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Contents
Almost all Cayley maps are mapical regular representations
Pablo Spiga, Dario Sterzi . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Factorizing the Rado graph and infinite complete graphs
Simone Costa, Tommaso Traetta . . . . . . . . . . . . . . . . . . . . . . . 11
Redundantly globally rigid braced triangulations
Qianfan Chen, Siddhant Jajodia, Tibor Jordán, Kate Perkins . . . . . . . . 31
Intersecting families of graphs of functions over a finite field
Angela Aguglia, Bence Csajbók, Zsuzsa Weiner . . . . . . . . . . . . . . . 45
Saturated 2-plane drawings with few edges
János Barát, Géza Tóth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Classification of thin regular map representations of hypermaps
Antonio Breda d’Azevedo, Domenico A. Catalano . . . . . . . . . . . . . 75
The covering lemma and q-analogues of extremal set theory problems
Dániel Gerbner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
On the beta distribution, the nonlinear Fourier transform
and a combinatorial problem
Pavle Saksida . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Products of subgroups, subnormality, and relative orders of elements
Luca Sabatini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Coincident-point rigidity in normed planes
Sean Dewar, John Hewetson, Anthony Nixon . . . . . . . . . . . . . . . . 137
Volume 24, Number 1, Spring/Summer 2024, Pages 1–153
iii

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.01 / 1–9
https://doi.org/10.26493/1855-3974.3071.37e
(Also available at http://amc-journal.eu)
Almost all Cayley maps are mapical regular
representations
Pablo Spiga *
Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca,
Via Cozzi 55, 20125 Milano, Italy
Dario Sterzi
Scuola Galileiana di Studi Superiori, Via Venezia 20, 35131, Padova, Italy and
Dipartimento di Matematica “Tullio Levi-Civita”,
University of Padova, Via Trieste 53, 35121 Padova, Italy
Received 13 February 2023, accepted 5 May 2023, published online 6 July 2023
Abstract
Cayley maps are combinatorial structures built upon Cayley graphs on a group. As
such the original group embeds in their group of automorphisms, and one can ask in which
situation the two coincide (one then calls the Cayley map a mapical regular representation
or MRR) and with what probability. The first question was answered by Jajcay. In this
paper we tackle the probabilistic version, and prove that as groups get larger the proportion
of MRRs among all Cayley Maps approaches 1.
Keywords: Regular representation, Cayley map, automorphism group, asymptotic enumeration, graph-
ical regular representation, GRR.
Math. Subj. Class. (2020): 05C25, 05C30, 20B25, 20B15
1 Introduction
In this first section we define Cayley graphs and maps, give some context and state our
main theorem. In the second section we prove the theorem. In the third one we prove a
slight variation of the result in which Cayley maps are considered up to isomorphism.
*Corresponding author.
E-mail addresses: pablo.spiga@unimib.it (Pablo Spiga), dario.sterzi@studenti.unipd.it (Dario Sterzi)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 24 (2024) #P1.01 / 1–9
1.1 Cayley graphs
We consider only finite groups and finite graphs in this paper. As usual a graph Γ is an
ordered pair (V, E) with V a finite non-empty set and with E a collection of 2-subsets of
V . An automorphism of a graph is a permutation on V that preserves the set E, and a path
on a graph is a sequence v1, v2, . . . , vn of adjacent vertices, i.e. {vi, vi+1} ∈ E for all i.
The neighbourhood of a vertex v is the set Γ(v) = {w ∈ V |{v, w} ∈ E} of all vertices
connected to it by an edge.
Let R be a group and let S be an inverse-closed subset of R, that is, S = {s−1 | s ∈ S}.
The Cayley graph Cay(R,S) is the graph with V = R and with {r, t} ∈ E if and only
if tr−1 ∈ S, i.e. E = {{r, sr}|s ∈ S, r ∈ R}. The condition S = S−1 is imposed to
guarantee that tr−1 ∈ S if and only if rt−1 ∈ S. A path r0, r1, . . . , rn in a Cayley graph
can be specified equivalently by its starting vertex r0 together with the unique sequence
of elements s1, s2, . . . , sn from S such that ri+1 = si+1ri. Usually one is interested in
connected Cayley graphs, where for any two vertices there is at least one path connecting
them. This is equivalent to the requirement that S is a set of generators for the group. We
shall assume so throughout this paper.
A graphical regular representation (GRR) for a group R is a graph whose automor-
phism group is the group R acting regularly on the vertices of the graph. (A permutation
group R is regular if it is transitive and if the identity element of R is the only element
fixing some point of the domain.) It is an easy observation that the right regular action of R
on itself preserves the edges, so R embeds in Aut(Cay(R,S)).1 A GRR for R is therefore
a Cayley graph on R that admits no other automorphisms.
The main thrust of much of the work through the 1970s was to determine which groups
admit GRRs. This question was ultimately answered by Godsil in [2]. It was conjectured
by Babai and Godsil that, except for two natural families of groups, GRRs not only exist,
but they are abundant, that is, with probability tending to 1 as |R| → ∞, a Cayley graph on
R is a GRR. The first author reported the recent progress in [7, 8, 9, 10] on the Babai-Godsil
conjecture at the SIGMAP 2022 conference at the University of Alaska Fairbanks. During
this conference, Robert Jajcay has suggested a similar investigation for Cayley maps.2 We
now give some background on Cayley maps, state Jajcay’s question and state our main
result.
1.2 Graph maps and Cayley maps
Let Γ := (V,E) be a graph. Given v ∈ V, we let Γ(v) denote the neighbourhood of v in Γ.
A rotation on Γ is a set ρ := (ρv)v∈V , where each ρv : Γ(v) → Γ(v) is a cyclic ordering 3
of Γ(v). A map is a pair (Γ, ρ), where Γ is a connected graph and ρ is a rotation of Γ.
The idea behind maps is that they represent a CW complex structure on an orientable
surface whose 1-skeleton is the given graph, essentially an embedding of the graph in an
orientable surface disconnecting it into disks. See for instance [3] for details. The ρv are
the cyclic orderings of the edges incident to v in the embedding.
1We let automorphisms act on the right, so we will write xφ to denote the image of the vertex x under the
automorphism φ, and we shall take xφψ to mean (xφ)ψ .
2During the preparation of this paper, Xia and Zheng have announced a solution to the Babai-Godsil conjec-
ture, see [11].
3A cyclic ordering on a (finite) set is a permutation with no fixed points and a single cycle in its cycle decom-
position.
P. Spiga and D. Sterzi: Almost all Cayley maps are mapical regular representations 3
Intuitively, an automorphism of a map (Γ, ρ) is a pair: an automorphism of the graph
and an oriented homeomorphism of the surface that are compatible through the embedding.
Combinatorially this translates to an automorphism of Γ (a permutation of the vertices
preserving the edges) which also preserves the rotation ρ. In order to make this idea precise,
we make a slight detour. Let Aut(Γ) be the automorphism group of Γ and let R(Γ) be the
collection of all rotations of Γ. Now, Aut(Γ) has a natural action on R(Γ):
R(Γ)×Aut(Γ) −→R(Γ)
(ρ, φ) 7−→ ρ(φ),
where ρ(φ)vφ = φ−1ρvφ, for all v ∈ V . In other words, the rotation ρ(φ) at the vertex vφ
takes uφ to wφ when ρv takes u to w. Now, an automorphism of a map M = (Γ, ρ) is an
automorphism φ of the graph Γ such that ρ(φ) = ρ, that is,
ρvφ = φ
−1ρvφ, for each vertex v of Γ. (1.1)
It is well known [1] that, if the underlying graph is connected, a map automorphism is
determined uniquely by its value on an oriented edge (i.e. an ordered pair of adjacent
vertices). We recall briefly the reason: suppose φ is a map automorphism, w0, w1 are
adjacent vertices mapped to wφ0 and w
φ
1 respectively and w0, w1, . . . , wt is a path in the
graph. We can describe the path as a sequence of left and right turns, or with a closer
analogy as the exits to take at consecutive roundabouts. There must be natural numbers ni
for i ∈ {1, . . . , t− 1} such that wi+1 = w
ρ
ni
wi
i−1. Thus the path φ(w0), φ(w1), . . . , φ(wt) is
uniquely determined by the relations
wφi+1 = w
φρ
ni
w
φ
i
i−1 for i ∈ {1, . . . , t− 1}.
In other words the automorphism group of a map on a connected graph acts semiregularly
on the set of oriented edges.
Let now R be a group and S as above an inverse-closed set of generators excluding the
identity. For every cyclic ordering r : S → S, we define the Cayley map CM(R,S, r) =
(Γ, ρ) as follows: Γ is the Cayley graph Cay(R,S) and, for every g ∈ R and for every x
lying in the neighbourhood Γ(g) of the vertex g,
ρg : Γ(g)−→Γ(g)
x 7−→ρg(x) := gr(g−1x).
This is the unique map with the prescribed rotation r around the identity vertex e ∈ R such
that the right regular action of the group on the Cayley graph preserves the rotation.
Combinatorially, we may think of a Cayley map as just a triple (R,S, r), where
• R is a finite group,
• S ⊆ R \ {e} is a generating set with S = S−1, and
• r : S → S is a cyclic ordering.
4 Ars Math. Contemp. 24 (2024) #P1.01 / 1–9
1.3 Mapical regular representations and the question of Jajcay
Given a Cayley map CM(R,S, r), the right regular representation of R is contained in the
automorphism group of CM(R,S, r). Analogously to GRRs, we say that CM(R,S, r) is
a mapical regular representation (or MRR for short) if
Aut(CM(R,S, r)) ∼= R.
As far as we are aware, this definition was coined by Robert Jajcay in [5]. Theorem 7
in [5] shows that each finite group not isomorphic to Z3 or Z22 possesses an MRR. Observe
that CM(R,S, r) is a MRR if and only if the only automorphism of CM(R,S, r) fixing a
vertex is the identity.
Once that the existence of MRRs is established it is fairly natural to investigate the
abundance of MRRs among Cayley maps. Indeed, Robert Jajcay has asked whether, as
|R| → ∞, the proportion of MRRs among Cayley maps on R tends to 1.
One could argue for different approaches in counting Cayley maps. In the present
paper we mainly first takle labelled Cayley maps, where two Cayley maps CM(R,S, r)
and CM(R,S′, r′) over the same group are considered to be the same if and only if S = S′
and r = r′. In the last section we show that our methods are trivially adapted to unlabelled
Cayley maps, which are reasonable isomorphism classes one might be interested in. In
both cases we manage to answer Jajcay’s question in the affirmative.
Theorem 1.1. As |R| → ∞, the proportion of MRRs among labelled Cayley maps on R
tends to 1.
Theorem 1.2. As |R| → ∞ the proportion of (equivalence classes of) MRRs among unla-
belled Cayley maps on R tends to 1.
Xia and Zheng [11] have recently announced a positive solution of the Babai-Godsil
conjecture. This means that, except for abelian groups of exponent greater than 2 and for
generalized dicyclic groups, with probability tending to 1 as |R| → ∞, a Cayley graph on
R is a GRR. There are some relations between our work and the work in [11]; for instance,
both results depend upon a theorem on group generation due to Lubotzky [6]. However,
there is no direct implication between our Theorem 1.1 and the main result in [11]; for
instance, a positive solution of the Babai-Godsil conjecture does not imply the veracity
of Theorem 1.1. Indeed, the number of Cayley maps on a fixed Cayley graph Cay(R,S)
is (|S| − 1)!, thus most Cayley maps have almost all the group as connection set of the
underlying Cayley graph, while a random Cayley graph has roughly |R|/2 elements in its
connection set. More precisely: the two questions consider different marginal probability
distributions on the space of Cayley graphs.
2 Proof of main theorem
In this section, we let R be a finite group and we let r denote its order.
We explore the inclusions R ≤ Aut(CM(R,S, r)) ≤ Sym(R). Our strategy is proving
a necessary condition for intermediate subgroups between R and Sym(R) to be automor-
phism groups of Cayley maps, bound the number of subgroups satisfying this condition
and then bound the number of pairs (S, r) compatible with at least one of them.
The following lemma is essentially a restatement of insights in [4].
P. Spiga and D. Sterzi: Almost all Cayley maps are mapical regular representations 5
Lemma 2.1. For any Cayley map CM(R,S, r), the stabilizer Aut(CM(R,S, r))e of the
identity vertex e is cyclic of order less than |R|. If Aut(CM(R,S, r))e = ⟨γ⟩, then Sγ = S
and the restriction γ|S has the same order as γ and it is a power of r.
Proof. An automorphism fixing e sends its neighbourhood Γ(e) = S to itself.
Since the action on oriented edges is semiregular, an element of the stabilizer is uniquely
determined by its action on S, i.e. the restriction mapping
Aut(CM(R,S, r))e −→Sym(S)
φ 7−→ φ|S
is injective.
Moreover, if φ ∈ Aut(CM(R,S, r))e, then from (1.1) we have r = φ−1rφ, i.e., φ|S ∈
CSym(S)(r). From standard computations in permutation groups, we have CSym(S)(r) =
⟨r⟩. Thus Aut(CM(R,S, r))e is isomorphic to a subgroup of a cyclic group, hence
Aut(CM(R,S, r))e is cyclic and all its elements restricted to S are powers of r.
Until now, we have adopted the view that a group R with r elements can be embedded
into Sym(r) using the usual right regular representation. It is convenient for our exposition
to consider the equivalent formulation “R is a regular subgroup of Sym(r)”, here regular
means that for any two points in {1, . . . , r} there exists a unique permutation in R sending
the first to the second.
Lemma 2.2. For every regular subgroup R of Sym(r), the number of subgroups G of
Sym(r) with
• R < G and
• G1 cyclic and |G1| ≤ r − 1 (where G1 is the stabiliser of 1 in G)
is at most 27(log2 r)
2+12 log2 r.
Proof. Given G0 and G1 two abstract groups and H0 ≤ G0, H1 ≤ G1, we write
(G0, H0) ∼ (G1, H1) if there exists a group isomorphism ϕ : G0 → G1 with Hϕ0 = H1.
Clearly, ∼ defines an equivalence relation. We denote by [(G,H)] the ∼-equivalence class
containing (G,H). Now consider
M = {[(G,H)] | G is (log2 r + 1)-generated, H ≤ G, |G| ≤ r(r − 1), and H is cyclic}.
CLAIM 1: We have
|M| ≤ 24(log2(r))
2+12 log2 r. (2.1)
Proof of Claim 1. From [6, Theorem 1] together with [6, Remark 3(1)] we get that the
number of isomorphism classes of groups of order N that are d-generated is at most
N2(d+1) log2(N) = 22(d+1)(log2(|N |))
2
. In particular, applying this theorem with d :=
log2(r) + 1 and with N ≤ r(r − 1), we get that the number of groups G that are
(log2(r) + 1)-generated and of order at most r(r − 1) is at most 24(log2(r)+2) log2 r · r2
(observe that the second factor counts the number of choices for N : the cardinality of G).
Now, let G be a group of order at most r(r − 1). Since G has at most |G| < r2 cyclic
subgroups H , our claim is proved.
6 Ars Math. Contemp. 24 (2024) #P1.01 / 1–9
Now, let R be a regular subgroup of Sym(r) and let SR be the set of subgroups G of
Sym(r) with R < G, with G1 cyclic and with |G1| ≤ r− 1. Since G = RG1 and since R,
as any group of order r, needs at most log2 r generators, we deduce that G needs at most
log2(r) + 1 generators.
CLAIM 2: We have
|SR| ≤ 23(log2 r)
2
|M|. (2.2)
Proof of Claim 2. Every G ∈ SR determines an element of M via the mapping φ : G 7→
[G,G1].
We show that there are at most 23(log2 r)
2
elements of SR having the same image
via φ, from which (2.2) immediately follows. We argue by contradiction and we let
G1, . . . , Gℓ ∈ SR with φ(Gi) = φ(G1), for every i ∈ {1, . . . , ℓ}, where ℓ > 23(log2 r)
2
.
Thus there exists a group isomorphism ϕi : G1 → Gi with (Gi)1 = ((G1)1)ϕi . Therefore
the permutation representation of G1 on the coset space G1/(G1)1 is permutation isomor-
phic to the permutation representation of Gi on the coset space Gi/(Gi)1. Thus G1 and
Gi are conjugate via an element of Sym(r), that is, G1 = (Gi)σi for some σi ∈ Sym(r).
Now, as G1 acts transitively on {1, . . . , r}, replacing σi by an element of the form giσi (for
some gi ∈ G1), we may assume that σi fixes 1, that is, 1σi = 1.
As R ≤ Gi for every i, we get that Rσ1 , . . . , Rσℓ are ℓ regular subgroups of G1.
Since R is log2(r)-generated, we see that G
1 contains at most |G1|log2(r) ≤ r2 log2 r =
22(log2 r)
2
distinct subgroups of order r. In particular, since ℓ > 23(log2 r)
2
, we see that
Rσi1 = · · · = Rσit for some t > 2(log2(r))2 and some subset {i1, . . . , it} of size t of
{1, . . . , ℓ}. Therefore σi1σ−1ij normalises R. As 1
σi1σ
−1
ij = 1, σi1σ
−1
ij
is an automorphism
of R, for every j ∈ {1, . . . , t}. Since R has at most |R|log2(r) = 2(log2(r))2 automorphisms,
we get σi1σ
−1
ij
= σi1σ
−1
ij′
for two distinct indices j and j′. Thus σij = σij′ and G
ij =
(G1)
σ−1ij = (G1)
σ−1i
j′ = Gij′ , which is a contradiction.
From (2.1) and (2.2), we have
|SR| ≤ 27(log2 r)
2+12 log2 r,
and the proof of this lemma immediately follows.
It remains to estimate the number of Cayley maps on a group R compatible with a fixed
intermediate subgroup G with cyclic point stabilizer H .
Lemma 2.3. For every pair of subgroups R and H of Sym(r) such that R is regular and
H = ⟨γ⟩ is non-identity, cyclic of order less than r and fixing the point 1, let
Rγ = {(S, r)|S ⊆ {2, . . . , r}, r cyclic ordering on S, γ ∈ Aut(CM(R,S, r))}
be the set of all Cayley maps on R admitting γ as an automorphism. Then |Rγ | ≤ (r −
1) r2 ⌊r/2⌋!2
r.
Proof. Let l be the order of γ. From Lemma 2.1, if (S, r) ∈ Rγ , then Sγ = S; thus S is
a union of H-orbits. Moreover, γ|S is a power of r; hence γ|S is a product of k disjoint
cycles all of the same length l fixing no point in S. Clearly kl = |S| < r. For a fixed
S (and hence k and l), r ∈ CSym(S)(γ|S). From routine computations, CSym(S)(γ|S) is
P. Spiga and D. Sterzi: Almost all Cayley maps are mapical regular representations 7
isomorphic to the wreath product Cl ≀Sym(k). Thus, given S, there are at most lkk! choices
for r.
If nl is the number of cycles of length l in the cycle decomposition of γ, then there are(
nl
l
)
choices for S such that γ|S decomposes in k cycles of length l.
Putting everything together, we have
|Rγ | ≤
r−1∑
l=2
nl∑
k=1
(
nl
k
)
k!lk. (2.3)
Of course lnl ≤ |S| < r and hence nl < r/l.
In what follows, we use the generalized binomial coefficient
(
x
k
)
= 1k!
∏k−1
i=0 (x − i).
Observe that
(
x
k
)
is increasing in the real variable x ≥ k. Elementary computations show
the inequality ( r
l+1
k
)
k!(l + 1)k( r
l
k
)
k!lk
=
k−1∏
i=0
r − i(l + 1)
r − il
≤ 1.
This gives that the summands appearing in (2.3) are non-increasing in l and hence they can
be estimated with l = 2. We deduce
|Rγ | ≤
r−1∑
l=2
⌊ rl ⌋∑
k=1
(
⌊ rl ⌋
k
)
k!lk ≤
r−1∑
l=2
⌊ rl ⌋∑
k=1
( r
l
k
)
k!lk ≤
r−1∑
l=2
⌊ rl ⌋∑
k=1
( r
2
k
)
k!2k.
Furthermore, an easy computation shows that (for 0 ≤ k ≤ x)
(
x
k+1
)
−
(
x
k
)
≥ 0 if and only
if k < x2 . Thus we can estimate generalized binomial coefficients with an “almost central
binomial coefficient”:
( r
2
k
)
≤
( r
2
⌊ r4 ⌋
)
. Thus
|Rγ | ≤
r−1∑
l=2
⌊ r2 ⌋∑
k=1
( r
2
⌊ r4⌋
)
k!2k ≤
r−1∑
l=2
⌊ r2 ⌋∑
k=1
( r
2
⌊ r4⌋
)⌊r
2
⌋
!2⌊
r
2 ⌋
≤ (r − 1)
⌊r
2
⌋
2
r
2
⌊r
2
⌋
!2⌊
r
2 ⌋ ≤ (r − 1)
⌊r
2
⌋ ⌊r
2
⌋
!2r.
Proof of Theorem 1.1. Notice that there are (r − 2)! Cayley maps with S = R \ {e} (this
is just the number of cyclic orderings r), the total number of Cayley maps must be greater
than that, so combining Lemmas 2.2 and 2.3, we deduce that the fraction of Cayley maps
on R admitting a group of automorphisms larger than R is less than
((r − 1) r2 ⌊r/2⌋!2
r)(27(log2 r)
2+12 log2 r)
(r − 2)!
