of M, in Cham(M) (see Figure 4). Hence, the set of faces of Cham(M) is in correspondence with the set of faces F(M) and the set of edges E(M) of M. That is, the set of faces of Cham(M) is
F(Cham(M)) = F(M) U E(M).
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Figure 4: Dodecahedron (left) and the chamfering of the dodecahedron (right).
It is straightforward to see that the map Cham(M) has two types of edges: those between hexagonal faces and those between a face $2 in F (M) and its adjacent hexagonal faces (corresponding to the incident edges on the face $2 in M). This is, the set of edges of Cham(M) is
E(Cham(M)) = {{$0, {$0, $2}}|$ € F(M)} U {{$1, $2}|$ € F(M)}.
In fact, Cham(M) has exactly 4|E(M)| edges. Finally, the set of vertices of M is a proper subset of the vertices of Cham(M), and the remaining 2|E(M)| vertices in V(Cham(M)) \ V(M) (each of these vertices are adjacent to exactly one vertex in V(M)), all have degree 3. Thus, the set of vertices of Cham(M) is
V(Cham(M)) = V(M) U {{$0, $2}|$ € F(M)}.
For an alternative definition of chamfering we refer the reader to [6].
Observe that the map on the left (dodecahedron) in Figure 4 is regular, while the map on the right is a 4-orbit map with symmetry type 4Dp (Figure 5). There is a single orbit of
• • • 0*2,1
0*2 0*2,1
Figure 5: Symmetry type graph 4Dp.
flags, O*, on a pentagon and three different flags on a hexagon. Note that by chamfering a non-degenerated map M, every flag $ := ($0, $1, $2) in F(M) is divided into four flags of Cham(M), as is depicted in Figure 6, and the corresponding four flags to $ € F(M) in Cham(M) can be written as
($, 0) :=($0, {$0, {$0, $2}}, $1), ($, 1) :=({$0, $2}, {$0, {$0, $2}}, $1), ($, 2) :=({$0, $2}, {$1, $2}, $1), ($, 3) :=({$0, $2}, {$1, $2}, $2).
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$2
Figure 6: The four respective flags of F(Cham(M)) to the flag $ = {$0, $2} € F (M).
It is then straightforward to see that the adjacencies of the flags of Cham(M) are closely related to those of the flags of M. In fact, we have that,
($, 0)0 = ($, 1),	($, 0)1 = ($s2, 0),
($, 1)0 = ($, 0),	($, 1)1 = ($, 2),
($, 2)0 = ($s0, 2),	($, 2)1 = ($, 1),
($, 3)0 = ($s0, 3),	($, 3)1 = ($S1, 3),
0)2 = ($S1,0), 1)2 = ($S1,1), ($, 2)2 = ($, 3), ($, 3)2 = ($, 2).
Thus, we define the algorithm in Figure 7 to construct the flag graph of Cham(M) out of qm.
Proposition 4.1. The flag graph GCham(M), of the chamfering map Cham(M) of any map M, can be quotient into a graph as the symmetry type graph 4Dp.
Proof. Let A = {($, i)|$ € F(M)} be the subset of F(Cham(M)) containing all flags of Cham(M) of the form ($, i), with i = 0,1, 2,3. Then, F(Cham(M)) = A0 U A1 U A2 U A3 and A n Aj = 0 whenever i = j. Hence, (A0, A1; A2, A3) is a partition of the set of flags F(Cham(M)). Based on Figure 7, it is straightforward to see that the quotient of GCham(M) over such partition, is isomorphic to the symmetry type graph of a map with symmetry type 4Dp (see Figure 5).	□
Note that for any flags T € A3, T2 € A2, T2'1 € A1 and T2'1'0 € A0, we can define a flag $Y € F(M), by assembling these four flags in Cham(M). Observe that an automorphism a € Aut(Cham(M)) that sends a flag T' € Aj to another flag also contained in Aj, with i = 0,1, 2,3, is induced by an automorphism a € Aut(M) that sends to the assembled flag $t s in M. Say this in other way, for each automorphism a € Aut(M), there is an automorphism a € Aut(Cham(M)) such that ($, i)a = ($a, i), with $ € F(M) and i = 0,1,2,3. Then, it follows that
|Orb(Cham(M))| < 4|Orb(M)|.
Motivated by Proposition 4.3 of [19], we are interested in studying the number of possible flag-orbits of the chamfering map Cham(M).
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Figure 7: Local representation of a flag in Gm, in the left. The image under the chamfering operation, locally obtained, in the right.
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Certainly, the chamfering map Cham(M), of a k-orbit map M, has 4k orbits on the set of flags F(Cham(M)), if for any $, ^ G F(M) there is no flag of the form ($, i) in the same orbit as a flag of the form j), with i, j g {0,1,2, 3} and i = j. In fact, if the chamfering map Cham(M) of a k-orbit map M is a 4k-orbit map then, the algorithm presented in Figure 7 works as an algorithm on the vertices of T(M) to obtain the symmetry type graph T(Cham(M)) with 4k vertices, of the chamfering map of M.
We denote by r0, ri and r2 the distinguished generators of Mon(Cham(M)). Observe that, in particular, ($s0, 3) = ($, 3) • r0, ($s1, 3) = ($, 3) • r1, and ($s2, 3) = ($, 3) • r2r1 r0r1r0r1r2, for any $ G F(M). This is, the action of the subgroup
D = (r0, r1, r2r1r0r1r0r1r2) < Mon(Cham(M))
over the subset of flags F(M) x {3} in Cham(M) is isomorphic to the action of the mon-odromy group Mon(M) over the set F(M), inducing the following action isomorphism.
(f, g) : (F(M), (s0, S1, S2)) ^ (F(M) x {3}, (r0, r1, r2r1r0r1r0r1r2)),
where f : $ ^ ($, 3) is a bijective function, and g : (s0,s1,s2) ^ (r0, r1; r2r1r0r1r0r1r2) is a group isomorphism, as that defined in [14]. Then, the action of D is transitive on the set of flags F(M) x {3}. Moreover, the action of D on F(Cham(M)) fixes the set A3 and permutes the sets A0, A1 and A2. Further on, because
($, 3) • r2 = ($, 2), ($, 3) • r2r1 = ($, 1) and ($, 3) • r2r1r0 = ($, 0),
conjugating D by the elements r2, r2r1 and r2r1r0 in Mon(Cham(M)), we obtain three different subgroups of Mon(Cham(M)), that act transitively on the set of flags F(M) x {2}, F(M) x {1} and F(M) x {0}, respectively. Therefore, we say that the conjugate subgroup Dai < Mon(Cham(M)) fixes the set Aj, for each i = 0,1,2, 3, and permutes the sets Aj1, Aj2 and Aj3, with j^ j2, j3 g {0,1, 2, 3} \ {i}, where a0 = r2r1r0, a1 = r2r1, «2 = r2, and 03 = id
With the following lemma we see that the chamfering map of a k-orbit map M, not necessarily has 4k flag-orbits. By an equivelar map with Schlafli type {6, 3} we mean a map that all its faces are 6-gons, and all its vertices have degree 3.
Lemma 4.2. Let Cham(M) be the chamfering map of a map M. If there is an automorphism a G Aut(Cham(M)) such that ($, i)a = (^,j) for some $, ^ G F(M) and i = j, with i, j G {0,1, 2, 3}. Then, M is an equivelar map with Schlafli type {6,3}.
Proof. Consider the partition (A0, A1, A2, A3) of the set F(Cham(M)), where A = {($, i)|$ G F(M)}, i = 0,1, 2,3, and recall that if we assemble the flags Y g A3, Y2 g A2, T2'1 g A1 and T2'1'0 g A0, we can define a flag $T G F(M).
Suppose that there is an automorphism a G Aut(Cham(M)) such that ($, i)a = j) for some $, ^ G F(M) and i = j, with i, j G {0,1, 2,3}. We shall verify the image, under a, of the assembled flags ($, 0), ($, 1), ($, 2), ($, 3), corresponding to $ G F(M), in terms of the adjacent flags of j). Note that $0 G ($, 0) and $2 G ($, 3), but they are neither in ($, 1) nor in ($, 2). Then, we have the following cases.
0) For i = 0.
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-	If ($, 0)a = (tf, 1), then $oa = jtf0, tf2} and $2a = (tf2-1)^ since ($, 0)a = (tf, 1) := (jtfo, tf 2}, jtfojtfo, tf 2}}, tf 1) and
($, 3)a = ($, 0)o-1-2a = (($, 0)a)0-1-2 = (tf, 1)0-1-2 = (tf2-1,0) := (tfo, jtfo, jtfo, (tf2)2}}, (tf2-1)^
-	If ($, 0)a = (tf, 2), then $oa = jtf0, tf2} and $2a = (tf0-1)1, since ($, 0)a = (tf, 2) := (jtfo, tf2}, jtf 1, tf2}, tf 1) and
($, 3)a = ($, 0)o-1-2a = (($, 0)a)0-1-2 = (tf, 2)0-1-2 = (tf0-1, 1) := (j(tf0)o, tf2}, j(tfo)o, j(tfo)o, tf2}}, (tf0'1)1).