,
which goes to 0 when r → ∞.
3 Unlabelled version
We have so far implicitly considered a probability distribution which is uniform on labelled
Cayley graphs on a fixed group R. But of course it can also make sense to not distinguish
between maps on the same group that are mapped to one another by a group automorphism.
We can show quite easily that Theorem 1.2 still holds. To be precise we consider two
8 Ars Math. Contemp. 24 (2024) #P1.01 / 1–9
Cayley maps CM(R,S0, r0) and CM(R,Sα1 , r1) on the same group R and we say that they
are equivalent if there exists a group automorphism α of R such that CM(R,S0, r0) =
CM(R,S1, α ◦ r1 ◦ α−1). We will call these equivalence classes unlabelled Cayley maps.
Proof of Theorem 1.2. This is a minor adaptation of the proof of Theorem 1.1. Of course
unlabelled Cayley maps are at most in the same number as their labelled counterparts, so
we can still apply Lemmas 2.2 and 2.3 to deduce that those admitting automorphisms other
than those given by the action of R are fewer than ((r−1) r2 ⌊r/2⌋!2
r)(27(log2 r)
2+12 log2 r),
where r = |R|. Moreover each equivalence class of labelled Cayley maps can contain at
most |Aut(R)| elements. Using nothing more than the classic estimate |Aut(R)| ≤ rlog2 r
we can deduce there are at least (r−2)!
rlog2 r
unlabelled Cayley maps with S = R \ {e}. Then
the fraction of non MRRs among unlabelled Cayley maps is bounded by
((r − 1) r2 ⌊r/2⌋!2
r)(27(log2 r)
2+12 log2 r)rlog2 r
(r − 2)!
,
which again goes to 0 when r → ∞.
ORCID iDs
Pablo Spiga https://orcid.org/0000-0002-0157-7405
Dario Sterzi https://orcid.org/0000-0002-0317-4394
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.02 / 11–29
https://doi.org/10.26493/1855-3974.2616.4a9
(Also available at http://amc-journal.eu)
Factorizing the Rado graph and
infinite complete graphs*
Simone Costa † , Tommaso Traetta
DICATAM - Sez. Matematica, Università degli Studi di Brescia,
Via Valotti 9, I-25123 Brescia, Italy
Received 30 April 2021, accepted 18 January 2023, published online 1 August 2023
Abstract
Let F = {Fα : α ∈ A} be a family of infinite graphs, together with Λ. The Factor-
ization Problem FP (F ,Λ) asks whether F can be realized as a factorization of Λ, namely,
whether there is a factorization G = {Γα : α ∈ A} of Λ such that each Γα is a copy of Fα.
We study this problem when Λ is either the Rado graph R or the complete graph Kℵ
of infinite order ℵ. When F is a countably infinite family, we show that FP (F , R) is
solvable if and only if each graph in F has no finite dominating set. We also prove that
FP (F ,Kℵ) admits a solution whenever the cardinality of F coincides with the order and
the domination numbers of its graphs.
For countable complete graphs, we show some non existence results when the dom-
ination numbers of the graphs in F are finite. More precisely, we show that there is no
factorization of KN into copies of a k-star (that is, the vertex disjoint union of k countable
stars) when k = 1, 2, whereas it exists when k ≥ 4, leaving the problem open for k = 3.
Finally, we determine sufficient conditions for the graphs of a decomposition to be
arranged into resolution classes.
Keywords: Factorization problem, resolution problem, Rado graph, infinite graphs.
Math. Subj. Class. (2020): 05C63, 05C70
1 Introduction
We assume that the reader is familiar with the basic concepts in (infinite) graph theory, and
refer to [10] for further details.
*The authors gratefully acknowledge support from GNSAGA of Istituto Nazionale di Alta Matematica.
†Corresponding author.
E-mail addresses: simone.costa@unibs.it (Simone Costa), tommaso.traetta@unibs.it (Tommaso Traetta)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
12 Ars Math. Contemp. 24 (2024) #P1.02 / 11–29
In this paper all graphs will be simple, namely, without multiple edges or loops. As
usual, we denote by V (Λ) and E(Λ) the vertex set and the edge set of a simple graph Λ,
respectively. We say that Λ is finite (resp. infinite) if its vertex set is so, and refer to the
cardinality of V (Λ) and E(Λ) as the order and the size of Λ, respectively. Note that in the
finite case |E(Λ)| ≤
(|V (Λ)|
2
)
, whereas if Λ is infinite, then its order, which is a cardinal
number, is greater than or equal to its size. We use the notation Kv for any complete graph
of order v, and denote by KV the complete graph whose vertex set is V .
Given a subgraph Γ of a simple graph Λ, we denote by Λ \ Γ the graph obtained from
Λ by deleting the edges of Γ. If Γ contains all possible edges of Λ joining any two of its
vertices, then Γ is called an induced subgraph of Λ (in other words, an induced subgraph is
obtained by vertex deletions only). Instead, if V (Γ) = V (Λ), then Γ is called a spanning
subgraph or a factor of Λ (hence, a factor is obtained by edge deletions only). If Γ is also h-
regular, then we speak of an h-factor. We recall that a set D of vertices of Λ is dominating if
all other vertices of Λ are adjacent to some vertex of D. The minimum size of a dominating
set of Λ is called the domination number of Λ. Finally, we say that Λ is locally finite if its
vertex degrees are all finite.
A decomposition of Λ is a set G = {Γ1,Γ2, . . .} of subgraphs of Λ whose edge-sets
partition E(Λ). If the graphs Γi are all isomorphic to a given subgraph Γ of Λ, then we
speak of a Γ-decomposition of Λ. When Γ and Λ are both complete graphs, we obtain
2-designs. More precisely, a Kk-decomposition of Kv is equivalent to a 2-(v, k, 1) design.
Classically, the graphs Γi and Λ are taken to be finite, and the same usually holds for the
parameters v and k of a 2-design. However, there has been considerable interest in designs
on a infinite set of v points, mainly when k = 3. In this case, we obtain infinite Steiner
triple systems whose first explicit constructions were given in [12, 13]. Further results
concerning the existence of rigid, sparse, and perfect countably Steiner triple systems can
be found in [6, 7, 11]. Results showing that any Steiner system can be extended are given
in [1, 15]. The existence of large sets of Steiner triple systems for every infinite v (and
more generally, of infinite Steiner systems) can be found in [4]. Also, infinite versions of
topics in finite geometry, including infinite Steiner triple systems and infinite perfect codes
are considered in [3]. A more comprehensive list of results on infinite designs can be found
in [9].
When each graph of a decomposition G of Λ is a factor (resp. h-factor), we speak of a
factorization (resp. h-factorization) of Λ. Also, when the factors of G are all isomorphic to
Γ, we speak of a Γ-factorization of Λ. A factorization of Kv into factors whose components
are copies of Kk is equivalent to a resolvable 2-(v, k, 1) design.
In this paper, we consider the Factorization Problem for infinite graphs, which is here
stated in its most general version.
Problem 1.1. Let Λ be a graph of order ℵ and let F = {Fα : α ∈ A} be a family
of (non-empty) infinite graphs, not necessarily distinct, each of which has order ℵ, with
ℵ ≥ |A|.
The Factorization Problem FP (F ,Λ) asks for a factorization G = {Γα : α ∈ A} of Λ
such that Γα is isomorphic to Fα, for every α ∈ A. If Λ is the complete graph of order ℵ,
we simply write FP (F). If in addition to this each Fα is isomorphic to a given graph F
and |A| = ℵ, we write FP (F ). 1
1Since in this case the factorization problem can be seen as a generalization of the Oberwolfach problem, in
[8] the problem FP (F ) was denoted by OP (F ).
S. Costa and T. Traetta: Factorizing the Rado graph and infinite complete graphs 13
Note that the graphs of F are allowed to have zero degree vertices. This means that if
Fα contains isolated vertices, then Γα is a copy of Fα covering all vertices of Kℵ (hence a
factor of Kℵ) and having the same number of isolated vertices as Fα. Otherwise, Fα has
no zero degree vertices, hence Γα is a factor of Kα in the usual sense, that is, a spanning
subgraph without isolated vertices.
As far as we know, there are only four papers dealing with the Factorization Problem
for infinite complete graphs, and two of them, concern classic designs. In [14] it is shown
that there exists a resolvable 2-design whenever v = |N| and k is finite; these designs have,
in addition, a cyclic automorphism group G acting sharply transitively on the vertex set;
briefly they are G-regular. In [9] it is shown that every infinite 2-design with k < v is
necessarily resolvable, and when k = v, both resolvable and non-resolvable designs exist.
We point out that in [9, 14] both these results are proven more generally for t-designs
whenever t ≥ 2 is finite.
Furthermore, in [2] the authors construct a G-regular 1-factorization of a countably
infinite complete graph for every finitely generated abelian infinite group G. Finally, [8]
proves the following.
Theorem 1.2. Let F be a graph whose order is the cardinal number ℵ. FP (F ) has a
G-regular solution whenever the following two conditions hold:
(1) F is locally finite,
(2) G is an involution free group of order ℵ.
Note that this result generalizes the one obtained in [14] to any complete graph of
infinite order ℵ, blocks of any size less than ℵ, and groups G not necessarily cyclic. Fur-
thermore, Theorem 1.2 can also be seen as a generalization of the result in [2] to complete
graphs of any infinite order.
When dealing with infinite graphs, a central role is played by the Rado graph R (see [16]),
named after Richard Rado who gave one of its first explicit constructions. Indeed, R is the
unique countably infinite random graph, and it can be constructed as follows: V (R) = N
and a pair {i, j} with i < j is an edge of R if and only if the i-th bit of the binary repre-
sentation of j is one. R shows many interesting properties, such as the universal property:
every finite or countable graph can be embedded as an induced subgraph of R.
When replacing the concept of induced subgraph with the dual one of factor, a weaker
result holds. Indeed, in [5] it is pointed out that a countable graph F can be embedded as
a factor of R if and only if the domination number of F is infinite. In the same paper, it is
further shown that FP (F , R) has a solution whenever F is infinite and each of its graphs is
locally finite. Note that a locally finite countable graph has infinite domination number, but
the converse is not true: for example, the Rado graph is not locally finite and it has no finite
dominating set (indeed, for every D = {i1, . . . , it} ⊂ N, there exists an integer j ∈ N
whose binary representation has 0 in positions i1, . . . , it, which means that j is adjacent
with no vertex of D).
In this paper, we extend this result to any countable family F of admissible graphs.
More precisely, we prove the following. We point out that throughout the paper, any count-
able family (or graph) is understood to be infinite.
Theorem 1.3. Let F be a countable family of countable graphs. Then, FP (F , R) has a
solution if and only if the domination number of each graph of F is infinite.
14 Ars Math. Contemp. 24 (2024) #P1.02 / 11–29
Furthermore, we prove the solvability of FP (F) whenever the size of F coincides with
the order and the domination number of its graphs.
Theorem 1.4. Let F be a family of graphs, each of which has order ℵ. FP (F) has a
solution whenever the following two conditions hold:
(1) |F| = ℵ, and
(2) the domination number of each graph in F is ℵ.
When F contains only copies of a given graph F satisfying condition (1) of Theo-
rem 1.2 (i.e., F is locally finite), then F satisfies both conditions (1) and (2) of Theo-
rem 1.4. Therefore, Theorem 1.4 can be seen as a generalization of Theorem 1.2, even
though it does not provide any information on the automorphisms of a solution to FP.
Note that if we just require that the domination number of each graph of F is ℵ, there
may exist factorizations with fewer factors than ℵ; this means that the two conditions in
Theorem 1.4 are independent. Indeed, the Rado graph R has no finite dominating set and
Corollary 2.4 shows that for every n ≥ 2 there exists a factorization of KN into n copies
of R. We point out that Theorem 1.4 constructs instead factorizations of KN into infinite
copies of R.
The paper is organized as follows. In Sections 2 and 3, we prove the main results
of this paper, Theorems 1.3 and 1.4. In Section 4, we deal with F -factorizations of KN
when F belongs to a special class of graphs with finite domination number (and hence not
satisfying condition (2) of Theorem 1.4): the countable k-stars (briefly, Sk), that is, the
vertex disjoint union of k countable stars. We prove that FP (Sk) has a solution whenever
k > 3, and there is no solution for k ∈ {1, 2}. This shows that there are families F of
graphs for which FP (F) is not solvable. We leave open the problem when k = 3. In the
last section, inspired by [9], we provide a sufficient condition for a decomposition F of Kℵ
to be resolvable (i.e., the graphs of F can be partitioned into factors of Kℵ).
2 Factorizing the Rado graph
In this section, we prove Theorem 1.3. Also, since the Rado graph R is self-complementary,
that is, KN \ R is isomorphic to R, we obtain as a corollary the countable version of
Theorem 1.4.
We start by recalling an important characterization of the Rado graph (see, for example,
[5]).
Theorem 2.1. A countable graph is isomorphic to the Rado graph if and only if it satisfies
the following property:
⋆ for every disjoint finite sets of vertices U and W , there exists a vertex z adjacent to
all the vertices of U and non-adjacent to all the vertices of V .
Property ⋆ is usually referred to as the existentially closed property. Therefore, Theo-
rem 2.1 states that, up to isomorphism, there is exactly one existentially closed countable
graph: the Rado graph.
Now we slightly generalize the construction of the Rado graph given in the introduction.
Definition 2.2. Given a set I ⊂ {0, . . . , q − 1}, with 1 ≤ |I| < q, we denote by RqI the
following graph: V (RqI) = N, and {x, y}, with x < y, is an edge of R
q
I whenever the x-th
digit of y in the base q expansion of y belongs to I .
S. Costa and T. Traetta: Factorizing the Rado graph and infinite complete graphs 15
Cleraly, when q = 2 and I = {1} we obtain our initial description of the Rado graph
(i.e. R = R2{1}).
Proposition 2.3. Every graph RqI is isomorphic to the Rado graph.
Proof. By Theorem 2.1, it is enough to show RqI is existentially closed. We assume, with-
out loss of generality, that 0 ∈ I while 1 ̸∈ I , and let U and V be two disjoint finite subsets
of N. Then there are infinitely many positive integers whose base q expansion has 0 in each
position u ∈ U and 1 in each position v ∈ V . Denoting by z one of these integers larger
than max(U ∪ V ), we have that z is adjacent to all the vertices of U but to none in V .
Note that KN =
⋃q−1
i=0 R
q
{i} and R
q
{0,...,q−2} =
⋃q−2
i=0 R
q
i . Considering that the R
q
{i}s
are pairwise edge-disjoint and isomorphic to the Rado graph, by taking n = q−1 we obtain
the following.
Corollary 2.4. For every positive integer n, the graphs R and KN can be factorized into n
and n+ 1 copies of R, respectively.
The following result is crucial to prove Theorem 1.3. It strengthens a result given in
[5] and allows us to suitably embed in the Rado graph R any countable graph with infinite
domination number.
Proposition 2.5. Let F be a countable graph with no finite dominating set. For every edge
e ∈ E(R), there exists an embedding σe of F in R such that:
(1) σe(F ) is a spanning subgraph of R containing the edge e;
(2) R \ σe(F ) is isomorphic to R.
Proof. By Proposition 2.3, the graphs R3{0,1}, R
3
{0} and R
3
{1} are isomoprhic to R. There-
fore, we can take R = R3{0,1}.
Let e be an edge of R = R3{0} ∪R
3
{1}. We can assume without loss of generality that e
lies in R3{0}. In [5, Proposition 8], it is shown that there exists an embedding σe of F into
the Rado graph R3{0} ⊂ R satisfying condition (1). It is then left to prove that condition (2)
holds. By Theorem 2.1, this is equivalent to saying that R \ σe(F ) satisfies ⋆.
Let U and V be two finite disjoint subsets of N. Clearly, there are infinitely many
positive integers whose base 3 expansion has 1 in each position u ∈ U and 2 in each
position v ∈ V . Let z be one of these integers larger than max(U ∪ V ). Since R \ σe(F )
contains R3{1} and it is edge-disjoint from R
3
{2}, it follows that z is adjacent in R\σe(F ) to
all the vertices of U and is non-adjacent to all the vertices of V . This means that R \σe(F )
is existentially closed.
We are now ready to prove Theorem 1.3, whose statement is recalled here, for clarity.
Theorem 1.3. Let F be a countable family of countable graphs. Then, FP (F , R) has a
solution if and only if the domination number of each graph of F is infinite.
Proof. Since the Rado graph has no finite dominating set, the same holds for its spanning
subgraphs. Hence, each graph of F must have infinite domination number. Under this
assumption, we are going to show that FP (F , R) has a solution.
16 Ars Math. Contemp. 24 (2024) #P1.02 / 11–29
By definition of Rado graph, it is easy to see that |E(R)| = ℵ0, which is also the car-
dinality of F . Let E(R) = {e1, . . . , en, . . .} and F = {F1, . . . , Fn, . . .}. By recursively
applying Proposition 2.5, we obtain a sequence of isomorphisms σei : Fi → Γi satisfying
for each i ∈ N the following properties:
• Γi is a spanning subgraph of R;
• R \ (Γ1 ∪ Γ2 ∪ · · · ∪ Γi−1) is isomorphic to R and contains Γi;
• ei lies in Γ1 ∪ Γ2 ∪ · · · ∪ Γi.
It follows that the Γis are pairwise edge-disjoint factors of R which partition E(R). There-
fore, {Γi : i ∈ N} is a solution to FP (F , R).
The proof of Theorem 1.3 allows us to construct solutions to FP (F , R) even when
the cardinality of F is finite, provided that F contains a copy of the Rado graph. In other
words, we have the following.
Corollary 2.6. Let F be a finite family of countable graphs such that
(1) F contains at least one graph isomorphic to the Rado graph;
(2) the domination number of each graph in F is infinite.
Then, FP (F , R) has a solution.
Recalling that R is self complementary, the countable version of Theorem 1.4 can be
easily obtained as a corollary to Theorem 1.3.
Corollary 2.7. Let F be a countable family of countable graphs. FP (F) has a solution
whenever the domination number of each graph in F is infinite.
Proof. Recall that R2{0} and R
2
{1} are copies of R which together factorize KN. Therefore,
it is enough to partition F into two countable families F1 and F2, and then apply Theo-
rem 1.3 to get a solution Gi to FP (Fi, R2{i}), for i = 0, 1. Clearly, G1 ∪ G2 provides a
solution to FP (F).
The natural generalization of property ⋆ to a generic cardinality ℵ is the following one:
⋆ℵ for every disjoint sets of vertices U and W whose cardinality is smaller than ℵ, there
exists a vertex z adjacent to all the vertices of U and non-adjacent to all the vertices
of V .
Then, using the transfinite induction (see Theorem 3.5 below), one could also prove the
following generalization of Proposition 2.1:
Proposition 2.8. Any two graphs of order ℵ that satisfy property ⋆ℵ are pairwise isomor-
phic.
Therefore, we can refer to any graph of order ℵ and satisfying property ⋆ℵ as the ℵ-
Rado graph Rℵ. Its existence is guaranteed under the Generalized Continuum Hypothesis
(GCH) which states that if ℵ′ ≺ ℵ then 2ℵ′ ⪯ ℵ. Under GCH, one can see that the set S of
all q-ary sequences of length ≺ ℵ has size ℵ. Indeed, for every ℵ′ ≺ ℵ, the set of all q-ary
S. Costa and T. Traetta: Factorizing the Rado graph and infinite complete graphs 17
sequences of length ℵ′ has cardinality 2ℵ′ , and by GCH we have that 2ℵ′ ⪯ ℵ; therefore,
|S| has size at most ℵ. Clearly, ℵ′ ≺ 2ℵ′ ⪯ |S| for every ℵ′ ≺ ℵ. It then follows that
|S| = ℵ.
This means that the construction of the countable Rado graph (Definition 2.2) based on
representing every natural number with a finite q-ary sequence (its base q expansion) can
be generalized to any order.
By assuming that GCH holds, we will prove as a corollary to Theorem 1.4 the following
generalization of Theorem 1.3.
Theorem 2.9. Let F be a family of graphs of order ℵ and assume that |F| = ℵ. Then
FP (F , Rℵ) has a solution if and only if the domination number of each graph in F is ℵ.
3 Factorizing infinite complete graphs
We say that a graph or a set of vertices is ℵ-small (resp. ℵ-bounded) if their order or
cardinality is smaller than ℵ (resp. smaller than or equal to ℵ). Given two graphs F and
Λ of order ℵ, we denote by Σℵ(F,Λ) the set of all graph embeddings between an induced
ℵ-small subgraph of F and a subgraph of Λ. A partial order on Σℵ(F,Λ) can be easily
defined as follows: if σ : G → Γ and σ′ : G′ → Γ′ are embeddings of Σℵ(F,Λ), we say
that σ ≤ σ′ whenever σ′ is an extension of σ, namely, G and Γ are subgraphs of G′ and Γ′,
respectively, and σ′|G = σ (where σ′|G is the restriction of σ′ to G).
Lemma 3.1. Let F be a graph of order ℵ and with no ℵ-small dominating set. Also, let Θ
be an ℵ-small subgraph of Kℵ, and let σ ∈ Σℵ(F,Kℵ \Θ).
(1) If v ∈ V (F ), then there is an embedding σ′ : G′ → Γ′ in Σℵ(F,Kℵ \Θ) such that
|V (G′)| ≤ |V (G)|+ 1, σ ≤ σ′ and v ∈ V (G′);
(2) If x ∈ V (Kℵ), then there is an embedding σ′′ : G′′ → Γ′′ in Σℵ(F,Kℵ \ Θ) such
that
|V (G′′)| ≤ |V (G)|+ 1, σ ≤ σ′′ and x ∈ V (Γ′′).
Proof. Let σ : G → Γ be an embedding in Σℵ(F,KN \ Θ), and let v ∈ V (F ) and
x ∈ V (Kℵ). Clearly, when v ∈ V (G) or x ∈ V (Γ), we can take σ′ = σ or σ′′ = σ,
respectively. Therefore, we can assume v ̸∈ V (G) and x ̸∈ V (Γ).
1. Let G′ be the subgraph of F induced by v and V (G). Since V (Θ) is ℵ-small, we
can choose a ∈ V (Kℵ) \ V (Θ) and let σ′ : V (G) ∪ {v} → V (Γ) ∪ {a} be the
extension of σ such that σ′(v) = a. Setting Γ′ = σ′(G′), we have that σ′ is the
required embedding of Σℵ(F,Kℵ \Θ).
2. Since F has no ℵ-small dominating set, V (G) (which is an ℵ-small set) cannot be
a dominating set for F . Hence, there is a vertex a ∈ V (F ) that is not adjacent to
any of the vertices of G. We denote by G′′ (resp., Γ′′) the graph obtained by adding
a to G (resp., x to Γ) as an isolated vertex. Clearly, G′′ is an induced subgraph of
F ; also, Γ′′ and Θ have no edge in common, since E(Γ′′) = E(Γ). Therefore, the
extension σ′′ : G′′ → Γ′′ of σ such that σ′′(a) = x is the required embedding of
Σℵ(F,Kℵ \Θ).
18 Ars Math. Contemp. 24 (2024) #P1.02 / 11–29
From now on, we will work within the Zermelo-Frankel axiomatic system with the
Axiom of Choice in the form of the Well-Ordering Theorem. We recall the definition of a
well-order.
Definition 3.2. A well-order ≺ on a set X is a total order on X with the property that every
non-empty subset of X has a least element.
The following theorem is equivalent to the Axiom of Choice.
Theorem 3.3 (Well-Ordering). Every set X admits a well-order ≺.
Given an element x ∈ X , we define the section X≺x associated to it:
X≺x = {y ∈ X : y ≺ x}.
Corollary 3.4. Every set X admits a well-order ≺ such that the cardinality of any section
is smaller than |X|.
Proof. Let us consider a well-order ≺ on X . Let x be the smallest element such that X≺x
has the same cardinality as X . The set Y = X≺x is such that all its sections with respect
to the order ≺ have smaller cardinality. Since Y instead has the same cardinality as X , the
order ≺ on Y induces an order ≺′ on X with the required property.
We recall now that well-orderings allow proofs by induction.
Theorem 3.5 (Transfinite induction). Let X be a set with a well-order ≺ and let Px denote
a property for each x ∈ X . Set 0 = minX and assume that:
• P0 is true, and
• for every x ∈ X , if Py holds for every y ∈ X≺x, then Px holds.
Then Px is true for every x ∈ X .
We are now ready to prove Theorem 1.4. The idea behind the proof can be better under-
stood by restricting our attention to the countable case, ℵ = N. To solve
FP ({Fα : α ∈ N}), we first order the edges of KN : {e0, e1, . . . , eγ , . . .}. Then, we define
embeddings σβα : G
β
α → Γβα where Gβα is an induced subgraph of Fα, and Γβα is a subgraph
of KN. These embeddings are obtained by recursively applying Lemma 3.1 which adds, at
each step, a vertex to Gβα and a vertex to Γ
β
α and makes sure that the vertex β belongs to
both these graphs (this procedure can be seen as a variation of Cantor’s “back-and-forth”
method). We also make sure that, for every γ, the graphs Γγ0 ,Γ
γ
1 , . . . ,Γ
γ
γ are pairwise edge-
disjoint and contain between them the edge eγ . The solution to FP ({Fα : α ∈ N}) will
be represented by G = {Γα : α ∈ N} where Γα =
⋃
β Γ
β
α.
Theorem 1.4 Let F be a family of graphs, each of which has order ℵ. FP (F) has a
solution whenever the following two conditions hold:
1. |F| = ℵ, and
2. the domination number of each graph in F is ℵ.
S. Costa and T. Traetta: Factorizing the Rado graph and infinite complete graphs 19
Proof. Let F = {Fα : α ∈ A}. We consider a well-order ≺ on A satisfying Corollary 3.4.
Since by assumption |V (Fα)| = |A| = ℵ, for every α ∈ A, we can take V (Fα) =
V (Kℵ) = A and index the edges of Kℵ over A: E(Kℵ) = {eα : α ∈ A}.
To prove the assertion, we construct a chain of families (Eγ)γ∈A, where
Eγ := {σβα : Gβα → Γβα | σβα ∈ Σℵ(Fα,Kℵ), (α, β) ∈ A⪯γ ×A⪯γ},
which satisfy the ascending property, that is, Eγ′ ⊆ Eγ if γ′ ⪯ γ, and the following three
conditions:
(1γ) for every (α, β) ∈ A⪯γ × A⪯γ and β′ ≺ β we have that σβ
′
α ≤ σβα and
β ∈ V (Gβα) ∩ V (Γβα);
(2γ) for every β ∈ A⪯γ , the graphs Γβα : α ⪯ β are pairwise edge-disjoint, and the edge
eβ belongs to their union;
(3γ) for every α, β ∈ A⪯γ , the graph Γβα is either finite or |A⪯γ |-bounded.
The desired factorization of Kℵ is then G = {Γα : α ∈ A}, where Γα =
⋃
β∈A Γ
β
α for
every α ∈ A. Indeed, property (1γ) guarantees that each Γα is a factor of Kℵ isomorphic
to Fα. Also, property (2γ) ensures that the Γαs are pairwise edge-disjoint and between
them contain all the edges of Kℵ.
We proceed by transfinite induction on γ.
BASE CASE. Let 0 = minA, choose an edge e ∈ E(F0) and let σ ∈ Σℵ(F0,Kℵ) be
the embedding that maps e to e0. By Lemma 3.1, there exists an embedding σ00 : G
0
0 → Γ00
in Σℵ(F0,Kℵ) such that Γ00 is a finite graph and
σ ≤ σ00 and 0 ∈ V (G00) ∩ V (Γ00).
Clearly, E0 := {σ00} satisfies properties (10), (20) and (30).
TRANSFINITE INDUCTIVE STEP. We assume that, for any γ′ ≺ γ, there is a family
Eγ′ satisfying properties (1γ′), (2γ′ ) and (3γ′ ), and prove that it can be extended to a family
Eγ that satisfies properties (1γ), (2γ) and (3γ). Clearly it is enough to provide the maps σβα
where either α = γ or β = γ.
We start by constructing the maps σγα for every α ≺ γ. We proceed by transfinite
induction on α.
• Base case. Set Θ0 :=
⋃
α,β≺γ Γ
β
α and note that, by property (3γ′ ), Θ0 is ℵ-small.
We also set σ≺γ0 :
⋃
β≺γ G
β
0 →
⋃
β≺γ Γ
β
0 to be the map of Σℵ(F0,Kℵ \Θ0) whose
restriction to Gβ0 is σ
β
0 . We note that property (3γ′ ) guarantees that the order of⋃
β≺γ G
β
0 is either finite or |A⪯γ |-bounded, hence ℵ-small.
Therefore, we can apply Lemma 3.1 (with σ = σ≺γ0 ) to obtain the map σ
γ
0 : G
γ
0 →
Γγ0 in Σℵ(F0,Kℵ \ Θ0) such that |V (Γ
γ
0)| ≤ |V (
⋃
β≺γ Γ
β
0 )| + 2 and, for every
γ′ ≺ γ,
σγ
′
0 ≤ σ
γ
0 and γ ∈ V (G
γ
0) ∩ V (Γ
γ
0).
20 Ars Math. Contemp. 24 (2024) #P1.02 / 11–29
• Inductive step. Assume we have defined the maps σγα′ for every α
′ ≺ α, and set
Θα :=
⋃
α′≺α
Γγα′ ∪
⋃
α≺α′≺γ,β≺γ
Γβα′ .
As before, by Lemma 3.1 there exists σγα : G
γ
α → Γγα in Σℵ(Fα,Kℵ \Θα) such that
|V (Γγα)| ≤ |V (
⋃
β≺γ Γ
β
α)|+ 2 and, for every γ′ ≺ γ,
σγ
′
α ≤ σγα and γ ∈ V (Gγα) ∩ V (Γγα).
Finally, we define the maps σβγ when β ⪯ γ. We set Θ :=
⋃
α≺γ Γ
γ
α and proceed by
transfinite induction on β.
• Base case. If eγ ∈ Θ, let σ be the empty map of Σℵ(Fγ ,Kℵ \Θ). Otherwise, chose
an edge e ∈ E(Fγ), and let σ ∈ Σℵ(Fγ ,Kℵ \ Θ) be the embedding that maps e to
eγ . By Lemma 3.1, there exists σ0γ : G
0
γ → Γ0γ in Σℵ(Fγ ,Kℵ \Θ) such that Γ0γ is a
finite graph and
σ ≤ σ0γ and 0 ∈ V (G0γ) ∩ V (Γ0γ).
• Inductive step. Assume we have defined the maps σβ
′
γ for any β
′ ≺ β. Again by
Lemma 3.1, there exists σβγ : G
β
γ → Γβγ in Σℵ(Fγ ,Kℵ \ Θ) such that |V (Γβγ )| ≤
|V (
⋃
β′≺β Γ
β′
γ )|+ 2 and, for any β′ ≺ β,
σβ
′
γ ≤ σβγ and β ∈ V (Gβγ ) ∩ V (Γβγ ).
It follows from the construction that the family
Eγ := {σβα : Gαβ → Γβα | σβα ∈ Σℵ(Fα,Kℵ), α, β ⪯ γ}
satisfies properties (1γ), (2γ) and (3γ).
Assuming that GCH holds, we obtain Theorem 2.9 as a corollary to Theorem 1.4.
Theorem 2.9. Let F be a family of graphs of order ℵ and assume that |F| = ℵ. Then
FP (F , Rℵ) has a solution if and only if the domination number of each graph in F is ℵ.
Proof. By property ⋆ℵ, one can easily check that Rℵ is self-complementary, that is Kℵ\Rℵ
is isomorphic to Rℵ, and the domination number of the ℵ-Rado graph is ℵ. Therefore, the
domination number of each graph of F must be ℵ.
To prove sufficiency, note that F ′ := F∪{Rℵ} satisfies the hypothesis of Theorem 1.4.
Therefore FP (F ′) admits a solution. This means that F factorizes Kℵ \Rℵ ≃ Rℵ.
4 The factorization problem for k-stars
Theorem 1.4 does not provide solutions to FP (F ) whenever the graph F has a dominating
set of cardinality less than its order. In particular, if F is countable with a finite dominating
set, then the existence of a solution to FP (F ) is an open problem. In this section, we
consider a special class of such graphs, the k-stars Sk. More precisely,
S. Costa and T. Traetta: Factorizing the Rado graph and infinite complete graphs 21
• the star S1, which we also call a 1-star, is the graph with vertex-set N whose edges
are of the form {0, i} for every i ∈ N \ {0};
• the k-star Sk is the vertex-disjoint union of k stars.
Note that Sk contains exactly k vertices of infinite degree, which we call centers and form
a finite dominating set of Sk.
In the following, we show that FP (Sk) has no solution whenever k ∈ {1, 2}, while it
admits a solution for every k > 3. Unfortunately, we leave open the problem for 3-stars.
4.1 The case k ∈ {1, 2}
Proposition 4.1. FP (S1) has no solution.
Proof. Assume for a contradiction that there is a factorization G of KN into 1-stars. Choose
any star Γ ∈ G and let g denote its center. Note that all the edges of KN incident with g
belong Γ. By recalling that G is a factorization of KN (and that a 1-star has no isolated
vertices), it follows that g cannot be a vertex in any other star of G. Therefore, every star of
G \ {Γ} is not spanning, contradicting the assumption.
With essentially the same proof, one obtains the following.
Remark 4.2. Let F be the vertex-disjoint union of S1 with a finite set of isolated vertices.
Then FP (F ) has no solution.
To prove the non-existence of a solution to FP (S2) it will be useful the following
lemma.
Lemma 4.3. If G is a factorization of KN into k-stars, then there is at most one vertex of
KN that is never a center in any k-star of G. It follows that |G| = |N|.
Proof. It is enough to notice that every pair {a, b} of vertices of KN is the edge of some
2-star Γ of G; hence, either a or b is a center of Γ.
Proposition 4.4. FP (S2) has no solution.
Proof. Assume for a contradiction that there is a factorization G of KN into 2-stars. For
every Γ ∈ G, letting c be a center of Γ, we denote by Γ(c) the set of vertices adjacent with
c in Γ (i.e., the neighborhood of c in Γ).
Choose any 2-star Γ ∈ G and let a and b denote its centers. Also, let Γ′ be the 2-star of
G \ {Γ} containing the edge {a, b}. Without loss of generality, we can assume that a is a
center of Γ′. Finally, by Lemma 4.3, we can choose x ∈ Γ′(a) \ {b} such that there exists
a 2-star Γ′′ ∈ G having x as one of its centers.
Since Γ is a factor of KN, it follows that x ∈ Γ(b). In other words, Γ ∪ Γ′ contains the
edges {x, a} and {x, b}. Therefore, a, b ̸∈ Γ′′(x). Since Γ′′ is a factor of KN and {a, b}
is an edge of Γ, it follows a, b ∈ Γ′′(y), where y is the other center of Γ′′. In other words,
{y, a} and {y, b} belong to Γ′′, hence y cannot lie in Γ, contradicting the fact that Γ is a
factor.
22 Ars Math. Contemp. 24 (2024) #P1.02 / 11–29
4.2 The case k ≥ 4
In this section we prove the solvability of FP (Sk) whenever k ≥ 4. For our constructions
we need to introduce the following notation.
Let D be an integral domain and set V = D × {0, 1, . . . , h}, for h ≥ 0. For the sake
of brevity, we will denote each pair (a, i) ∈ V by ai. Given a graph Γ with vertices in V ,
for every a, b ∈ D we denote by aΓ + b the graph obtained by replacing each vertex xi of
Γ with (ax+ b)i; further, if {xi, yi} is an edge of Γ, then {(ax+ b)i, (ay+ b)i} is an edge
of aΓ + b. Also, we denote by OrbD(Γ) = {Γ + d : d ∈ D} the D-orbit of Γ, that is, the
set of all translates of Γ by the elements of D.
Proposition 4.5. For every k ≥ 4, there exists a k-star Γ with vertex set V = Z × {0, 1}
such that OrbZ(Γ) is a factorization of KV into k-stars.
Proof. We first deal with the case k = 4. Set Γ =
⋃4
i=1 Γi, where each Γi is the 1-star
with vertices in V = Z× {0, 1} and center xi defined as follows (see Figure 1):
• x1 = 00 and Γ1(x1) = {i0 : i ≥ 1};
• x2 = −11 and Γ2(x2) = {i1 : i ≥ 0} ∪ {−10};
• x3 = −20 and Γ2(x3) = {i1 : i ≤ −3};
• x4 = −21 and Γ4(x4) = {i0 : i ≤ −3}.
−61
−60
−51
−50
−41
−40
−31
−30
−21
−20
01
00
11
10
21
20
31
30
41
40
−11
−10
Γ1
Γ2
Γ3 ∪ Γ4
Figure 1: The graph Γ when k = 4.
We claim that G := OrbZ(Γ) is a factorization of KV into 4-stars. Denote by KU,W the
complete bipartite graph whose parts are U = Z × {0} and W = Z × {1}, and consider
the 1-factor I = {{i0, i1} : i ∈ Z} of KU,W . Clearly, KV decomposed into KU , KW ∪ I
and KU,W \ I . One can check that
• OrbZ(Γ1) decomposes KU ,
• OrbZ(Γ2) decomposes KW ∪ I , and
• OrbZ(Γ3 ∪ Γ4) decomposes KU,W \ I .
S. Costa and T. Traetta: Factorizing the Rado graph and infinite complete graphs 23
Γ1
Γ′1,2
∆0
∆∗1
00 10 20 30 40 50 60 70 80
00 10 20 40 60 80
30 50 70
Figure 2: Replacing Γ1 with Γ′1,2 produces Γ when k = 5.
Hence, G is a decomposition of KV . Considering that the Γis are pairwise vertex-
disjoint and their vertex-sets partition V , we have that Γ and each of its translates (under
the action of Z) are factors of KV isomorphic to a 4-star. Therefore, G is a factorization of
KV into 4-stars.
To deal with the case k ≥ 5, it is enough to replace the component Γ1 of Γ with a
(k − 3)-star Γ′1 satisfying the following conditions:
V (Γ′1) = V (Γ1), and (4.1)
OrbZ(Γ
′
1) decomposes KU . (4.2)
Indeed, letting Γ′ = (Γ \ Γ1) ∪ Γ′1, by condition (4.1) we have that Γ′ is a k-star with
vertex-set V . Recalling that OrbZ(Γ1) decomposes KU , by condition (4.2) it follows that
OrbZ(Γ
′) and OrbZ(Γ) decompose the same graph, that is, KV . Hence, OrbZ(Γ′) is a
factorization of KV into k-stars.
Let k = h + 3 with h ≥ 2. It is left to construct an h-star Γ′1,h satisfying conditions
(4.1) and (4.2), for every h ≥ 2. For sake of clarity, in the rest of the proof we identify
U = Z× {0} with Z. Therefore, Γ1 is the 1-star centered in 0 with Γ1(0) = {i : i ≥ 1}.
Let ∆j and ∆∗j be the 1-stars centered in cj = 2(2
j − 1) such that
∆j(cj) = {cj + i : 0 < i ≡ 2j (mod 2j+1)},
∆∗j (cj) = {cj + i : 0 < i ≡ 0 (mod 2j)},
for j ≥ 0, and set Γ′1,h = ∆0 ∪ ∆1 ∪ . . . ∪ ∆h−2 ∪ ∆∗h−1 for h ≥ 2. It is not difficult
to check that {∆j − cj : 0 ≤ j ≤ h− 2} ∪ {∆∗h−1 − ch−1} decomposes Γ1. Therefore,
OrbZ(Γ
′
1,h) and OrbZ(Γ1) decompose the same graph, that is, KU . Hence, Γ
′
1,h satisfies
condition (4.2).
We show that Γ′1,h is an h-star satisfying condition (4.1) by induction on h. If h = 2,
then V (∆0) = {0, 1, 3, 5, . . .} and V (∆∗1) = {2, 4, 6, . . .}. Therefore, Γ′1,2 = ∆0 ∪ ∆∗1 is
24 Ars Math. Contemp. 24 (2024) #P1.02 / 11–29
a 2-star with the same vertex-set as Γ1. Now assume that Γ′1,h is an h-star satisfying condi-
tion (4.1) for some h ≥ 2. Recalling the definition of Γ′1,h and Γ′1,h+1, and considering that
the vertex-sets of ∆h−1 and ∆∗h partition V (∆
∗
h−1), we have that Γ
′
1,h+1 is an (h+1)-star
with the same vertex-set as Γ′1,h, that is, V (Γ1), and this concludes the proof.
Propositions 4.1, 4.4 and 4.5 leave open FP (Sk) only when k = 3. In this case, an
approach similar to Theorem 4.5 cannot work, as shown in the following.
Proposition 4.6. There is no 3-star Γ with vertex-set V = Z× {0, 1, . . . , k} such that the
Z-orbit of Γ is an S3-factorization of KV .
Proof. Assume for a contradiction that there exists a 3-star Γ with vertex-set V = Z ×
{0, 1, . . . , k} such that G = OrbZ(Γ) is a factorization of KV .
We first notice that Γ must have at least a center in Z×{i}, for every i ∈ {0, 1, . . . , k}.
Indeed, if Γ has no center in Z × {i} for some i ∈ {0, 1, . . . , k}, then no edge of KZ×{i}
can be covered by G. Since Γ has 3 centers, it follows that k ≤ 2. Note that if k = 2,
the centers of Γ must be x0, y1, z2 for some x, y, z ∈ Z, but in this case the edge {x0, y1}
cannot lie in any translate of Γ. Therefore k ≤ 1.
If k = 1, without loss of generality we can assume that the centers of Γ are 00, x1 and
y1 with x ̸= y. Since the edge {00, x1} does not belong to Γ, it lies in some of its translates,
say Γ + z with z ̸= 0. This is equivalent to saying that {(−z)0, (x− z)1} ∈ Γ. It follows
that x− z = y, hence {(y−x)0, y1} ∈ Γ. Similarly, we can show that {(x− y)0, x1} ∈ Γ.
It follows that Γ cannot contain the edges {00, (x− y)0} and {00, (y − x)0}. This implies
that no edge of the form {w0, (x− y + w)0} lies in any translate of Γ, contradicting again
the assumption that G is a factorization of KV . Therefore k = 0.
Let V = Z and denote by ∆Γ the multiset of all differences y − x between any two
adjcent vertices x and y of Γ, with x < y:
∆Γ = {y − x : {x, y} ∈ E(Γ), x < y}.