-	If ($, 0)a = (tf, 3), then $oa = jtf0, tf2} and $2a = (tf0-1)1, since ($, 0)a = (tf, 3) := (jtfo, tf2}, jtf 1, tf2}, tf2) and
($, 3)a = ($, 0)o-1-2a = (($, 0)a)0-1-2 = (tf, 3)0-1-2 = (tf0-1,2) := (j(tf0)o, tf2}, j(tf0-1)1, tf2}, (tf0-1)!).
Similarly, we follow the same analysis in the next cases.
1)	For i = 1.
-	If ($, 1)a = (tf, 0), then $oa = jtf 0, tf2} and $2a = (tf2-1)1.
-	If ($, 1)a = (tf, 2), then $oa = j(tf0)0, tf2} and $2a = (tf 1)1.
-	If ($, 1)a = (tf, 3), then $oa = j(tf0)0, tf2} and $2a = (tf 1)1.
2)	For i = 2.
-	If ($, 2)a = (tf, 0), then $oa = jtf 0, (tf2)2} and $2a = (tf 1)1.
-	If ($, 2)a = (tf, 1), then $oa = j(tf0)0, tf2} and $2a = (tf 1)1.
-	If ($, 2)a = (tf, 3), then $oa = j(tf 1-0)0, tf2} and $2a = tf 1.
3)	For i = 3.
-	If ($, 3)a = (tf, 0), then $oa = jtf 0, (tf 1-2)2} and $2a = tf 1.
-	If ($, 3)a = (tf, 1), then $oa = j(tf 1-2)0, (tf 1-2)2} and $2a = tf 1.
-	If ($, 3)a = (tf, 2), then $oa = j(tf 1-2)0, tf2} and $2a = (tf 0-1)1.
Observe from the cases above, that all the vertices jtf0, tf2}, j(tf0)0, tf2}, jtfo, (tf2)2}, j(tf1-o)o, tf2}, jtfo, (tf1-2)2}, j(tf1-2)o, (tf1-2)2} and j(tf1-2)o, tf2} are vertices with degree 3 in Cham(M). So as all the faces (tf2-1)1, (tf0-1)1, (tf 1)1 and tf 1, correspond to 6-gons in Cham(M). Thus, the vertex $0 has degree 3 and the face $2 is a 6-gon in M, with $ G F(M). Furthermore, let = A G F(M), with w G Mon(M). Then we have that
(A, i)a = , i)a = ($, i)wa = (($, i)a)w = (tf, j)w,
with w G Mon(Cham(M)). Recall that the conjugated subgroup Dai of Mon(Cham(M)) fixes the set A and permutes the sets Aj, Aj2 and Aj3, with j^ j2, j3 G j0,1,2,3} \ ji}, where a0 = r2r1r0, a1 = r2r1, a2 = r2, and a3 = id. Since (A, i) = ($,i)w, it follows that w g Dai, and henceforth (A, i)a = (tf,j)w G Ajk, with j,jfc Gj0,1, 2, 3} \ ji}.	_
Thus, we follow with a similar analysis as the previous one for (A, i)a = (tf, j)w, and we conclude that the vertex Ao has degree 3 and the face A2 is a 6-gon in M. This latter
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was for arbitrary A G F(M) and w G Mon(M). Therefore, we have that each vertex in V(M) has degree 3 and every face F(M) is a 6-gon. Consequently, the map M is an equivelar map with Schafli type {6, 3}.	□
By the Euler characteristic of a map, the surface of an equivelar map with Schlafli type {6, 3} is either the torus or Klein bottle. In the following subsection we find the number of flag-orbits of the chamfering of an equivelar map of type {6, 3}.
4.1 Chamfering of equivelar maps of type {6, 3}
In [16] Hubard, Orbanic, Pellicer and Weiss studied the symmetry types of equivelar maps in the torus, described as {6,3}vi V2, where v1 and v2 are two linearly independent vectors. In [27] Wilson shows that there are two kinds of maps of type {6,3} in the Klein bottle, and denotes them by {6,3}|m n| and {6, 3}\m n\ respectively, where the two glide reflections of these maps are on axes that are at distance a multiple of n and have length a multiple of m.