It is not difficult to see that G = OrbZ(Γ) is a factorization of KZ if and only if ∆Γ =
N \ {0}. Denoting by Γ + i the translate of Γ obtained by replacing each vertex x ∈ V (Γ)
with x + i, one can easily see that ∆(Γ + i) = ∆Γ for every i ∈ Z. Therefore, up to
a translation, we can assume that the centers of Γ are 0, x, n with 0 < x < n. Now,
for every i ≥ n, denote by Γi the induced subgraph of Γ with vertex-set {0, 1, . . . , i}.
Also, let Γ∗ be the induced subgraph of Γ on the vertices {−3,−2,−1, 0, x, n}. Clearly,
|∆Γ∗| = 3, |∆Γi| = i− 2 and ∆Γi ⊂ {1, 2, . . . , i}. Also, since the multiset ∆Γ contains
all positive integers with no repetition, it follows that ∆Γ∗ and ∆Γi are disjoint, hence
∆Γi ⊂ {1, 2, . . . , i} \ ∆Γ∗ for every i ≥ n. Then, for i = max(∆Γ∗), we obtain the
following contradiction: i− 2 = |∆Γi| ≤ |{1, 2, . . . , i} \∆Γ∗| = i− 3.
5 The resolvability problem
Theorem 1.4 allows us to construct decompositions of Kℵ into ℵ graphs of specified type.
More precisely, we have the following.
Corollary 5.1. Let F = {Fα : α ∈ A} be an infinite family of (non-empty) ℵ-bounded
graphs, where ℵ = |A|. Then there exists a decomposition G = {Γα : α ∈ A} of Kℵ such
that each Γα is isomorphic to Fα.
S. Costa and T. Traetta: Factorizing the Rado graph and infinite complete graphs 25
Furthermore, for any β ∈ A such that the domination number of Fβ is less than ℵ, we
have that |V (Kℵ) \ V (Γβ)| = ℵ. Otherwise, for every 0 ⪯ ℵ′ ⪯ ℵ, the decomposition G
can be constructed so that |V (Kℵ) \ V (Γβ)| = ℵ′.
Proof. For every α ∈ A, set ℵα = ℵ if the domination number of Fα is less than ℵ;
otherwise, let 0 ⪯ ℵα ⪯ ℵ. By adding to each graph Fα a set of ℵα isolated vertices we
obtain a graph F ′α whose order and domination number are ℵ. Since the assumptions of
Theorem 1.4 are satisfied, there exists a factorization G′ = {Γ′α : α ∈ A} of Kℵ such that
each Γ′α is isomorphic to F
′
α. By replacing Γ
′
α with the isomorphic copy of Fα, we obtain
the desired decomposition G.
Inspired by [9], we ask under which conditions a decomposition G of Kℵ is resolvable,
namely, its graphs can be partitioned into factors of Kℵ, also called resolution classes. It
follows that a resolvable decomposition G of Kℵ must satisfy the following two conditions:
N1. if Γ ∈ G is not a factor of Kℵ, then |V (Kℵ) \ V (Γ)| ≥ min{|Γ| : Γ ∈ G};
N2.
G(z) ⊆ G(x) ∪ G(y) ⇒ G(z) ⊇ G(x) ∩ G(y),
where G(v) = {Γ ∈ G : v ∈ V (Γ)} is the set of all graphs of G passing through v.
In the following, we easily construct decompositions of Kℵ that do not satisfy the above
conditions, and therefore they are non-resolvable.
Example 5.2. Let F = {Fα : α ∈ A} be an infinite family of (non-empty) ℵ-bounded
graphs, where ℵ = |A|. Also, assume that the domination number of at least one of its
graphs, say Fβ , is ℵ. Then, by applying Corollary 5.1 with ℵ′ ≺ min{|Γα| : α ∈ A}, we
construct a decomposition that does not satisfy condition N1.
For instance, if ℵ = |N|, each Fα is a countable locally finite graph (hence, its dom-
ination number is ℵ) and ℵ′ = 1 for every β ∈ N, then we construct a decomposition
G = {Gβ : β ∈ N} of KN into connected regular graphs where V (Gβ) = N \ {xβ} for
some xβ ∈ N. Clearly, no graph of G is a factor of KN, and any two graphs of G have
common vertices. Therefore, G is not resolvable.
Example 5.3. Let G be any decomposition of the infinite complete graph KV (for example,
one of those constructed by Corollary 5.1). Let y and z be vertices not belonging to KV
and set W = V ∪ {y, z}. We can easily extend G to a non-resolvable decomposition G′
of KW in the following way.
Choose x ∈ V and let C be the following family of paths of length 1 or 2:
C = {[y, v, z] : v ∈ V \ {x}} ∪ {[x, z, y], [x, y]}.
Clearly, C decomposes KW \KV , hence G′ = G ∪ C is a decomposition of KW . Also,
x, y and z do not satisfy condition N2, since G′(z) ⊆ G′(x) ∪ G′(y), while [x, y] belongs
to G′(x) ∩ G′(y), but not to G′(z). Therefore, G′ is non-resolvable. Indeed, any resolution
class of G′ could cover the vertex z only with graphs passing through x or y. This means
that the graph [x, y] cannot belong to any resolution class of G′.
The following result provides sufficient conditions for a decomposition G to be resolv-
able.
26 Ars Math. Contemp. 24 (2024) #P1.02 / 11–29
Theorem 5.4. Let G be a decomposition of the infinite complete graph Kℵ satisfying the
following properties for some ℵ′ ≺ ℵ:
R1. each graph in G is ℵ′-bounded;
R2. |G(x) ∩ G(y)| ⪯ ℵ′ for every distinct x, y ∈ V (Kℵ).
Then G is resolvable.
Proof. Let G = {Gα : α ∈ A}. We consider a well-order ≺ on A satisfying Corollary 3.4.
Since the graphs of G are ℵ′-bounded, we have that |A| = ℵ and we can assume V (Kℵ) =
A. Here we need to construct an ascending chain (Gγ)γ∈A of families Gγ := {Γγα :
α ∈ A⪯γ} (where Γγ
′
α is a subgraph of Γ
γ
α whenever γ
′ ⪯ γ) that satisfy the following
proprieties:
(1γ) each Γγα is a vertex-disjoint union of graphs of G;
(2γ) for every α ∈ A⪯γ , γ ∈ V (Γγα);
(3γ) Gγ is contained in exactly one Γγα where α ∈ A⪯γ ;
(4γ) for every α ∈ A⪯γ , Γγα is either a finite graph or (ℵ′ · |A⪯γ |)-bounded.
The desired resolution of Kℵ is then R = {Γα : α ∈ A}, where Γα =
⋃
γ∈A Γ
γ
α for
every α ∈ A. Indeed, due to properties (1γ) and (2γ), each Γα is a resolution class of G
and, by property (3γ), R is a partition of G into resolution classes.
We proceed by transfinite induction on γ.
BASE CASE. Let 0 = minX . By condition R2, if 0 is not a vertex of G0, |G(0) ∩
G(x)| ⪯ ℵ′ for any x ∈ V (G0). Since, due to condition R1, |G(0)| = ℵ, there exists
G ∈ G(0) disjoint from G0. Therefore we can define G0 = {Γ00} where Γ00 is either G0∪G
or, if 0 belongs to V (G0), G0.
TRANSFINITE INDUCTIVE STEP. For every γ′ ≺ γ, we assume there is a family
Gγ′ satisfying (iγ′) for 1 ≤ i ≤ 4. We show that Gγ′ can be extended to a family Gγ that
satisfies the same properties, (iγ) for 1 ≤ i ≤ 4.
We are going to define, recursively, the graphs Γγα whenever α ⪯ γ. First, we consider
the case α ≺ γ. We start by setting Γ≺γα :=
⋃
γ′≺γ Γ
γ′
α . Note that property (4γ′ ) guarantees
that Γ≺γα is either finite or |Γ≺γα | ⪯ ℵ′ · |A⪯γ |; hence, Γ≺γα is ℵ-small.
• Base case. If γ ∈ V (Γ≺γ0 ), set Γ
γ
0 = Γ
≺γ
0 .
If γ ̸∈ V (Γ≺γ0 ), by condition R2 we have |G(γ)∩G(x)| ⪯ ℵ′ for every x ∈ V (Γ
≺γ
0 ).
Since Γ≺γ0 is ℵ-small, this means that the family of graphs of G(γ) that intersect
V (Γ≺γ0 ) is ℵ-small.
Moreover, any Γ≺γα is either finite or (ℵ′ · |A⪯γ |)-bounded (note that ℵ′ · |A⪯γ | ≺ ℵ,
since |A⪯γ | ≺ ℵ). Hence, the set of graphs in G(γ) that are contained in some Γ≺γα
is ℵ-small.
Finally, by condition R1, we have that |G(γ)| = ℵ. Therefore, there exists a graph
G ∈ G(γ) that is not contained in any Γ≺γα and such that V (G) ∩ V (Γ
≺γ
0 ) = ∅.
Then, we set Γγ0 = Γ
≺γ
0 ∪G.
S. Costa and T. Traetta: Factorizing the Rado graph and infinite complete graphs 27
• Recursive step. Let α ≺ γ. If γ ∈ V (Γ≺γα ), set Γγα = Γ≺γα . Otherwise, by proceed-
ing as in the previous case, we obtain the existence of a graph G ∈ G(γ) that is not in
any Γ≺γα′ or any Γ
γ
α′′ (where α
′ ≺ γ and α′′ ≺ α), and such that V (G)∩V (Γ≺γα ) = ∅.
In this case, we set Γγα = Γ
≺γ
α ∪G.
It is left to define Γγγ . We proceed by constructing, recursively, an ascending chain of
graphs Γαγ , for α ∈ A⪯γ , that are either finite or (ℵ′ · |A⪯γ |)-bounded.
• Base case. Let us first suppose that Gγ is not contained in any Γ
γ
α′ (where α
′ ≺ γ).
Again, by conditions R1 and R2, there exists G ∈ G(0) that is also not contained in
any Γγα′ such that G is either Gγ or is disjoint from Gγ . We set Γ
0
γ to be Gγ ∪ G.
Otherwise, we set Γ0γ to be any graph G in G(0) that is not contained in any Γ
γ
α′ .
• Recursive step. Let us suppose that α ̸= 0 and that we have defined Γα′γ for every
α′ ≺ α. Here we set Γ≺αγ to be
⋃
α′≺α Γ
α′
γ . Note that, by construction, Γ
≺α
γ is
either a finite graph or |Γ≺αγ | ⪯ ℵ′ · |A⪯γ |. If α belongs to V (Γ≺αγ ), we set Γαγ to
be Γ≺αγ . Otherwise, proceeding as in the previous case, we obtain that there exists
G ∈ G(α) disjoint from Γ≺αγ that does not belong to any of the Γ
γ
α′ . Now we set Γ
α
γ
to be G ∪ Γ≺αγ .
Then the family Gγ = {Γγα : α ∈ A⪯γ} satisfies the properties (1γ), (2γ), (3γ) and (4γ) by
construction.
Remark 5.5. A cardinal ℵ is said to be regular if any ℵ-small union of ℵ-small sets (resp.
graphs) is still an ℵ-small set (resp. graph) otherwise it is said to be singular. It is easy to
see that, for regular cardinals, conditions R1 and R2 of Theorem 5.4 can be relaxed to:
R1’. each graph in G is ℵ-small;
R2’. |G(x) ∩ G(y)| ≺ ℵ for every distinct x, y ∈ V (Kℵ).
However, if ℵ is a singular cardinal, then conditions R1’ and R2’ are no longer sufficient.
Indeed, we can construct a decomposition G of Kℵ into ℵ-small graphs such that
a. |G| is ℵ-small,
b. G satisfies conditions R1’ and R2’,
c. there are two (possibly isolated) vertices x and y belonging to every graphs of G, that
is, G = G(x) ∩ G(y).
Then, choosing any vertex z such that G(z) ̸= G, we have that
G(z) ⊆ G(x) ∪ G(y) = G but G(z) ̸⊇ G(x) ∩ G(y) = G.
This means that condition N2 does not hold, therefore the decomposition G is not resolv-
able.
We conclude by showing that there is always a resolution for an ‘almost’ 2-design with
blocks that are ℵ′-bounded for some ℵ′ ≺ ℵ, that is, a decomposition of Kℵ whose graphs
are almost all ℵ′-bounded complete graphs. This extends some results on the resolvability
of 2-designs given in [9].
28 Ars Math. Contemp. 24 (2024) #P1.02 / 11–29
Proposition 5.6. Let G be a decomposition of the infinite complete graph Kℵ into ℵ′-
bounded graphs for some ℵ′ ≺ ℵ, where ℵ′ is not necessarily infinite. If the subset of G
consisting of all non-complete graphs is ℵ′-bounded, then G has a resolution.
Proof. By assumption, condition R1 of Theorem 5.4 holds. To prove that G satisfies con-
dition R2 for some ℵ′′ ≺ ℵ, we assume for a contradiction the existence of vertices x and y
such that |G(x) ∩ G(y)| ≻ ℵ′′ := (ℵ′ + 1). It follows that there are at least two complete
graphs in G(x) ∩ G(y), meaning that the edge {x, y} is covered more than once by graphs
in G, and this is a contradiction. The assertion follows from Theorem 5.4.
ORCID iDs
Simone Costa https://orcid.org/0000-0003-3880-6299
Tommaso Traetta https://orcid.org/0000-0001-8141-0535
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.03 / 31–43
https://doi.org/10.26493/1855-3974.2800.d12
(Also available at http://amc-journal.eu)
Redundantly globally rigid braced
triangulations*
Qianfan Chen
Brown University, Providence, RI, USA
Siddhant Jajodia
University of California, Irvine, CA, USA
Tibor Jordán †
Department of Operations Research, ELTE Eötvös Loránd University, and
ELKH-ELTE Egerváry Research Group on Combinatorial Optimization, Eötvös Loránd
Research Network (ELKH), Pázmány Péter sétány 1/C, 1117 Budapest, Hungary
Kate Perkins
Harvey Mudd College, Claremont, CA, USA
Received 8 January 2022, accepted 23 January 2023, published online 9 August 2023
Abstract
By mapping the vertices of a graph G to points in R3, and its edges to the corresponding
line segments, we obtain a three-dimensional realization of G. A realization of G is said to
be globally rigid if its edge lengths uniquely determine the realization, up to congruence.
The graph G is called globally rigid if every generic three-dimensional realization of G is
globally rigid.
We consider global rigidity properties of braced triangulations, which are graphs ob-
tained from maximal planar graphs by adding extra edges, called bracing edges. We show
that for every even integer n ≥ 8 there exist braced triangulations with 3n− 4 edges which
remain globally rigid if an arbitrary edge is deleted from the graph. The bound is best pos-
sible. This result gives an affirmative answer to a recent conjecture. We also discuss the
connections between our results and a related more general conjecture, due to S. Tanigawa
and the third author.
*This paper is based on the results of a research opportunities project of the Budapest Semesters in Mathemat-
ics (BSM) programme.
†Corresponding author. The third author was supported by the Hungarian Scientific Research Fund grant no.
K135421 and the project Application Domain Specific Highly Reliable IT Solutions which has been implemented
with the support provided from the National Research, Development and Innovation Fund of Hungary, financed
under the Thematic Excellence Programme TKP2020-NKA-06 (National Challenges Subprogramme) funding
scheme.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
32 Ars Math. Contemp. 24 (2024) #P1.03 / 31–43
Keywords: Triangulation, globally rigid graph, braced triangulation, rigidity.
Math. Subj. Class. (2020): 52C25, 05C10
1 Introduction
A d-dimensional framework (or geometric graph) is a pair (G, p), where G is a simple
graph and p : V (G) → Rd is a map. We also call (G, p) a realization of G in Rd. The
length of an edge uv in the framework is defined to be the distance between the points p(u)
and p(v). The framework is said to be rigid in Rd if every continuous motion of its vertices
in Rd that preserves all edge lengths preserves all pairwise distances. It is globally rigid
in Rd if the edge lengths uniquely determine all pairwise distances. A realization (G, p)
is generic if the set of the d|V (G)| coordinates of the vertices is algebraically independent
over the rationals. It is known that for generic frameworks rigidity and global rigidity in
Rd depend only on the graph of the framework, for every d ≥ 1. So we may call a graph G
rigid (resp. globally rigid) in Rd if every (or equivalently, if some) generic d-dimensional
realization of G is rigid (resp. globally rigid). The characterization of rigid and globally
rigid graphs is known for d = 1, 2. For d ≥ 3 these are major open problems. We refer the
reader to [8, 10] for more details on the theory of rigid and globally rigid frameworks and
graphs.
Rigid and globally rigid graphs occur in several applications, including sensor network
localization [4], molecular conformation [3], formation control [13], and statics [9]. In
some applications it is desirable to have a graph which remains rigid or globally rigid even
if some vertices or edges are removed. In this paper we study graphs G for which G − e
is globally rigid in Rd for each edge e of G. They are called redundantly globally rigid in
Rd. In the rest of the paper we focus on the three-dimensional case, i.e. d = 3, and the
following two conjectures concerning redundant global rigidity.
A triangulation T = (V,E) is a maximal planar graph on at least three vertices. A
braced triangulation G = (V,E ∪B) is a graph obtained from a triangulation T = (V,E)
by adding a set B of new edges, called the bracing edges. If |B| = 1 (resp. |B| = 2) then
we say that G is a uni-braced (resp. doubly braced) triangulation. The characterization of
globally rigid braced triangulations in R3 is known, see Theorem 2.6 below. A conjectured
sufficient condition for redundant global rigidity is as follows.
Conjecture 1.1 ([7]). Every 5-connected braced triangulation G = (V,E∪B) with |B| ≥
2 is redundantly globally rigid in R3.
A related extremal problem is to determine the smallest number of edges in a redun-
dantly globally rigid graph in R3 on n vertices, as a function of n, for all (sufficiently large)
n. By a theorem of B. Hendrickson [3] every globally rigid graph G in Rd on n ≥ d + 2
vertices remains rigid in Rd after removing any edge of G. It is well-known that a rigid
graph in R3 on n ≥ 3 vertices has at least 3n − 6 edges. These facts imply that 3n − 4 is
a lower bound for the extremal value, and n ≥ 6 must hold. It was conjectured in [6] that
this lower bound is tight.
E-mail addresses: qianfan chen@alumni.brown.edu (Qianfan Chen), jajodias@uci.edu (Siddhant Jajodia),
tibor.jordan@ttk.elte.hu (Tibor Jordán), kperkins@hmc.edu (Kate Perkins)
Q. Chen et al.: Redundantly globally rigid braced triangulations 33
Conjecture 1.2 ([6]). For every integer k there exists a redundantly globally rigid graph
G in R3 on n ≥ k vertices with 3n− 4 edges.
The truth of Conjecture 1.1, combined with the fact that there exist arbitrarily large
5-connected triangulations, would imply Conjecture 1.2. We remark that W. Whiteley [12]
conjectured that every 5-connected doubly braced triangulation G remains rigid in R3 after
removing any pair of its edges. The truth of Conjecture 1.1, together with Hendrickson’s
theorem, would imply an affirmative answer to his conjecture.
In the rest of the paper – after introducing the results from rigidity theory that we shall
use – we consider doubly braced triangulations in which both bracing edges are dihedral
(i.e. they connect non-adjacent vertices that belong to edge sharing faces). We shall prove
sufficient conditions that guarantee that a specific edge can be removed from such a trian-
gulation while preserving global rigidity.
Based on these results we can analyse special families of such triangulations which will
lead to the proof of (a stronger form of) Conjecture 1.2. We shall prove that for every even
integer n ≥ 8 there exist redundantly globally rigid graphs in R3 on n vertices with 3n− 4
edges1. In the last section we prove necessary conditions for the redundant global rigidity
of braced triangulations and formulate a couple of conjectures.
2 Rigid and globally rigid graphs
We shall use the following results in order to verify the rigidity or global rigidity of a graph.
Let G = (V,E) be a graph. For a vertex v ∈ V let NG(v) (resp. dG(v)) denote the set
(resp. the number) of neighbours of v in G. For a set X ⊆ V the graph obtained from G
by adding a complete graph on vertex set X (that is, by adding new edges connecting the
vertex pairs x, y ∈ X which are not adjacent in G) is denoted by G+K(X).
Theorem 2.1 ([11]). Let G = (V,E) be a graph and v ∈ V with dG(v) ≥ d+1, for some
d ≥ 1. If G− v is rigid and G− v +K(NG(v)) is globally rigid in Rd then G is globally
rigid in Rd.
A bracing edge uv in a braced triangulation G is called dihedral if it connects two non-
adjacent vertices u, v of two edge sharing triangles on vertices uab and vab, respectively,
of the triangulation.
A block and hole graph is obtained from the graph of an (embedded) plane triangula-
tion by removing the interiors of some discs, defined by their boundary cycles, and then
rigidifying the vertex sets of some of these cycles by adding new edges. This operation
creates some holes and blocks. We shall only consider special block and hole graphs. By
removing a single edge or a vertex of degree five from an (embedded) triangulation, we
may create a face whose boundary is a 4-cycle or a 5-cycle, respectively. We shall say that
such a cycle is a 4-hole or 5-hole in (some planar embedding of) the graph. The addition
of a dihedral bracing edge creates a K4 subgraph, which can be viewed as a 4-block that
rigidifies the cycle aubv, provided the two edge sharing triangles uab, vab are both faces
in the embedding. Since we shall only consider 5-connected triangulations, these triangles
will always be faces (in any embedding) and the resulting 4-block will be uniquely defined.
For simplicity we shall call a braced triangulation with dihedral bracing edges and a re-
moved edge or degree-five vertex a block and hole graph. See [2, 12] for a more general
1We can extend our result to odd values of n by using different techniques. We do not discuss this extension
in this paper.
34 Ars Math. Contemp. 24 (2024) #P1.03 / 31–43
definition and results on rigid block and hole graphs in three-space. We need the following
corollaries of their results.
Theorem 2.2 ([12]). Let G be a 4-connected block and hole graph which has a single
4-hole and a single 4-block. Then G is rigid in R3.
Theorem 2.3 ([2]). Let G′ be a 5-connected block and hole graph with two 4-blocks and
let G = G′ − v, where v is a vertex of degree five in G′ which is disjoint from the blocks.
Then G is rigid in R3.
Let G be a graph and let uv, vw be a pair of incident edges in G. Let Evuw be the set of
the remaining edges incident with v and let Evuw = F ∪ F ′ be a bipartition of Evuw. The
(3-dimensional) vertex splitting operation (at v, on edges uv, vw) adds a new vertex v′ to
the graph, adds the new edges uv′, v′w, vv′, and then replaces every edge xv in F ′ by an
edge xv′. The edges in F stay incident to v. See Figure 1. The vertex splitting is said to be
non-trivial if F and F ′ are both non-empty.
u
w
v
u
w
v
v′
Figure 1: A non-trivial vertex splitting operation on edges uv, vw.
An important conjecture in rigidity theory is that non-trivial vertex splitting preserves
global rigidity in Rd, for all d ≥ 1, see [1]. The next result verifies a special case.
Theorem 2.4 ([7]). A graph is globally rigid in R3 if it can be obtained from K5 by a
sequence of non-trivial vertex splitting operations.
This theorem can be used in the analysis of globally rigid braced triangulations, due to
the following combinatorial result.
Theorem 2.5 ([7]). Every 4-connected uni-braced triangulation can be obtained from K5
by a sequence of non-trivial vertex splitting operations.
Thus 4-connected uni-braced triangulations are globally rigid. A complete characteri-
zation, with no bounds on the number of bracing edges, is the following.
Theorem 2.6 ([7]). A braced triangulation G = (V,E ∪B) with |V | ≥ 5 is globally rigid
in R3 if and only if G is 4-connected and |B| ≥ 1.
The inverse operation of vertex splitting is the contraction of an edge uv for which
u and v have exactly two common neighbours. This operation takes a triangulation to a
smaller triangulation. We shall also use the fact that an edge contraction decreases the
vertex connectivity of a graph by at most one.
Q. Chen et al.: Redundantly globally rigid braced triangulations 35
3 Redundant edges in braced triangulations
In this section we fix the dimension d = 3. Every 5-connected braced triangulation with at
least one bracing edge is globally rigid by Theorem 2.6. We shall describe several situations
in which the removal of an edge from a 5-connected braced triangulation preserves global
rigidity. The first lemma is an immediate corollary of Theorem 2.6.
Lemma 3.1. Let G = (V,E ∪ B) be a 5-connected braced triangulation with |B| ≥ 2.
Then G− e is globally rigid for every e ∈ B.
In the rest of this section we shall assume that G is a 5-connected graph obtained from
an (embedded) triangulation by adding exactly two dihedral bracing edges that create two
4-blocks, with at most two vertices in common.
Lemma 3.2. Let G = (V,E ∪ B) be a 5-connected doubly braced triangulation with two
4-blocks. Suppose that e = uv ∈ E is an edge with dG(v) = 5 and v is disjoint from the
4-blocks. Then G− e is globally rigid.
Proof. We shall prove that v satisfies the conditions of Theorem 2.1 in graph G − e. The
inequality dG−e(v) ≥ 4 is clearly satisfied. Since v is disjoint from the 4-blocks of G, the
graph (G− e)− v (= G− v) is a block and hole graph with one 5-hole and two 4-blocks.
The 5-connectivity of G and Theorem 2.3 imply that (G − e) − v is rigid. Next consider
the graph H = (G − e) − v +K(NG−e(v)). By 5-connectivity the four neighbours of v
in G− e induce three edges in G− e. Thus three new edges are added to G− e to obtain
H . Notice that H is a braced triangulation: two new edges can be used to triangulate the
graph obtained from T = (V,E) by removing v, while the third one becomes a bracing
edge. See Figure 2.
v
G - e G - e - v + K(N(G - e)(v))
K4
K4
K4
K4
Figure 2: The neighbourhood of v in G − e and the edges they induce in H . The dashed
edge is a bracing edge.
Since G is 5-connected, (G − e) − v = G − v is 4-connected. This implies that H
is 4-connected. Hence H is globally rigid by Theorem 2.6. The lemma now follows from
Theorem 2.1, applied to G− e and v.
36 Ars Math. Contemp. 24 (2024) #P1.03 / 31–43
Lemma 3.3. Let G = (V,E ∪ B) be a 5-connected doubly braced triangulation with two
4-blocks. Suppose that e = uv ∈ E is an edge with dG(v) = 5 and v belongs to exactly
one of the 4-blocks. Then G− e is globally rigid.
Proof. Suppose that the 4-blocks are C1 and C2, and v is part of C1, say. Then the deletion
of v from G − e creates a block and hole graph with a 4-block (namely, C2) and a 4-hole.
Note that if v is not incident with the bracing edge f of C1 then f becomes an edge of the
underlying (almost) triangulation of (G−e)−v. Since G is 5-connected, (G−e)−v = G−v
is 4-connected. Thus (G − e) − v is rigid by Theorem 2.2. Furthermore, it follows that
G−v+K(NG−e(v)) is a 4-connected braced triangulation with two bracing edges. Hence
it is globally rigid by Theorem 2.6.
The Lemma now follows from Theorem 2.1, applied to G− e and v.
Lemma 3.4. Let G = (V,E ∪ B) be a 5-connected doubly braced triangulation with two
4-blocks C1, C2 and let v ∈ V (C1) − V (C2). Suppose that vw ∈ E ∩ E(C1) for which
there is a triangular face uvw of T = (V,E) with u /∈ V (C1). Let e = uv. Then G− e is
globally rigid.
Proof. We show that G − e can be obtained from K5 by a sequence of non-trivial vertex
splitting operations. Observe that G − e has a 4-hole and two 4-blocks in which v and w
have exactly two common neighbours (the two other vertices of C1) by 5-connectivity. Let
H be the graph obtained from G−e by contracting the edge vw. It is easy to see that H is a
4-connected uni-braced triangulation. Thus H (and hence also G−e) can be obtained from
K5 by a sequence of non-trivial vertex splitting operations by Theorem 2.5. The Lemma
now follows from Theorem 2.4.
The last lemma of this section is concerned with the case when the two 4-blocks share
two vertices.
Lemma 3.5. Let G = (V,E∪B) be a 5-connected doubly braced triangulation with two 4-
blocks C1, C2 with V (C1)∩V (C2) = {a, b}, where V (C1) = {a, b, c, d}, and the dihedral
bracing edge in C1 is ad. Then G− ab and G− ac are globally rigid. Furthermore, if v is
a vertex which is disjoint from the blocks and cv, av ∈ E then G− av is globally rigid.
Proof. We have V (C1) − V (C2) = {c, d}. Let us consider the removal of edge e = ab.
Observe that in G−e the vertices c and a have exactly two common neighbours. Moreover,
the graph H obtained from G − e by contracting the edge ca is a 4-connected uni-braced
triangulation. Thus H (and hence also G − e) can be obtained from K5 by a sequence of
non-trivial vertex splitting operations by Theorem 2.5. Thus G − ab is globally rigid by
Theorem 2.4.
The proof for edge ac is similar. In this case we delete the edge ac, contract the edge
cd, and apply the same argument. Finally, to show that G − av is globally rigid, we use a
similar proof again in which we delete av and then contract ac.
4 Two families of graphs
In this section, we define two infinite families of redundantly globally rigid doubly braced
triangulations in R3.
Q. Chen et al.: Redundantly globally rigid braced triangulations 37
Definition 4.1 (Belted bipyramid). For every n ≥ 3, an n-gonal belted bipyramid, denoted
by G(n), is a graph on 2n+2 vertices that is constructed as follows. Take two n-gonal pyra-
mids with poles N and S, respectively, and label the vertices on the base of one pyramid 1
to n and on that of the other 1’ to n’ consecutively. Insert edges between the corresponding
pairs of vertices (i.e. between 1 and 1’, 2 and 2’, and so on) and insert an edge between k
and (k + 1)’ for every 1 ≤ k ≤ (n − 1). Finally, insert an edge between n and 1’. See
Figure 3.
It is easy to see that G(n) is a triangulation. Let G(n, k) denote the graph obtained by
inserting the edges 1n′ and k(k−1)′ to G(n). Then G(n, k) is a doubly braced triangulation
with two dihedral bracing edges. See Figure 3.
N
S
3
21
5
4
5’
1’ 2’
3’
4’
N
S
3
21
5
4
5’
1’ 2’
3’
4’
Figure 3: The graphs G(5) and G(5, 4).
Lemma 4.2. For every n ≥ 5, G(n) (and hence, G(n, k) for every 2 ≤ k ≤ n) is 5-
connected.
Proof. By using the structure and the symmetry of G(n) it is not hard to check that it is
5-connected. A simple argument is as follows: consider the base cycle C of one of the
38 Ars Math. Contemp. 24 (2024) #P1.03 / 31–43
pyramids on vertex set (1, 2, ..., n). It is easy to verify that for every v ∈ V − V (C)
there exist 5 paths from v to V (C) that are vertex-disjoint, apart from v. Furthermore, for
every u, v ∈ V (C) there exist 5 u-v-paths that are vertex-disjoint apart from u, v. Since
|V (C)| ≥ 5, this implies that G(n) cannot have a vertex separator of size less than 5.
Theorem 4.3. For every n ≥ 5 and 2 ≤ k ≤ n the graph G(n, k) is redundantly globally
rigid in R3.
Proof. Theorem 2.6 implies that G(n, k) is globally rigid in R3. It remains to show that
the removal of any edge preserves global rigidity. First suppose that 3 ≤ k ≤ n − 1, in
which case the two 4-blocks are disjoint.
Each bracing edge is redundant by Lemma 3.1. Note that each vertex has degree five in
G(n, k), except for the two poles (when n ≥ 6) and the end-vertices of the bracing edges.
Thus we can use Lemmas 3.2 and 3.3 to show that most of the edges are redundant. The
edges that do not satisfy the conditions of at least one of these two lemmas are the edges
from the poles to the end-vertices of the bracing edges and, possibly, an edge that connects
the end-vertices of different bracing edges. These edges are redundant by Lemma 3.4. So
every edge is redundant and the graph is redundantly globally rigid, as required.
We can also show that G(n, 2) and G(n, n) are redundantly globally rigid by a similar
argument. In these two special cases the two 4-blocks share two vertices, so we also need
Lemma 3.5 in order to handle some of the edges incident with the intersection of the blocks.
A slightly different construction is the following.
Definition 4.4 (Flat belted bipyramid). For every n ≥ 4, an n-gonal flat belted bipyramid,
denoted by F (n), is a graph on 2n vertices that is constructed as follows. Take G(n)
and delete its two poles. Retaining the vertex labels described in Definition 1, for every
1 ≤ k ≤ n, insert an edge between vertex 3 and vertex k (unless 3 is already adjacent to
k). Then, for every 1 ≤ k ≤ n, insert an edge between vertex 2′ and vertex k′ (unless 2’ is
already adjacent to k′). See Figure 4.
It is easy to see that F (n) is a triangulation. Let H(n) be the graph obtained from F (n)
by inserting edges 1′2 and 3′4. See Figure 4. Thus H(n) is a doubly braced triangulation
with two dihedral bracing edges that create two disjoint 4-blocks. Although F (n) is not
5-connected, a proof strategy similar to that of Lemma 4.2 can be used to show that H(n)
is 5-connected.
Lemma 4.5. For every n ≥ 4 the graph H(n) is 5-connected.
In fact we can show that H(4) is the smallest 5-connected doubly braced triangulation2.
Theorem 4.6. H(n) is redundantly globally rigid in R3 for n ≥ 4.
2The minimum degree condition implies that the number of vertices is at least eight, and equality holds only
if the graph is 5-regular. Thus the complement of the graph is isomorphic to one of the following: (i) the disjoint
union of a three-cycle and a five-cycle, (ii) the disjoint union of two four-cycles, (iii) a cycle on eight vertices. In
the first two cases a simple analysis shows that the graph cannot be made planar by removing at most two edges.
In the third case the graph is H(4).
Q. Chen et al.: Redundantly globally rigid braced triangulations 39
H5
3
21
5
4
5’
1’ 2’
3’
4’
H(5)
3
21
5
4
5’
1’ 2’
3’
4’
Figure 4: The graphs F (5) and H(5).
Proof. Theorem 2.6 implies that H(n) is globally rigid in R3. It remains to show that the
removal of any edge preserves global rigidity. The rest of the proof is similar to that of
Theorem 4.3, using the lemmas of the previous section. Note that in the case of H(n) the
two 4-blocks are disjoint.
The results of this section provide an affirmative answer to Conjecture 1.2.
Theorem 4.7. For every even integer n ≥ 8 there exist redundantly globally rigid graphs
in R3 on n vertices with 3n− 4 edges.
A simple degree count shows that there are no such graphs for n ≤ 7.
As we noted earlier, redundantly globally rigid graphs are “doubly redundantly rigid”,
that is, they remain rigid after the removal of any pair of edges. Thus the graphs defined in
this section are also examples of doubly redundantly rigid graphs with the smallest number
of edges for every even n ≥ 8. They are different from the ones constructed in [6], and
easier to analyse.
40 Ars Math. Contemp. 24 (2024) #P1.03 / 31–43
5 Concluding remarks and conjectures
A natural question is whether the 5-connectivity condition in Conjecture 1.1 can be weak-
ened. The next example shows that 5-connectivity is not necessary.
Example 5.1. Consider the graph G in Figure 5. It is a 4-connected (but not 5-connected)
doubly braced triangulation, and hence it is globally rigid by Theorem 2.6. We sketch a
proof which shows that G− e is globally rigid for every edge e. By the symmetry of G we
have four cases to consider: the deleted edge e is
(i) a cross edge in the top K4,
(ii) a side in the top K4,
(iii) an edge from the K4 to the 4-cycle of the 4-separator,
(iv) an edge of the 4-cycle of the separator.
In case (i) G− e is a 4-connected braced triangulation. In cases (ii) and (iii) we can apply
(the proof of) Lemma 3.3 by noting that its proof works here by using the specific structure
of G (rather than 5-connectivity). In case (iv) we perform two contractions and obtain a
4-connected uni-braced triangulation as follows. Suppose, by symmetry, that e = cd. Then
first contract an edge between c and the top K4. Next contract one of the edges from c to
the remainder of the top K4. By contracting the appropriate edge we obtain a 4-connected
uni-braced triangulation. Then global rigidity follows by Theorem 2.4.
This leads us to the next question: is it possible to obtain a complete characterization
of redundantly globally rigid braced triangulations, at least in some special cases (say, for
doubly braced triangulations with two dihedral bracing edges)?
a b c d
G
C2
C1
Figure 5: A redundantly globally rigid doubly braced triangulation G with a 4-separator
S = {a, b, c, d}.
In this section we prove some necessary conditions and then formulate a conjecture.
A k-separator S in a connected graph G = (V,E) is a set of vertices with |S| = k for
which G − S is disconnected. For some X ⊆ V we use G[X] to denote the subgraph of
Q. Chen et al.: Redundantly globally rigid braced triangulations 41
G induced by vertex set X . It is known that for a minimal separator S in a triangulation G
we have |S| ≥ 3, the graph G − S has exactly two connected components, and G[S] is a
cycle (see e.g. [7, Section 5]). For a separator S and connected component C of G− S we
say that G[C ∪ S] is an extended component of S in G.
Lemma 5.2. Let G = (V,E ∪ B) be a redundantly globally rigid braced triangulation
and let S be a 4-separator in G. Suppose that S is a minimal separator in the underlying
triangulation (V, T ). Then for every component C of G − S there exists a bracing edge
incident with C.
Proof. Let T = (V,E). Since S is a minimal separator in T , the graph T − S (and hence
also G−S) has exactly two connected components C,D. For a contradiction suppose that
there is no bracing edge incident with C. Since T [S] induces a 4-cycle the graph K ob-
tained from the extended component G[C ∪S] of S by adding the edges that connect those
vertex pairs of S which are not adjacent in G, is a 4-connected uni-braced triangulation in
which S induces a K4. Let e be an edge of K incident with C. Then K − e is a minimally
rigid graph on at least five vertices. By Hendrickson’s theorem K − e is not globally rigid.
The fact that G− e can be obtained from K − e by merging K − e and the other extended
component G[D ∪ S] along a complete graph (and, possibly, by deleting edges) implies
that G− e is not globally rigid. This contradiction completes the proof.
The proof shows that the lemma holds even if redundantly globally rigid is weakened to
doubly redundantly rigid in the condition. If the underlying triangulation T is 4-connected,
then every 4-separator of G is obviously a minimal separator in T , so the conditions of
Lemma 5.2 are satisfied.
Let us consider the case when T is not 4-connected and G is doubly braced. Then for
every 3-separator S of T , and corresponding components C,D of T − S, both bracing
edges must connect C and D (for otherwise S is a 3-separator in G − e for some bracing
edge e, contradicting redundant global rigidity). Call a component C arising by the removal
of a 3-separator of T a 3-separator component of T . It is not hard to see that this implies
that T has exactly two minimal 3-separator components C1 and C2, both bracing edges
connect C1 and C2, and that T can be made 4-connected by adding a single edge (from
C1 to C2). We believe that in this rather special case G is redundantly globally rigid.
Otherwise, when T is 4-connected, the necessary condition of Lemma 5.2, together with
Hendrickson’s connectivity condition, might be sufficient.
Conjecture 5.3. Let G = (V,E ∪B) be a doubly braced triangulation. Then G is redun-
dantly globally rigid in R3 if and only if
(i) G− e is 4-connected for all e ∈ E ∪B, and
(a) either T = (V,E) has a 3-separator, or
(b) for every 4-separator S of G and component C of G − S there is a bracing
edge incident with C.
Note that if G is doubly braced and the bracing edges induce two disjoint 4-blocks then
T must be 4-connected. Thus in this case the conjecture can be simplified.
We close this section by noting that an interesting related open problem is to charac-
terize globally rigid block and hole graphs with a single block (with no constraints on the
size of the block and the number of holes - see [2] for the definition). It is possible that the
global rigidity of these graphs can be characterized by Hendrickson’s necessary conditions.
42 Ars Math. Contemp. 24 (2024) #P1.03 / 31–43
Conjecture 5.4. A block and hole graph with a single block is globally rigid in R3 if and
only if it is 4-connected and redundantly rigid in R3 .
A characterization of redundantly rigid block and hole graphs with a single block can
be obtained from a recent result in [5].
ORCID iDs
Siddhant Jajodia https://orcid.org/0009-0008-6644-9851
Tibor Jordán https://orcid.org/0000-0003-3662-5558
Kate Perkins https://orcid.org/0000-0003-3596-7505
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1007/s00454-009-9220-0.
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.04 / 45–66
https://doi.org/10.26493/1855-3974.2903.9ca
(Also available at http://amc-journal.eu)
Intersecting families of graphs of functions over
a finite field*
Angela Aguglia , Bence Csajbók †
Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari,
Via Orabona 4, I-70125 Bari, Italy
Zsuzsa Weiner
ELKH–ELTE Geometric and Algebraic Combinatorics Research Group,
1117 Budapest, Pázmány P. stny. 1/C, Hungary and
Prezi.com, H-1065 Budapest, Nagymező utca 54-56, Hungary
Received 10 June 2022, accepted 30 December 2022, published online 9 August 2023
Abstract
Let U be a set of polynomials of degree at most k over Fq , the finite field of q elements.
Assume that U is an intersecting family, that is, the graphs of any two of the polynomials in
U share a common point. Adriaensen proved that the size of U is at most qk with equality
if and only if U is the set of all polynomials of degree at most k passing through a common
point. In this manuscript, using a different, polynomial approach, we prove a stability
version of this result, that is, the same conclusion holds if |U | > qk − qk−1. We prove a
stronger result when k = 2.
For our purposes, we also prove the following results. If the set of directions determined
by the graph of f is contained in an additive subgroup of Fq , then the graph of f is a line.
If the set of directions determined by at least q −√q/2 affine points is contained in the set
of squares/non-squares plus the common point of either the vertical or the horizontal lines,
then up to an affinity the point set is contained in the graph of some polynomial of the form
αxp
k
.
Keywords: Direction problem, Erdős-Ko-Rado, finite field, polynomial.
Math. Subj. Class. (2020): 11T06
*We are extremely grateful for the reviewer’s thorough reading and valuable comments. An inaccuracy spotted
out by the reviewer led us to the discovery of Theorem 2.13. The second and the third author acknowledge the
support of the National Research, Development and Innovation Office – NKFIH, grant no. K 124950. This
work was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications
(GNSAGA– INdAM).
†Corresponding author.