Regarding equivelar toroidal maps of type {6,3}, from Theorem 8 in [16], we obtain the following proposition.
Proposition 4.3. Equivelar toroids with Schlafli type {6,3} are either regular, chiral, or have symmetry type 302 or 6H .
302
¡h 0 >
Figure 8: Symmetry type graphs of regular, chiral, 302 and 6Hp maps.
An equivelar toroidal map of type {6,3} is described as {6,3}V1,V2, where the linearly independent vectors v1 and v2 are a linear combination of the basis {\/3ei, ^e1 + §e2}, with the origin in the centre of an hexagon in the {6, 3}-tessellation of the plane.
In Figures 9-12 examples of equivelar toroids and their corresponding chamfering maps are depicted. Note that by chamfering a toroidal map M := {6, 3}V1,V2 we replace the edges of M by the corresponding hexagonal faces in Cham(M). Thus, the centres of adjacent faces of Cham(M) are at half distance than in the centres of adjacent hexagons of M. This implies that the chamfering map Cham(M) is the equivelar toroidal map {6, 3}2vi 2v2. Thus, we have the following lemma.
Lemma 4.4. Let M be an equivelar toroidal map of type {6, 3}. Then the symmetry type graph T(Cham(M)) is isomorphic to T(M).
As what it concerns to equivelar maps of type {6,3} in the Klein bottle. Following [27], the two kinds of maps of type {6,3} in the Klein bottle are denoted by {6, 3}|m n| and {6,3}\m n\ respectively, where m and n are measured in
respect to the centres of the hexagons. The map in the Klein bottle, described as {6,3} |m n|,
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Figure 10: Chamfering of chiral toroids of type {6, 3}.
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{6, 3}(2,0),(-1,2)
Cham(M)
{6, 3}(4,0),(-2,4)
{6, 3}(2,1),(3,-1)
Cham(M)
{6, 3}(4,2),(6,-2)
Figure 11: Chamfering of 3-orbit toroids of type {6,3}.
{6, 3}(2,1),(2,0)
{6, 3}(4,2),(4,0)
Figure 12: Chamfering of 3-orbit toroids of type {6,3}.
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results by using two glide reflections of length y on axes of type (a) or (b), as in Figure 13, that are n^3 apart. And, the map in the Klein bottle, described as {6, 3}\m n\, results by using two glide reflections of length m^3 on axes of type (c) or (d) as in Figure 13, that are n apart. In both cases, the generating glide reflections are symmetries of the regular hexagonal tessellation of the plane. Since the glide reflection axes (a), (b), (c) and (d) are either
parallel to the edges of the hexagons or cross the edges in their midpoint, by chamfering an equivelar map M in the Klein bottle, of type either {6, 3}|m n| or {6,3}\m n\, the distance between both glide reflection axes and their length are the half than for those in M. This is, in Cham(M), the values of m and n are the half as those for M. Therefore, the chamfering map Cham(M) is an equivelar map in the Klein bottle described as {6,3}|2m,2n|, or as {6, 3}\2m 2n\, with glide reflection axes of type (a) or (d), respectively.
Hence, we obtain the following lemma.
Lemma 4.5. If M is the toroidal map {6, 3}V1,V2 or a map in the Klein bottle of type either {6, 3}|m n|, or {6,3}\m n\, then Cham(M) is a map on the same surface of type {6, 3}2vi,2v2, {6, 3}|2m,2n|, or {6,3}\2m,2n\, respectively.
Following [27] we can see that maps {6,3}|m,n| and {6, 3}\m,n\ have 3mn edges and thereby 12mn flags. Moreover, the automorphism group of these maps have 4m elements. Thus, the maps {6,3}|m n| and {6,3}\m n\ are 3n-orbit maps. Hence, Cham({6, 3}K„|) = {6, 3}|2m,2n| and Cham({6, 3}\m,„\) = {6, 3}\2m,2„\ have 48mn flags and their respective automorphism group have 8m elements. Therefore, {6, 3}|2m 2n| and {6,3}\2m 2n\ are 6n-orbit maps. In Figures 14 and 15 are depicted examples of maps of type {6, 3}|m1| and {6, 3}\m1\, with m even and odd, and its chamfering maps. Note that both maps of type {6, 3}|m1| and {6, 3}\m1\ have symmetry type 302, while their chamfering maps {6,3}|2m 2| and {6, 3}\2m 2\ have symmetry type 6Hp.