E-mail addresses: angela.aguglia@poliba.it (Angela Aguglia), bence.csajbok@poliba.it (Bence Csajbók),
zsuzsa.weiner@gmail.com (Zsuzsa Weiner)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
46 Ars Math. Contemp. 24 (2024) #P1.04 / 45–66
1 Introduction
In 1961, Erdős et al. [6] proved that if F is a k-uniform intersecting family of subsets of
an n-element set X , then |F | ≤
(
n−1
k−1
)
when 2k ≤ n. Furthermore, they proved that if
2k + 1 ≤ n, then equality holds if and only if F is the family of all subsets containing a
fixed element x ∈ X . There are several versions of the Erdős-Ko-Rado theorem. For a
survey of this type of results, see [7, 13] or [5].
In this manuscript, we investigate an Erdős-Ko-Rado type problem for graphs of func-
tions over a finite field. The idea of this work comes from the recent manuscript [1] by
Adriaensen, where the author studies intersecting families of ovoidal circle geometries and,
as a consequence, of graphs of functions over a finite field.
Definition 1.1. If f is an Fq → Fq function, then the graph of f is the affine q-set:
Uf = {(x, f(x)) : x ∈ Fq}.
The set of directions determined by (the graph of) f is
Df =
{
f(x)− f(y)
x− y
: x, y ∈ Fq, x ̸= y
}
.
Definition 1.2. For a family of polynomials, U , we say that U is t-intersecting if for any
two polynomials f1, f2 ∈ U , the graphs of f1 and f2 share at least t points, that is,
|{(x, f1(x)) : x ∈ Fq} ∩ {(x, f2(x)) : x ∈ Fq}| ≥ t.
Instead of 1-intersecting, we will also use the term “intersecting”.
Note that if U is a t-intersecting family of polynomials of degree at most k, then also
|{(x, f1(x)) : x ∈ Fq} ∩ {(x, f2(x)) : x ∈ Fq}| ≤ k
holds for any pairs f1, f2 ∈ U , since (x, f1(x)) = (x, f2(x)) implies that x is a root of
f1 − f2 which has degree at most k.
In this note, we improve a result due to Adriaensen [1, Theorem 6.2] by using different
techniques. Adriaensen’s proof goes through association schemes and circle geometries,
our proof does not use these, as we rely on two classical results (Results 2.3 and 2.8) about
polynomials over finite fields.
To be more precise, our main results are the following theorems.
Theorem 1.3. Let U be a set of intersecting polynomials of degree k ≤ 2 over Fq . Assume
that q ≥ 53, when q is odd and q ≥ 8 when q is even. If
|U | > q2 −
q
√
q
4
+
cq
8
+
√
q
8
,
where c = 1 for q even and c = 3 for q odd, then the graphs of the functions in U share a
common point.
A. Aguglia et al.: Intersecting families of graphs of functions over a finite field 47
We will prove Theorem 1.3 separately for q odd (Theorem 3.11) and for q even (Theo-
rem 3.15). For k > 2, the proof can be finished by induction as in [1, page 33]. We obtain
the following result.
Theorem 1.4. If U is a set of more than qk − qk−1 intersecting polynomials over Fq ,
q ≥ 53 when q is odd and q ≥ 8 when q is even, and of degree at most k, k > 1, then
there exist α, β ∈ Fq such that g(α) = β for all g ∈ U . Furthermore, U can be uniquely
extended to a family of qk intersecting polynomials over Fq and of degree at most k.
While finalizing our manuscript, a stronger stability version of the above mentioned
result for k = 2 was published by Adriaensen; see [2]. Our proof is different, based on
polynomials and hence might be of independent interest.
2 Preliminaries
Throughout this paper, q = pn for some prime p and a positive integer n. The algebraic
closure of the finite field Fq will be denoted by Fq .
The absolute trace function is defined as Trq/p : Fq → Fp, Trq/p(x) = x+ xp + . . .+
xp
n−1
. Recall that for q even and c ̸= 0, a+ bx+ cx2 ∈ Fq[x] has a root in Fq if and only
if b = 0, or b ̸= 0 and
Trq/2
(ac
b2
)
= 0. (2.1)
When q is a square, we will also use the notation N: Fq → F√q , x 7→ x
√
q+1, which is the
norm of x over F√q .
We will frequently need the following result of Ball, Blokhuis, Brouwer, Storme, Szőnyi
and Ball.
Result 2.1 (Part of [3, 4]). Let f be an Fq → Fq function such that |Df | ≤ (q + 1)/2.
Then Uf is affinely equivalent to the graph of a linearised polynomial, that is, a polynomial
of the form
∑n−1
i=0 aix
pi ∈ Fq[x].
Theorem 2.2. LetU denote a proper Fp-subspace of Fq , q = pn > 2, p prime and consider
a function σ : Fq → Fq . If the set of directions
Dσ =
{
σ(x)− σ(y)
x− y
: x, y ∈ Fq, x ̸= y
}
is contained in U , then σ(x) = ax+ b for some a, b ∈ Fq .
Proof. First note that U is contained in an (n− 1)-dimensional Fp-subspace V and hence
|Dσ| ≤ pn−1. Then by Result 2.1, σ(x) = α + g(x), where α ∈ Fq and g(x) =∑n−1
i=0 bix
pi ∈ Fq[x], thus
Dσ =
{
g(x)
x
: x ∈ Fq \ {0}
}
.
It is well-known that βq/p
∏
γ∈V (x− γ) = Trq/p(βx) for some β ∈ Fq \ {0}. Next define
f(x) := βg(x). Then Dσ ⊆ V implies
Trq/p
(
f(x)
x
)
= 0
48 Ars Math. Contemp. 24 (2024) #P1.04 / 45–66
for each x ∈ Fq \{0}. To prove the assertion, it is enough to prove that f(x) is linear. With
f(x) =
∑n−1
i=0 aix
pi ∈ Fq[x],
Trq/p
(
f(x)
x
)
= Trq/p
(
n−1∑
i=0
aix
pi−1
)
=
n−1∑
j=0
n−1∑
i=0
ap
j
i x
pi+j−pj ,
and because of our assumption, this polynomial vanishes at every element of Fq \ {0}.
The p > 2 case:
If we multiply this polynomial by x1+p+p
2+...+pn−1 , then we obtain
n−1∑
j=0
n−1∑
i=0
ap
j
i x
1+p+...+pn−1+pi+j−pj
and this polynomial vanishes at every element of Fq . As a function, this polynomial re-
mains the same if we consider it modulo xp
n − x, so it is the same function as the poly-
nomial we obtain when we replace the exponents pi+j with pi+j−n for each i + j ≥ n.
Denote this new polynomial with f̃ . The fact that we multiplied f by x1+p+p
2+...+pn−1
ensures that the exponents of f̃ are larger than 0 and smaller than q. We claim that each
monomial has different degree in f̃ . It is clear that f̃ is the sum of at most n2 monomials
and the set of degrees of these monomials is contained in the set
A := {1 + p+ p2 + . . .+ pn−1 + pc − pd : c, d ∈ {0, 1, . . . , n− 1}}.
Assume that for some c1, c2 ∈ {0, 1, . . . , n − 1}, d1, d2 ∈ {0, 1, . . . , n − 1} and
(c1, d1) ̸= (c2, d2)
1 + p+ p2 + . . .+ pn−1 + pc1 − pd1 = 1 + p+ p2 + . . .+ pn−1 + pc2 − pd2 ,
or equivalently pc1 + pd2 = pc2 + pd1 . Since the base p-digits of an integer are uniquely
determined, this implies {c1, d2} = {c2, d1}, so either c1 = c2 and d1 = d2, or c1 = d1
and c2 = d2. We conclude that two distinct monomials of f̃ have the same degree d if any
only if d = 1 + p+ . . .+ pn−1, that is, when in
1 + p+ . . .+ pn−1 + pi+j − pj
we have i = 0.
Note that the degree of f̃ is at most
m := 1 + p+ . . .+ pn−1 + pn−1 − 1.
Since p > 2, m is clearly smaller than q, but f̃ has q roots (the elements of Fq). Thus it is
the zero polynomial, all of its coefficients are zero. The coefficients of f̃ are the pj-powers
of a1, . . . , an−1 and Trq/p(a0). So f(x) = a0x.
The p = 2 case:
Note that when i ̸= 0, then
2i+j − 2j = 2j + 2j+1 + . . .+ 2i+j−1.
A. Aguglia et al.: Intersecting families of graphs of functions over a finite field 49
If 2i+j − 2j ≥ 2n = q then write this number as
2i+j−2j = 2j+2j+1+ . . .+2i+j−1 = 2j+2j+1+ . . .+2n−1+2n(1+ . . .+2i+j−1−n).
Clearly, as Fq → Fq functions,
x 7→ x2
i+j−2j
are the same as
x 7→ x2
j+2j+1+...+2n−1x1+...+2
i+j−1−n
=
x1+...+2
i+j−1−n+2j+2j+1+...+2n−1 .
When 2i+j − 2j ≥ 2n = q, then substitute these exponents in
Trq/2
(
f(x)
x
)
=
n−1∑
j=0
n−1∑
i=0
a2
j
i x
2i+j−2j
with the exponents
Bij := 1 + . . .+ 2
i+j−1−n + 2j + 2j+1 + . . .+ 2n−1 < q,
and denote this new polynomial with f̃ . Note that in this case i+ j− 1−n < j− 1. When
2i+j − 2j < q, i ̸= 0, then define
Aij := 2
i+j − 2j = 2j + 2j+1 + . . .+ 2i+j−1 < q.
If i = 0 then put A0j = 0. Since base 2-digits of an integer are uniquely determined,
we have the following: (1) If i1 ̸= 0, then Ai1j1 = Ai2j2 iff (i1, j1) = (i2, j2), (2)
A0j1 = A0j2 for any pair (j1, j2), (3) Bi1j1 = Bi2j2 iff (i1, j1) = (i2, j2), finally (4)
Bi1j1 = Ai2j2 iff i1+j1−n = j1 and j2 = 0 and i2 = n (otherwiseBi1j1 in base 2 has the
form 11 . . . 110 . . . 011 . . . 11, whileAi2j2 has the form 11 . . . 1100 . . . 00, a contradiction);
but i1, i2 < n. It follows that the only exponent which appears in more than one monomial
is the 0. The degree of f̃ is at most q − 2 (obtained in xA(n−1) 1 ) and it has q − 1 roots
(the elements of Fq \ {0}), so it is the zero polynomial. Hence all of its coefficients are
zero. These coefficients are the 2j-powers of a1, . . . , an−1 and Trq/2(a0). It follows that
f(x) = a0x.
We will need the following two results regarding functions over finite fields.
Result 2.3 ([9, Theorem 5.41], Weil’s bound). Let ψ be a multiplicative character of Fq
of order m > 1 and let f ∈ Fq[x] be a monic polynomial of positive degree that is not an
m-th power of a polynomial. Let d be the number of distinct roots of f in Fq . Then for
every a ∈ Fq we have ∣∣∣∣∣∣
∑
c∈Fq
ψ(af(c))
∣∣∣∣∣∣ ≤ (d− 1)√q.
We will also consider polynomials of degree
√
q+1 admitting square values for almost
every element of Fq . In this case, the inequality above seems to be useless. In Lemma 2.6,
we show a way how to derive information from Weil’s bound also in this case. When
m = d = 2 then the following, stronger result holds which can be easily proved by counting
Fq-rational points of a conic of PG(2, q):
50 Ars Math. Contemp. 24 (2024) #P1.04 / 45–66
Result 2.4 ([11, Exercise 5.32]). Let q be an odd prime power, f(x) = ax2+bx+c ∈ Fq[x]
with a ̸= 0, and let ψ denote the quadratic character Fq → {−1, 1, 0}. Then∑
x∈Fq
ψ(ax2 + bx+ c)
equals −ψ(a) if b2 − 4ac ̸= 0 and (q − 1)ψ(a) if b2 − 4ac = 0.
To use Result 2.3, we will need the following.
Lemma 2.5. Put f(x) = axp
k+1 + dxp
k
+ bx + c ∈ Fq[x], k ̸= 0. If q is odd and
f(x) = g(x)2, then dp
k
a = bap
k
and dp
k+1a = cap
k+1, or a = b = d = 0.
Proof. If a = 0, then b = d = 0 otherwise the degree of f was odd. Assume a ̸= 0 and
suppose f(x) = g(x)2. Then the roots of f have multiplicities at least 2 and hence they
are also roots of f ′(x) = axp
k
+ b = (ap
−k
x+ bp
−k
)p
k
. It follows that f(x) has a unique
root, −(b/a)p−k , so
f(x) = a(x+ γ)p
k+1 = a(xp
k
+ γp
k
)(x+ γ) = axp
k+1 + γaxp
k
+ aγp
k
x+ γp
k+1a,
with γ = (b/a)p
−k
. It follows that f has the listed properties.
Lemma 2.6. If for some odd, square q > 9 there is a subset D of Fq of size larger than
q −√q/2 + 1/2 such that the Fq → Fq function x 7→ ℓ(x) := ax
√
q+1 + dx
√
q + bx+ c,
a ̸= 0, has the property that ℓ(x) is a square of Fq for each x ∈ D, then a
√
qb = d
√
qa.
Proof. Suppose a
√
qb ̸= d
√
qa. Then the value set of ℓ clearly does not change if we
replace x with g(y) = ((b/a)
√
q − (d/a))y − d/a, since g is a permutation polynomial.
Also, C := g−1(D), will have the properties that |C| > q − √q/2 + 1/2 and for each
y ∈ C,
f(y) := ℓ(g(y))
is a square of Fq . One can easily verify f(y) = ℓ(g(y)) = βy
√
q+1 + βy + α, where
β =
(a
√
qb− ad
√
q)
√
q+1
a2
√
q+1
and α = c− bd/a. We will show that this is not possible.
Since the norm x 7→ N(x) takes (√q − 1) distinct non-zero values in Fq , and
(
√
q − 1)− (√q/2− 1/2) ≥ 2 (here we use q > 9 square), we may take t1, t2 ∈ Fq \ {0}
such that N(t1) ̸= N(t2) and N(t1),N(t2) /∈ {N(d) : d ∈ Fq \ C}. We show that if f(x)
is a square for each x ∈ C then also the polyomials
f(t1y
√
q−1) = N(t1)βy
q−1 + βt1y
√
q−1 + α ∈ Fq[y],
f(t2y
√
q−1) = N(t2)βy
q−1 + βt2y
√
q−1 + α ∈ Fq[y],
have only square values for each y ∈ C. Indeed, this follows from the fact that N(tiy
√
q−1) /∈
{N(d) : d ∈ Fq \ C} and hence tiy
√
q−1 ∈ C for i = 1, 2.
A. Aguglia et al.: Intersecting families of graphs of functions over a finite field 51
Then the polynomials
G1(y) := N(t1)β + βt1y
√
q−1 + α ∈ Fq[y],
G2(y) := N(t2)β + βt2y
√
q−1 + α ∈ Fq[y],
take only square values on the non-zero elements of C. Denote by ψ the multiplicative
character of Fq of order two. The polynomialGi has at most
√
q−1 roots, and ψ(Gi(x)) =
1 for every element x of C \ {0} if x is not a root of Gi. Define ε to be ψ(Gi(0)) if 0 ∈ C
and to be 0 otherwise. Then |C \ {0}| − (√q − 1) + ε ≤
∑
x∈C ψ(Gi(x)). On the other
hand, −(q − |C|) ≤
∑
x∈Fq\C ψ(Gi(x)), and hence
2|C| − q −√q − 1 ≤
∣∣∣∣∣∣
∑
y∈Fq
ψ(Gi(y))
∣∣∣∣∣∣ .
Since
(
√
q − 2)√q < 2|C| − q −√q − 1,
by Result 2.3 (with m = 2) this can only happen if Gi = g2i for some polynomials gi,
i = 1, 2. Then the roots of Gi (in the algebraic closure of Fq) are multiple roots of Gi
and hence also roots of gcd(Gi, G′i). The only root of G
′
i is 0, thus Gi(0) = 0 and hence
N(ti)β + α = 0. Since a
√
qb ̸= d
√
qa, we have β ̸= 0. It follows that N(ti) = −α/β for
i = 1, 2, a contradiction because of the choice of t1 and t2.
The next example shows that ℓ(x) = ax
√
q+1 + dx
√
q + bx + c can have only square
values if a
√
qb = d
√
qa holds.
Example 2.7. For t, r ∈ Fq , the polynomial
f(x) = r
√
q+1x
√
q+1 + r
√
qtx
√
q + rt
√
qx+ t
√
q+1 = (t+ rx)
√
q+1
has only square values in Fq .
We will need a generalisation of the following result by Göloğlu and McGuire.
Result 2.8 ([8, Theorem 1.2]). Let q be odd and consider a non zero polynomial L(x) =∑n−1
i=0 aix
pi ∈ Fq[x]. Denote by □q the set of non-zero squares in Fq . Then
Im
(
L(x)
x
)
⊆ □q ∪ {0}
if and only if L(x) = axp
d
for some a ∈ □q and 0 ≤ d ≤ n.
Definition 2.9. If U is a point set of AG(2, q), then the set of directions defined by U is
DU =
{(
a− b
c− d
)
: (a, b), (c, d) ∈ U, (a, b) ̸= (c, d)
}
.
(If the denominator is zero then
(
a−b
0
)
= (∞), the ideal point of vertical lines.)
52 Ars Math. Contemp. 24 (2024) #P1.04 / 45–66
In the proof of Theorem 2.13 the following result of Szőnyi will be crucial.
Result 2.10 ([10, Theorem 4 and Proposition 6]). Let U be a point set of AG(2, q) of size
at least q −√q/2 and let DU be the set of directions determined by U .
1. If U determines less than (q + 1)/2 directions, then U can be extended to a q-set
determining the same set of directions as U .
2. If U determines exactly (q + 1)/2 directions, one of them is (∞) and there is no
point P ∈ AG(2, q) \ U such that U ∪ {P} determines the same set of directions as
U , then the (q + 1)/2-set
{d ∈ Fq, (d) /∈ DU}
is the set of Y coordinates of the points of an irreducible conic C of AG(2, q) and the
direction (0) is an internal point of C.
Remark 2.11. By [12, Remark 3.3] a blocking set of size at most 2q contains a unique
minimal blocking set. Let U denote an affine point set of size at least q−√q/2 such that U
determines less than (q + 1)/2 directions. Assume that P and P ′ are two affine point sets
of size q−|U | which extend U to a q-set determining the same set of directions as U . Then
B := U ∪P ∪P ′ ∪DU is a blocking set of size at most ⌊q+
√
q/2+ (q+1)/2⌋ ≤ 2q and
hence B contains a unique minimal blocking set. But both U ∪ P ∪DU and U ∪ P ′ ∪DU
are minimal blocking sets and this proves P = P ′, that is, the unicity of the extension of
U in Result 2.10.
Lemma 2.12. Let S denote the set of non-zero squares or non-squares in GF(q). If the
set of Y coordinates of the points of an irreducible conic C of AG(2, q), q ≥ 53 odd, is
contained in S ∪ {0} then C is a parabola with equation
Y = α(a′X + b′Y + c′)2,
where α ∈ S.
Proof. Note that horizontal translations of C does not affect the properties that we are
examining, so after substituting X by X − β for a suitable β ∈ Fq we may assume that
(0, 0) is not a point of C and hence the equation of the conic is
aX2 + bXY + cY 2 + dX + eY + 1 = 0.
The direction (0) cannot be a point of the projective extension of C since otherwise we
would get at least q − 1 > (q + 1)/2 different Y coordinates. It follows that there are at
most 2 horizontal lines meeting C in 1 point and at least (q− 3)/2 horizontal lines meeting
C in 2 points. Fix some α ∈ S. At least (q − 5)/2 horizontal lines meet C in 2 points
(Ai, αB
2
i ) and (A
′
i, αB
2
i ) with Bi ̸= 0; and C has at most 2 points on the X axis. Next
define the quartic Q (which might as well be of smaller degree if c = 0):
aX2 + αbXY 2 + α2cY 4 + dX + αeY 2 + 1 = 0.
Points of C on the X axis are points of Q as well, and if (Ai, αB2i ), (A′i, αB2i ), Bi ̸= 0,
were two points of C then (Ai,±Bi), (A′i,±Bi) are 4 points of Q. It follows that
A = {(x, y), (x,−y) : (x, αy2) ∈ C}
A. Aguglia et al.: Intersecting families of graphs of functions over a finite field 53
is a subset of the point set of Q of size at least 2 + 2(q − 5) = 2q − 8. Note that 2q − 8 >
q + 1 + 6
√
q (here we use q ≥ 53). It follows instantly from the Hasse-Weil bound that Q
cannot be an irreducible cubic or an irreducible quartic.
First we show that Q cannot be a quartic curve which is the product of an irreducible
cubic and a line. Vertical and horizontal lines meet A in at most 2 points and hence if such
a line would be a factor of Q then the remaining at least 2q − 10 points of A should lie
on the cubic, a contradiction again by the Hasse-Weil bound now applied to cubic curves.
Similarly, if Y = mX + n was a factor of Q, for some m ̸= 0, then
aX2 + αbX(mX + n)2 + α2c(mX + n)4 + dX + αe(mX + n)2 + 1
was the zero polynomial. The coefficient of X4 is α2cm4, so c has to be zero but then Q is
not a quartic curve, a contradiction.
From now on we may assume that
aX2 + αbXY 2 + α2cY 4 + dX + αeY 2 + 1 = F ·G,
where F and G are of degree at most 2. Put
F = (a1Y
2 + b1X + c1Y + 1 + d1X
2 + e1XY ),
G = (a2Y
2 + b2X + c2Y + 1 + d2X
2 + e2XY ).
In F · G the coefficient of Y is c1 + c2, while it is 0 in the equation of Q, so clearly
c2 = −c1 and we will use this from now on. The coefficient of X4 is d1d2 and it has to be
zero, so from now on we may assume d1 = 0. Then the coefficient of X3 is d2b1 and it has
to be zero.
In this paragraph assume d2 ̸= 0 , then b1 = 0 and the coefficient of X3Y is d2e1 so
e1 = 0. The coefficient of Y X2 is c1d2, so c1 = 0. Then the coefficient of XY is e2,
so e2 = 0. The coefficient of X2Y 2 is d2a1, so a1 = 0. We arrived to the conclusion
that the equation of Q is a2Y 2 + b2X + 1 + d2X2. It follows that the equation of C is
Y = −α(b2X+1+d2X2)/a2. Then by Result 2.4, −(b2X+1+d2X2)/a2 = (a′X+b′)2
for some a′, b′ ∈ Fq and this finishes the proof of the d2 ̸= 0 case.
Now assume d2 = 0 (recall also d1 = 0). Then the coefficient of X2Y 2 is e1e2. We
may assume e1 = 0. The coefficient ofX2Y is e2b1. First assume e2 ̸= 0, so b1 = 0. Then
the coefficient of Y 3X is a1e2, so a1 = 0. Then the coefficient of Y 3 is c1a2. We cannot
have c1 = 0 since then the coefficient of XY would be e2 ̸= 0, so a2 = 0. The coefficient
of XY is c1b2 + e2 = 0 and hence the equation of Q is: (1 + c1Y )(1 − c1Y )(1 + b2X),
a contradiction since vertical and horizontal lines contain at most 2 points of A. So e2 = 0
and from now on we may assume that Q has equation
(a1Y
2 + b1X + c1Y + 1)(a2Y
2 + b2X − c1Y + 1) = 0.
The fact that Y 3 and XY should have zero coefficient yields a1 = a2 and b1 = b2, or
c1 = 0. In the former case the equation of Q is
(a1Y
2 + b1X + c1Y + 1)(a1Y
2 + b1X − c1Y + 1) = 0,
so C had equation
(α−1a1Y + b1X + 1)
2 − α−1c21Y = 0,
54 Ars Math. Contemp. 24 (2024) #P1.04 / 45–66
which proves the assertion.
In the latter case the equation of Q is
(a1Y
2 + b1X + 1)(a2Y
2 + b2X + 1) = 0,
so C had equation
(a1Y/α+ b1X + 1)(a2Y/α+ b2X + 1) = 0,
a contradiction since C was irreducible.
The following can be considered as a generalisation of Result 2.8.
Theorem 2.13. Let U denote a point set of AG(2, q), q ≥ 53 odd, of size at least q−√q/2.
Let S denote the set of non-zero squares or non-squares in Fq and let (d) denote one of
the directions (0) or (∞). If DU is contained in {(s) : s ∈ S} ∪ {(d)}, then U is affinely
equivalent to a subset of the graph of a function of the form
f(x) = αxp
k
,
where α ∈ S.
Proof. If |DU | < (q + 1)/2 then by Result 2.10 U can be extended to a q-set U ′ deter-
mining the same set of directions. According to Result 2.1 U ′ is affinely equivalent to the
graph of a linearised polynomial f . Then Result 2.8 shows that f has the requested form.
Now assume |DU | = (q + 1)/2. If (d) = (0), then apply the affinity φ : (x : y : z) 7→
(y : x : z). Clearly, DUφ = (DU )φ and U can be extended if and only if Uφ can be
extended. We have (0)φ = (∞) and if m ̸= 0 then (m)φ = (1/m), so
{(s)φ : s ∈ S} = {(s) : s ∈ S}.
By Result 2.10, if Uφ (or U , if (d) = (∞)) cannot be extended, then the set of non-
zero squares or non-squares together with the zero equals the set of Y coordinates of an
irreducible affine conic C and (0) is an internal point of C. Then the line at infinity is not
a tangent to C, thus C is not a parabola (and not a hyperbola because then the size of the
set of Y coordinates would be (q − 1)/2; but we don’t need this) but this is not possible
because of the Lemma 2.12. It follows that U can be extended to a q-set determining the
same set of directions as U and the proof can be finished as in the previous paragraph.
3 On intersecting families of graphs of functions
Our first aim is to prove Theorem 1.3 which we will do separately in the odd and even case.
Lemma 3.1. If U is a set of t-intersecting polynomials of degree at most k over Fq , then
the (k− t+1)-ple of coefficients of the monomials xt, . . . , xk in elements of U are distinct
elements of Fk−t+1q .
Proof. If the coefficients of xt, . . . , xk coincide in f1, f2 ∈ U , then f1 − f2 would have
degree at most t − 1, and hence at most t − 1 roots, thus the graphs of f1 and f2 would
share at most t− 1 points, a contradiction.
A. Aguglia et al.: Intersecting families of graphs of functions over a finite field 55
Next, we report below Lemma 6.1 from [1] with an alternative proof.
Lemma 3.2 ([1, Lemma 6.1]). Assume t ≤ k < q. Let U be a set of polynomials of degree
at most k over Fq .
(1) If for any f, g ∈ U there exist x1, . . . , xt ∈ Fq such that f(xi) = g(xi) for i =
1, . . . , t, then |U | ≤ qk−t+1.
(2) If for any f, g ∈ U there are no x1, . . . , xt ∈ Fq such that f(xi) = g(xi) for
i = 1, . . . , t, then |U | ≤ qt.
Proof. Proof of Part (1): It is a direct consequence of Lemma 3.1.
Proof of Part (2): Take any t distinct field elements, say, x1, . . . , xt. For any polynomial
f over Fq , (f(x1), . . . , f(xt)) can take at most qt distinct values of Ftq and hence if |U | >
qt then there will be at least 2 polynomials in U which have the same values on the set
{x1, . . . , xt}.
Lemma 3.3. LetU be a set of intersecting polynomials of degree at most 2 over Fq . Assume
that there are more than ⌊(q+1)/2⌋ polynomials hi in U , so that their x2 coefficients are c,
for some fixed c ∈ Fq and suppose also that there exist values α and β so that hi(α) = β.
Then for every polynomial f ∈ U , whose coefficient in x2 is not c, f(α) = β.
Proof. First assume that α = 0. Then the constant term in the polynomials hi is always β
and we want to show that for any polynomial f ∈ U , if the coefficient of x2 is not c, the
constant term must be β. Assume to the contrary, that there is a polynomial g ∈ U , whose
constant term is not β. Consider the polynomials:
{g − hi}.
Since g and hi are intersecting, g − hi must have a root in Fq . Also, by the assumptions
of the lemma, (g − hi)(x) = dx2 + vx + w, where d ̸= 0 and w ̸= 0 are fixed. We
claim that there are at most ⌊(q + 1)/2⌋ such polynomials, hence a contradiction. Indeed,
if (g− hi)(x) has a root in Fq , then it can be written as d(x− u)(x− wdu ). So to bound the
number of possible polynomials, we have to bound the number of different (u, wdu ) pairs,
where the order does not matter.
First assume q to be odd. If w/d is not a square, then we get (q − 1)/2 such pairs. If it
is a square, then we see 2 + (q − 3)/2 pairs, which is (q + 1)/2.
Now, assume q to be even. In this case the number of different pairs (u, wdu ) is
(q − 2)/2 + 1, that is, q/2.
Finally, if α ̸= 0, then instead of the polynomials f in U , consider the polynomials
f̄(x) := f(x + α). This new family is clearly an intersecting family, the h̄i polynomials
will still have the same leading coefficients and h̄i(0) = β, so we are in the previous
case.
Lemma 3.4. Let q be even and U be a set of intersecting polynomials of degree at most 2
over Fq . Assume that |U | > q
2+q
2 and assume also that H is a subset of U with more than
q2
2 polynomials hi, so that there exist values α and β for which hi(α) = β. Then for every
polynomial f ∈ U , f(α) = β.
56 Ars Math. Contemp. 24 (2024) #P1.04 / 45–66
Proof. By the pigeon hole principle, there exists a value c such that there are more than
q
2 polynomials in H with x
2 coefficient c. Let U c denote the polynomials in U with x2
coefficient c. Then Lemma 3.3 implies that for any polynomial f ∈ (U \ U c), f(α) = β.
Note that |U \ U c| > q
2+q
2 − q. Again by the pigeon hole principle, there exists a value
c′ ̸= c so that there are more than q
2−q
2(q−1) =
q
2 polynomials in (U \ U
c) with x2 coefficient
c′. Lemma 3.3 yields that for any polynomial g ∈ U c, g(α) = β.
The next result can be proved in exactly the same way as Lemma 3.4.
Lemma 3.5. Let q be odd and U be a set of intersecting polynomials of degree at most 2
over Fq . Assume that |U | > q
2+2q−1
2 and suppose also that H is a subset of U with more
than q
2+q
2 polynomials hi, so that there exist values α and β for which hi(α) = β. Then
for every polynomial f ∈ U , f(α) = β.
Lemma 3.6. Let U be a set of intersecting polynomials of degree at most k > 1 over
Fq . Assume that there are more than (q − 1)qk−2 polynomials hi in U , so that their xk
coefficients are c, for some fixed c ∈ Fq and suppose also that there exist values α and β
so that hi(α) = β. Then for every polynomial f ∈ U , whose coefficient of xk is not c, it
holds that f(α) = β.
Proof. First assume that α = 0. Then the constant term in the polynomials hi is always β
and we want to show that for any polynomial f ∈ U , if the coefficient of xk is not c, the
constant term must be β. Assume to the contrary, that there is a polynomial g ∈ U , whose
constant term is not β. Consider the polynomials:
{g − hi}.
Since g and hi are intersecting, g − hi must have a root in Fq . Also, by the assumptions of
the lemma, (g − hi)(x) = dxk + v1xk−1 + v2xk−2 + . . .+ vk−1x+w, where d ̸= 0 and
w ̸= 0 are fixed. We claim that there are at most (q − 1)qk−2 such polynomials, hence a
contradiction. Indeed, such polynomials can be written in the form (x− u)(dxk−1 + . . .−
w/u). Note that u ̸= 0, because w ̸= 0, hence u can take q − 1 values. The second term
is a polynomial of degree k − 1, its coefficient in xk−1 and its constant term are fixed, so
there are at most qk−2 different such polynomials.
As before, if α ̸= 0, then instead of the polynomials f in U , consider the polynomials
f̄(x) := f(x+α). This new family is clearly an intersecting family, the h̄i polynomials will
still have the same leading coefficients and h̄i(0) = β, so we are in the previous case.
3.1 Intersecting families of polynomials of degree at most 2, over Fq , q odd
According to Lemma 3.1, the members of an intersecting family of polynomials of degree
at most 2 are of the form f(b, c) + bx+ cx2 for some function f . More precisely:
Definition 3.7. Suppose that U is a set of intersecting polynomials. Put D = {(b, c) ∈
Fq × Fq : a+ bx+ cx2 ∈ U} and define f : D → Fq as f(b, c) = a, where a ∈ Fq is the
unique field element such that a+ bx+ cx2 ∈ U .
Lemma 3.8. Let U be a set of intersecting polynomials of degree at most 2 and for b ∈ Fq ,
q ≥ 53 odd, define
Cb := {c ∈ Fq : f(b, c) + bx+ cx2 ∈ U}
A. Aguglia et al.: Intersecting families of graphs of functions over a finite field 57
and
domo := {b ∈ Fq : |Cb| > q −
√
q/2 + 1/2}.
There exist functions s, t : domo → Fq and h : domo → {0, 1, . . . , n− 1} (where q = pn)
such that for every c ∈ Cb
f(b, c) = s(b)cp
h(b)
+ t(b),
and −s(b) is square in Fq .
Proof. If f(b, c)+ bx+ cx2 and f(d, e)+ dx+ ex2 are members of U , then the difference
of the two polynomials must have a root in Fq and hence
F (b, c, d, e) := (b− d)2 − 4(f(b, c)− f(d, e))(c− e)
is a square. For c ∈ Cb put fb(c) for f(b, c).
For each b ∈ Fq and C,E ∈ Cb, C ̸= E, consider F (b, C, b, E) = −4(fb(C) −
fb(E))(C − E), which has to be a square, or, equivalently, after dividing by (C − E)2,
−fb(C)− fb(E)
C − E
is in □q ∪ {0} for each C,E ∈ Cb.
If b ∈ domo, then by Theorem 2.13, fb can be uniquely extended to a function f̃b : Fq →
Fq determining the same set of directions as fb and for each c ∈ Fq f̃b(c) = s(b)cp
h(b)
+t(b)
for some domo → Fq functions s, t such that −s(b) is a square, and a function h : domo →
{0, 1, . . . , n− 1}. Then for c ∈ Cb we have fb(c) = s(b)cp
h(b)
+ t(b).
Lemma 3.9. If q ≥ 53 is odd, U is a set of intersecting polynomials of degree at most 2
such that |domo| > 1, then for b, d ∈ domo and c ∈ Cb, e ∈ Cd recall that
F (b, c, d, e) = (b− d)2 − 4(f(b, c)− f(d, e))(c− e)
= (b− d)2 − 4(s(b)cp
h(b)
+ t(b)− s(d)ep
h(d)
− t(d))(c− e).
For b ∈ domo, one of the following holds
1. s(b) = s(d) = 0 and t(d) = t(b) for each d ∈ domo,
2. s(b) = s(d) ̸= 0, h(d) = h(b) = 0 and (t(b) − t(d))2 = −s(b)(b − d)2 for each
d ∈ domo,
3. s(b) = s(d) ̸= 0, h(d) = h(b) = n/2 and t(b) = t(d) for each d ∈ domo.
Proof. For b, d ∈ domo and c ∈ Cb, e ∈ Cd recall that F (b, c, d, e) is a square in Fq .
Define the function Gb,d,e : Fq → Fq , as
c 7→ (b− d)2 − 4(s(b)cp
h(b)
+ t(b)− s(d)ep
h(d)
− t(d))(c− e) =
−4s(b)cp
h(b)+1 + 4es(b)cp
h(b)
− 4(t(b)− s(d)ep
h(d)
− t(d))c+
(b− d)2 + 4e(t(b)− s(d)ep
h(d)
− t(d)).
58 Ars Math. Contemp. 24 (2024) #P1.04 / 45–66
First assume 0 < h(b) < n/2 for some b ∈ domo.
Denote by ψ the quadratic character of Fq and apply Result 2.3 to the function Gb,d,e.
Then we have
q −√q − ph(b) < −(q − |Cb|) + (|Cb| − ph(b) − 1) ≤
∑
c∈Fq
ψ(Gb,d,e(c)),
as Gb,d,e is a polynomial of degree ph(b) + 1 in c, so the number of its roots is at most
ph(b) + 1.
Thus, we cannot have ∣∣∣∣∣∣
∑
c∈Fq
ψ(Gb,d,e(c))
∣∣∣∣∣∣ ≤ ph(b)√q.
It follows that Gb,d,e is the square of a polynomial in c. And hence by Lemma 2.5, one of
the following holds:
(i) s(b) = 0 and t(b) − s(d)eph(d) − t(d) = 0 for each d ∈ domo, e ∈ Cd. If we fix d
as well and let e run through Cd then we obtain s(d) = 0 and t(b) = t(d), for each
d ∈ domo.
(ii) s(b) ̸= 0 and
(4es(b))p
h(b)
(−4s(b)) = −4(t(b)− s(d)ep
h(d)
− t(d))(−4s(b))p
h(b)
, (3.1)
(4es(b))p
h(b)+1(−4s(b)) = ((b−d)2+4e(t(b)−s(d)ep
h(d)
−t(d)))(−4s(b))p
h(b)+1.
(3.2)
Then (3.1) yields s(d)ep
h(d) − s(b)eph(b) = t(b)− t(d), for each d ∈ domo, e ∈ Cd.
Fix d as well and let e run through Cd. PutK for the dimension over Fp of the kernel
of the Fp-linear Fq → Fq function e 7→ s(d)ep
h(d) − s(b)eph(b) . Then
|Cd|
pK
≤
∣∣∣{s(d)eph(d) − s(b)eph(b) : e ∈ Cd}∣∣∣ = |{t(b)− t(d)}| = 1,
thus K = n (q = pn) and hence s(d) = s(b), h(d) = h(b) and t(d) = t(b) for each
d ∈ domo.
Then (3.2) reads as 0 = (b − d)2 for each d ∈ domo, a contradiction since
|domo| > 1.
We proved that 0 < h(b) < n/2 implies s(d) = 0 and t(b) = t(d) for each d ∈ domo.
Next assume n/2 < h(b) < n for some b ∈ domo.
Apply Result 2.3 to the function c 7→ (Gb,d,e(c))p
n−h(b)
(mod cq − c) and continue as
above. It turns out that s(d) = 0 and t(b) = t(d) for each d ∈ domo also in this case.
A. Aguglia et al.: Intersecting families of graphs of functions over a finite field 59
Now assume h(b) = n/2 for some b ∈ domo.
If s(b) = 0, then
−4(t(b)− s(d)ep
h(d)
− t(d))c+ (b− d)2 + 4e(t(b)− s(d)ep
h(d)
− t(d))
is a square for each d ∈ domo and c ∈ Cb, e ∈ Cd. If we consider d and e fixed as well,
then it follows that as a function of c it has to be a constant, so t(b)− s(d)eph(d) − t(d) = 0
for each e ∈ Cd and hence s(d) = 0 and t(b) = t(d) for each d ∈ domo.
If s(b) ̸= 0, then Lemma 2.6 applied to Gb,d,e gives (3.1) and hence, as before,
s(d) = s(b), t(d) = t(b) and h(d) = h(b) for each d ∈ domo.
Finally, consider the case when h(b) = 0 for some b ∈ domo.
Then again from Result 2.3, one obtains Gb,d,e(c) = (b − d)2 − 4(s(b)c + t(b) −
s(d)ep
h(d) − t(d))(c− e) = (α+ βc)2, for some α, β ∈ Fq .
If s(b) = 0, that is, when Gb,d,e is a constant, then t(b) − s(d)ep
h(d) − t(d) = 0 for
each d ∈ domo, e ∈ Cd, so s(d) = 0 and t(b) = t(d) for each d ∈ domo.
If s(b) ̸= 0, that is, when Gb,d,e is of degree two, then the discriminant of Gb,d,e has to
be zero, i.e.
s(b)(b− d)2 + (s(b)e− s(d)ep
h(d)
+ t(b)− t(d))2 = 0.
For d ∈ domo let εd be an element of Fq for which ε2d = −s(b)(b − d)2. Consider
d ∈ domo fixed as well, then for e ∈ Cd:
s(b)e− s(d)ep
h(d)
∈ {εd + t(d)− t(b),−εd + t(d)− t(b)}.
Put K for the dimension over Fp of the kernel of the Fp-linear Fq → Fq function e 7→
s(b)e− s(d)eph(b) . Then
|Cd|
pK
≤
∣∣∣{s(b)e− s(d)eph(d) : e ∈ Cd}∣∣∣ ≤ 2,
which is a contradiction, unless K = n. It follows that s(b)e − s(d)eph(b) = 0 for each
e ∈ Fq , so h(d) = 0, s(d) = s(b) and t(d)− t(b) is one of εd and −εd.
Lemma 3.10. If q ≥ 53 is odd, U is a set of intersecting polynomials of degree at most 2
such that |domo| > (q+1)/2, then there exist α, β ∈ Fq such that g(α) = β for all g ∈ U
with g = f(b, c) + bx+ cx2 where b ∈ domo.
Proof. According to Lemma 3.9, we consider the following two cases.
Suppose that there exists some b′ ∈ domo such that s(b′) = 0.
Then s(d) = 0 and t(d) = t(b) for each d ∈ domo. Put T for t(b). It follows
that for b ∈ domo and c ∈ Cb the polynomials f(b, c) + bx + cx2 ∈ U have the shape
T + bx+ cx2 ∈ U and hence (0, T ) is a common point of their graphs.
Suppose that s(b) ̸= 0 for each b ∈ domo.
Then s(b) = s(d) and h(d) = h(b) for each d ∈ domo. We will denote these values by
S and h, respectively. Note that h = 0, or h = n/2.
60 Ars Math. Contemp. 24 (2024) #P1.04 / 45–66
When h = 0.
Then (t(b)− t(d))2 = −S(b− d)2 for each b, d ∈ domo, and so{
t(b)− t(d)
b− d
: b, d ∈ domo, b ̸= d
}
⊆ {s,−s},
where s2 = −S.
It follows that the point set {(b, t(b)) : b ∈ domo} determines at most two directions.
But there is no point set determining exactly two directions, thus t determines a unique
direction, i.e.
t(d) = γd+ T, for d ∈ domo,
where γ is a constant satisfying γ2 = s2 = −S. Then for b ∈ domo and c ∈ Cb the
polynomials f(b, c) + bx+ cx2 ∈ U have the shape Sc+ γb+ T + bx+ cx2, so (−γ, T )
is the common point of their graphs.
When h = n/2.
Then also t(b) = t(d) for each d ∈ domo. Then
(b− d)2 − 4S(c− e)
√
q+1
has to be a square for each b, d ∈ domo and c ∈ Cb, e ∈ Cd.