Corollary 4.6. If M is a k-orbit toroidal equivelar map of Schlafli type {6,3}, then Cham(M) is a k-orbit map, with k = 1,2,3, 6. If M is a k-orbit equivelar map of Schlafli type {6, 3} in the Klein bottle, then 3|k and Cham(M) is a 2k-orbit map.
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{6, 3}|4,2|
Figure 14: Chamfering of a 3-orbit map of type {6,3}|m1| in the Klein bottle.
Figure 15: Chamfering of a 3-orbit map of type {6, 3}\m1\ in the Klein bottle.
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5 Conclusion
Putting our results together we see that lemma 4.2 implies that if M is a k-orbit map such that Cham(M) is nota 4k-orbitmap, then it is of type {6,3}. Hence, Corollary 4.6 implies the following theorem.
Theorem 5.1. Let M be a k-orbit map. Then, Cham(M) has either k, 2k or 4k flag-orbits.
We denote as T(Cham(T')) the chamfering symmetry type graph with 4k vertices that results from applying the algorithm in Figure 7 to the symmetry type graph T' of a k-orbit map. (See for instance Figures 16). As a consequence of the above discussion we have the following corollary.
Corollary 5.2. Let M be a k-orbit map with symmetry type either 1, 2, 302 or 6Hp, and Cham(M) its chamfering map. Then the following holds.
(1)	If M is a regular map, then Cham(M) is either regular of type {6, 3} (and hence toroidal), or has symmetry type 4Dp.
(2)	If M is a chiral map, then Cham(M) is either chiral of type {6, 3} (and hence toroidal), or has symmetry type graph T(Cham(2)) with 8 vertices. (See Figure 16.)
(3)	If M has symmetry type 302, then Cham(M) is either a toroidal map of type {6,3} with symmetry type graph 302, or Cham(M) is a 6-orbit map in the Klein bottle and has symmetry type graph 6Hp, or it has symmetry type graph T(Cham(302)) with 12 vertices. (See Figure 16.)
(4)	If M has symmetry type 6Hp, then Cham(M) is either a toroidal map of type {6,3} and has symmetry type graph 6Hp, or Cham(M) is a 12-orbit map in the Klein bottle, or it has symmetry type graph T(Cham(6Hp )) with 24 vertices. (See Figure 16.)
In [6] A. Deza, M. Deza and V. Grishukhin denote by Chamt(M) the t-times chamfering of M. It is straightforward to see that Chamt(M) of a k-orbit equivelar map M on the torus is a k-orbit map described as {6,3}2tvi ,2tv2. Similarly, Chamt(M) of a k-orbit equivelar map M on the Klein bottle is a 2k-orbit map denoted either {6,3}12(m 2én| or
{6, 3}\2 2tn\.
Finally, based on the results obtained in the previous section, we conclude with the following theorem.
Theorem 5.3. Let M be a k-orbit map and Chamt (M ) the t-times chamfering map of M having s flag-orbits. Then at least one of the following holds.
1.	s = 4fk, 2fk or k.
2.	If s = 4*k, then x(M) = 0 (M is on the torus or on the Klein bottle) and M is of type {6, 3}.
3.	If M is on the torus of type {6, 3} then s = k and k = 1, 2, 3,4.
4.	If M is on the Klein bottle of type {6,3} then s = 2lk and 3|k.
Furthermore, joining the results obtained for the medial and truncation operations on korbit maps, in [19], that motivated the work done for this paper, with the results in obtained for the chamfering operation on k-orbit maps, we obtain the following table.
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M'	Me(M)	Tr(M)	Cham(M)		
|Orb(M')|	2k or k	3k, f or k	4k,	2k	or k
				k|3	k = 1, 2, 3, 6
Table 1: Possible number of possible flag-orbits of a map M' with regard to k = |Orb(M) |, where M' is the medial, truncation or chamfering map of M .
Acknowledgments
The author would like to thank Isabel Hubard and Tomaz Pisanski for many valuable discussions, as well as their support and orientation for the completion of this work. I further acknowledge the partial support from the Slovenian Research Agency (ARRS) for the scholarship granted for my PhD. Part of this work was done during my visit to the Instituto de Matematicas of the Universidad Nacional Autónoma e Mexico that was supported by PAPIIT-UNAM, grant IN106811.
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