Fix b, c, d. Note that for k ∈ F√q \ {0} there are
√
q + 1 elements x in Fq such that
x
√
q+1 = k. Since e runs through more than q −√q/2 + 1/2 values, we have
F√q \ {0} ⊆ {(c− e)
√
q+1 : e ∈ Cd}
so
(b− d)2 − Sk = b2 − 2db+ d2 − Sk (3.3)
has to be a square in Fq for each b, d ∈ domo, k ∈ F√q \ {0}. As a polynomial in b, the
discriminant of (3.3) is
4d2 − 4(d2 − Sk) = Sk.
Recall S = s(b) ̸= 0, so this discriminant cannot be zero. By Result 2.4, for fixed d ∈
domo and k ∈ F√q \ {0} and for the character ψ of order 2,∑
b∈Fq
ψ(b2 − 2db+ d2 − Sk) = −ψ(1) = −1.
On the other hand, a lower bound for this sum is
|domo| − 2− |Fq \ domo|,
which is at least 2|domo| − q − 2, a contradiction when |domo| > (q + 1)/2.
Next, we prove Theorem 1.3 when q is odd.
Theorem 3.11. If q ≥ 53 is odd and U is a set of q2 − ε, ε < q
√
q
4 −
3q
8 −
√
q
8 , intersecting
polynomials of degree at most 2 over Fq , then the graphs of the functions in U share a
common point.
A. Aguglia et al.: Intersecting families of graphs of functions over a finite field 61
Proof. Let domo denote the set of values as before and let us call a polynomial f(b, c) +
bx+ cx2 ∈ U good if b ∈ domo. According to the previous lemma, if |domo| > (q+1)/2
the graphs of the good polynomials share a common point. If there are more than q
2+q
2
good polynomials then Lemma 3.5 finishes the proof.
Clearly,
|U | ≤ |domo|q + (q − |domo|)(q −
√
q/2 + 1/2) = q2 − (q − |domo|)(
√
q/2− 1/2).
Assume to the contrary that the number of good polynomials is at most q
2+q
2 , then |domo| <
q2+q
2(q−√q/2+1/2) <
q
2 +
√
q
4 +
1
2 . Hence:
|U | ≤ q2 −
(
q
√
q
4
− 3q
8
−
√
q
8
+
1
4
)
,
which is a contradiction.
3.2 Intersecting families of polynomials of degree at most 2, over Fq , q even
Lemma 3.12. Let U be a set of intersecting polynomials of degree at most 2 and for t ∈ Fq ,
q > 2 even, define
Bt := {b ∈ Fq : f(b, b+ t) + bx+ (b+ t)x2 ∈ U}
and
dome := {t ∈ Fq : |Bt| ≥ q −
√
q/2}.
There exist functions A,B : dome → Fq such that for every b ∈ Bt
f(b, b+ t) = A(t)b+B(t),
and A(t) ∈ kerTrq/2.
Proof. Consider F (x) = f(b, c) + bx + cx2 and G(x) = f(d, e) + dx + ex2. Then the
graphs of F and G share a common point if and only if F − G has a root in Fq , that is,
b = d or
H(b, c, d, e) := Trq/2
(
(c+ e)(f(b, c) + f(d, e))
(b+ d)2
)
= 0.
Then for each t ∈ Fq , b, d ∈ Bt, b ̸= d,
H(b, b+ t, d, d+ t) = Trq/2
(
(b+ d)(f(b, b+ t) + f(d, d+ t))
(b+ d)2
)
= 0.
Simplifying by b+ d yields
Trq/2
(
f(b, b+ t) + f(d, d+ t)
b+ d
)
= 0.
Define Rt : Bt → Fq as Rt(x) = f(x, x+ t). For each x, y ∈ Bt, x ̸= y, it holds that
Trq/2
(
Rt(x) +Rt(y)
x+ y
)
= 0.
62 Ars Math. Contemp. 24 (2024) #P1.04 / 45–66
In particular, the set of directions determined by the graph of Rt is contained in kerTrq/2,
and hence it has size at most q/2.
From now on assume t ∈ dome and hence |Bt| ≥ q −
√
q/2. By results of Szőnyi
[10], there exists a unique extension R̃t : Fq → Fq of Rt such that the set of direc-
tions determined by R̃t is the same as the set of directions determined by the point set
{(x,Rt(x)) : x ∈ Bt} ⊆ AG(2, q). So the set of directions determined by R̃t is
contained in kerTrq/2 and hence by Theorem 2.2 there exist A(t), B(t) ∈ Fq such that
R̃t(x) = A(t)x+B(t) with Trq/2(A(t)) = 0. It follows that for b ∈ Bt we have
Rt(b) = f(b, b+ t) = A(t)b+B(t).
Lemma 3.13. Let U be a set of intersecting polynomials of degree at most 2 and define
Bt, dome and the functions A and B as in the previous lemma. Then there exist α, β ∈ Fq ,
q ≥ 8, such that A(t) = αq/2 + α and B(t) = αt+ β for each t ∈ dome.
Proof. If |dome| = 1, then the assertion is trivial, so assume |dome| ≥ 2 and take any
s, t ∈ dome. Fix some b ∈ Bs. Then for each d ∈ Bt \ {b},
H(b, b+ s, d, d+ t) = Trq/2
(
(b+ s+ d+ t)(f(b, b+ s) + f(d, d+ t))
(b+ d)2
)
= 0,
that is,
Trq/2
(
(b+ s+ d+ t)(f(b, b+ s) +A(t)d+B(t))
(b+ d)2
)
= 0,
i.e.,
Trq/2
(
A(t) +
f(b, b+ s) +B(t) +A(t)b+A(t)(s+ t)
b+ d
+
(s+ t)
f(b, b+ s) +B(t) +A(t)b
(b+ d)2
)
= 0.
Applying Trq/2(A(t)) = 0 and Trq/2(z) = Trq/2(z2) for each z ∈ Fq , we obtain for each
d ∈ Bt \ {b}, d ̸= b
Trq/2
(
f2(b, b+ s) +B2(t) +A2(t)b2 +A2(t)(s+ t)2+
(b+ d)2
(s+ t)(f(b, b+ s) +B(t) +A(t)b)
(b+ d)2
)
= 0.
The numerator does not depend on d, while the denominator ranges over a subset of
F∗q of size |Bt \ {b}| > degTrq/2 = q/2 and hence this is possible only if
f2(b, b+ s)+B2(t)+A2(t)b2+A2(t)(s+ t)2+(s+ t)(f(b, b+ s)+B(t)+A(t)b) = 0.
Since f(b, b+ s) = A(s)b+B(s), this is equivalent to
(A(s)b+B(s))2 +B2(t) +A2(t)b2 +A2(t)(s+ t)2+
(s+ t)(A(s)b+B(s) +B(t) +A(t)b) = 0,
A. Aguglia et al.: Intersecting families of graphs of functions over a finite field 63
that is,
b2(A2(s) +A2(t)) + b(A(s) +A(t))(s+ t)+
(B(s) +B(t))(B(s) +B(t) + s+ t) +A2(t)(t+ s)2 = 0.
Since this holds for every b ∈ Bs, and |Bs| > 2, it follows that as a polynomial of b, this is
the zero polynomial, so A(s) = A(t) and
(B(s) +B(t))(B(s) +B(t) + s+ t) +A2(t)(t+ s)2 = 0. (3.4)
Since Trq/2(A(t)) = 0, and A(t) is a constant function, this proves the existence of α′ ∈
Fq such thatA(x) = α′q/2+α′ for each x ∈ dome. If |dome| = 2, then clearlyB is linear,
so assume |dome| ≥ 3 and take some t′ ∈ dome \ {s, t}. The same arguments show
(B(s) +B(t′))(B(s) +B(t′) + s+ t′) +A2(t′)(t′ + s)2 = 0. (3.5)
Summing up (3.4) and (3.5) we obtain
B(s)(t+t′)+s(B(t)+B(t′))+B2(t)+B2(t′)+B(t)t+B(t′)t′+(α′2+α′)(t+t′)2 = 0,
so for x ∈ dome \ {t, t′} it holds that
B(x) = x
B(t) +B(t′)
t+ t′
+
B2(t) +B2(t′) +B(t)t+B(t′)t′
t+ t′
+ (α′2 + α′)(t+ t′),
and from (3.4) (with s = t′) one obtains the same for x ∈ {t, t′}, so B is linear. Put
B(x) = γx+ β, then from (3.4)
γ(s+ t)(γ(s+ t) + (s+ t)) = (α′2 + α′)(s+ t)2,
so γ2 + γ = α′2 + α′ which proves γ = α′ or γ = α′ + 1. Now, if γ = α′ then we set
α := α′ whereas if γ = α′+1 we set α := α′+1. Since α′q/2+α′ = (α′+1)q/2+α′+1,
our lemma follows.
Corollary 3.14. If g(x) = f(b, c) + bx+ cx2 ∈ U and b+ c ∈ dome, then g(αq/2) = β.
Proof. Put t = b+ c. Then t ∈ dome and hence
f(b, c) = f(b, b+ t) = A(t)b+B(t) = αq/2b+ αb+ αt+ β,
hence
g(αq/2) = αq/2b+ αb+ αt+ β + bαq/2 + (b+ t)α = β.
Finally, we prove Theorem 1.3 for q even.
Theorem 3.15. If q ≥ 8 is even and U is a set of q2 − ε, ε < q
√
q
4 −
q
8 −
√
q
8 , intersecting
polynomials of degree at most 2 over Fq , then the graphs of the functions in U share a
common point.
64 Ars Math. Contemp. 24 (2024) #P1.04 / 45–66
Proof. Let dome denote the set of values as before and let us call a polynomial f(b, c) +
bx + cx2 ∈ U good if b + c ∈ dome. According to the previous corollary, the graphs of
the good polynomials share a common point. If there are more than q
2
2 good polynomials
then Lemma 3.4 finishes the proof.
Clearly,
|U | ≤ |dome|q + (q − |dome|)(q −
√
q/2) = q2 − (q − |dome|)
√
q/2.
Assume to the contrary that the number of good polynomials is at most q
2
2 , then |dome| ≤
q2
2(q−√q/2) <
q
2 +
√
q
4 +
1
4 . Hence:
|U | < q2 −
(
q
√
q
4
− q
8
−
√
q
8
)
,
which is a contradiction.
3.3 Intersecting families of polynomials of degree at most k > 2
Theorem 1.4. If U is a set of more than qk − qk−1 intersecting polynomials over Fq ,
q ≥ 53 when q is odd and q ≥ 8 when q is even, and of degree at most k, k > 1, then
there exist α, β ∈ Fq such that g(α) = β for all g ∈ U . Furthermore, U can be uniquely
extended to a family of qk intersecting polynomials of degree at most k over Fq .
Proof. Let U be a set of more than qk − qk−1 intersecting polynomials over Fq and of
degree at most k, k > 1. First we show that there exist α, β ∈ Fq such that g(α) = β for
all g ∈ U . We prove this by induction.
For k = 2, this is true by Theorem 1.3. Now assume that it is true for k − 1 and we
want to prove it for k. By the pigeon hole principle there must be a value c, such that there
are more than qk−1−qk−2 polynomials hi in U whose coefficient in xk is c. Now consider
the family of polynomials in the form of {hi − cxk}. Clearly, this is an intersecting family
of polynomials of degree at most k − 1. So by the induction hypothesis, there are values α
and β so that for every i, (hi − cxk)(α) = β and hence of course hi(α) = β + cαk and so
Lemma 3.6 finishes the proof of the first part.
Next, we will prove that U can be uniquely extended to a family of qk intersecting
polynomials of degree at most k over Fq . Hence, let F and F ′ be two intersecting families
of size qk, both of them containing U . Then, there exist α, α′, β, β′ ∈ Fq such that g(α) =
β for all g ∈ F and g(α′) = β′ for all g ∈ F ′. The polynomials in U are in F ∩ F ′, a
contradiction unless (α, β) = (α′, β′), since there are at most qk−1 < |U | polynomials of
degree at most k, whose graph contains two distinct, fixed points. Theorem 1.4 follows.
4 Large intersecting families whose graphs do not share a common
point
The following construction was drawn to our attention in a talk by Sam Adriaensen. Note
that it shows the sharpness of the lower bound on |U | in Lemma 3.4.
Example 4.1 (Hilton-Milner type). Pick a point P := (α, β) and a line e := {(x, vx+w) :
x ∈ Fq} in AG(2, q), so that β ̸= vα + w. Let U ′ be the set of those polynomials over
Fq , which are of the form h(x) = cx2 + bx + a and for which h(α) = β and there exist
A. Aguglia et al.: Intersecting families of graphs of functions over a finite field 65
values α′ and β′ so that h(α′) = β′ and β′ = vα′ + w. The set U = U ′ ∪ {e} is a set
of intersecting polynomials of degree at most 2 over Fq . The size of U is q
2+q
2 and clearly
there exist no values s, t ∈ Fq so that for every polynomial f ∈ U , f(s) = t.
Proof. Clearly, we may assume that P = (0, 1) and e = {(x, 0) : x ∈ Fq}. Then a = 1
for the polynomials in U ′. Pick a point R := (u, 0) from e. The number of polynomials
g in U ′, so that g(u) = 0 is 0 if u = 0, q otherwise. Hence if we count the polynomials
of U ′ corresponding to R when R runs on the points of e, we see q(q − 1) polynomials.
But most of the polynomials in U ′ will correspond to two different points R and R′ of e.
Actually, only the polynomials which are of the form bx + 1 (b ∈ F∗q) and polynomials
of the form c−2(x + c)2 (c ∈ F∗q) in U ′ will correspond to exactly one point in e. Hence
|U ′| = q(q−1)−2(q−1)2 + 2(q − 1) =
q2+q
2 − 1 and so |U | =
q2+q
2 .
Example 4.2. Let q be odd. There is a family M of intersecting polynomials of degree at
most 2 such that |M| = q
2−q+1
2 and there exists f ∈ M with the property that |Uf ∩Ug| =
1 for each g ∈ M, g ̸= f .
Proof. Choose a polynomial f(x) = Ax2 + Bx + C and let □q be the set of non-zero
squares in Fq . Let
P =
{
aix
2 + bix+ C −
(B − bi)2
4(A− ai)
: A− ai ∈ □q, bi ∈ Fq
}
.
Note that |P| = q(q − 1)/2.
If (ai, bi) and (aj , bj) correspond to two elements of P then the graphs of the corre-
sponding polynomials meet each other if and only if
(bi − bj)2 − 4(ai − aj)
(
(B − bj)2
4(A− aj)
− (B − bi)
2
4(A− ai)
)
=
(aiB − ajB −Abi + ajbi +Abj − aibj)2
(A− ai)(A− aj)
is a (possibly zero) square in Fq . This certainly holds since both (A − ai) and (A − aj)
are squares. Hence P is an intersecting family. Finally, we prove that M = P ∪ {f} is
also an intersecting family. We will do this by proving that for each g ∈ P , Ug meets Uf
in a unique point. So assume g(x) = ax2 + bx + C − (B−b)
2
4(A−a) . It is easy to see that the
discriminant of f − g is zero and hence the result follows.
ORCID iDs
Angela Aguglia https://orcid.org/0000-0001-6854-8679
Bence Csajbók https://orcid.org/0000-0003-2801-452X
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.05 / 67–74
https://doi.org/10.26493/1855-3974.2805.b49
(Also available at http://amc-journal.eu)
Saturated 2-plane drawings with few edges
János Barát *
Department of Mathematics, University of Pannonia and
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Géza Tóth †
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Received 12 January 2022, accepted 24 May 2023, published online 18 August 2023
Abstract
A drawing of a graph is k-plane if every edge contains at most k crossings. A k-plane
drawing is saturated if we cannot add any edge so that the drawing remains k-plane. It is
well-known that saturated 0-plane drawings, that is, maximal plane graphs, of n vertices
have exactly 3n−6 edges. For k > 0, the number of edges of saturated n-vertex k-plane
graphs can take many different values. In this note, we establish some bounds on the
minimum number of edges of saturated 2-plane graphs under various conditions.
Keywords: Saturated drawing, 2-planar, graphs, discharging.
Math. Subj. Class. (2020): 05C10, 05C35
1 Preliminaries
In a drawing of a graph in the plane, vertices are represented by points, edges are repre-
sented by curves connecting the points, which correspond to adjacent vertices. The points
(curves) are also called vertices (edges). We assume that an edge does not go through any
vertex, and three edges do not cross at the same point. A graph together with its drawing
is a topological graph. A drawing or a topological graph is simple if any two edges have at
most one point in common, that is either a common endpoint or a crossing. In particular,
there is no self-crossing. In this paper, we assume the underlying graph has neither loops
nor multiple edges.
For any k ≥ 0, a topological graph is k-plane if each edge contains at most k crossings.
A graph G is k-planar if it has a k-plane drawing in the plane.
*Corresponding author. Supported by NKFIH grant K-131529 and ERC Advanced Grant “GeoScape”
No. 882971.
†Supported by NKFIH grant K-131529 and ERC Advanced Grant “GeoScape” No. 882971.
E-mail addresses: barat@mik.uni-pannon.hu (János Barát), toth.geza@renyi.mta.hu (Géza Tóth)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
68 Ars Math. Contemp. 24 (2024) #P1.05 / 67–74
There are several versions of these concepts, see e.g. [4]. The most studied one is when
we consider only simple drawings. A graph G is simple k-planar if it has a simple k-plane
drawing in the plane.
A simple k-plane drawing is saturated if no edge can be added so that the obtained
drawing is also simple k-plane. The 0-planar graphs are the well-known planar graphs. A
plane graph of n vertices has at most 3n − 6 edges. If it has exactly 3n − 6 edges, then it
is a triangulation of the plane. If it has fewer edges, then we can add some edges so that
it becomes a triangulation with 3n− 6 edges. That is, saturated plane graphs have 3n− 6
edges.
Pach and Tóth [6] proved the maximum number of edges of an n-vertex (simple)
1-planar graph is 4n − 8. Brandenburg et al. [3] noticed that saturated simple 1-plane
graphs can have much fewer edges, namely 4517n + O(1) ≈ 2.647n. Barát and Tóth [2]
proved that a saturated simple 1-plane graph has at least 20n9 −O(1) ≈ 2.22n edges.
For any k, n, let sk(n) be the minimum number of edges of a saturated n-vertex simple
k-plane drawing. With these notations, 45n17 + O(1) ≥ s1(n) ≥
20
9 n − O(1). For k > 1,
the best bounds known for sk(n) are shown by Auer et al [1] and by Klute and Parada [5].
Interestingly for k ≥ 5 the bounds are very close.
In this note, we concentrate on 2-planar graphs on n vertices. Pach and Tóth [6] showed
the maximum number of edges of a (simple) 2-planar graph is 5n− 10. Auer et al [1] and
Klute and Parada [5] proved that 4n3 +O(1) ≥ s2(n) ≥
n
2 −O(1). We improve the lower
bound.
Theorem 1.1. For any n > 0, s2(n) ≥ n− 1.
A drawing is l-simple if any two edges have at most l points in common. By definition
a simple drawing is the same as a 1-simple drawing. Let slk(n) be the minimum number of
edges of a saturated n-vertex l-simple k-plane drawing. In [5] it is shown that 4n5 +O(1) ≥
s22(n) ≥ n2 − O(1) and
2n
3 + O(1) ≥ s
3
2(n) ≥ n2 − O(1). We make the following
improvements:
Theorem 1.2. (i) s22(3) = 3, and ⌊3n/4⌋ ≥ s22(n) ≥ ⌊2n/3⌋ for n ̸= 3,
(ii) s32(3) = 3, and s
3
2(n) = ⌊2n/3⌋ for n ̸= 3.
The saturation problem for k-planar graphs has many different settings, we can allow
self-crossings, parallel edges, or we can consider non-extendable abstract graphs. See [4]
for many recent results and a survey.
2 Proofs
Definition 2.1. Let G be a topological graph and u a vertex of degree 1. For short, u is
called a leaf of G. Let v be the only neighbor of u. The pair (u, uv) is called a flag. If there
is no crossing on uv, then (u, uv) is an empty flag.
Definition 2.2. Let G be an l-simple 2-plane topological graph. If an edge contains two
crossings, then its piece between the two crossings is a middle segment. The edges of G
divide the plane into cells. A cell C is special if it is bounded only by middle segments
and isolated vertices. Equivalently, C is special, if there is no vertex on its boundary, apart
from isolated vertices. An edge that bounds a special cell is also special.
J. Barát and G. Tóth: Saturated 2-plane drawings with few edges 69
Let G be a saturated l-simple 2-plane topological graph, where 1 ≤ l ≤ 3. Suppose
a cell C contains an isolated vertex v. Since G is saturated, C must be a special cell and
there is no other isolated vertex in C. Now suppose C is an empty special cell. Each
boundary edge contains two crossings. Therefore, if we put an isolated vertex in C, then
the topological graph remains saturated. So if we want to prove a lower bound on the
number of edges, we can assume without loss of generality that each special cell contains
an isolated vertex.
Claim 2.3. A special edge can bound at most one special cell.
Proof. Suppose uv is a special edge and let pq be its middle segment. If uv bounds more
than one special cell, then there is a special cell on both sides of pq, C1 and C2 say. Let
p be a crossing of the edges uv and xy. There is no crossing on xy between p and one
of the endpoints, x say. Therefore, one of the cells C1 and C2 has x on its boundary, a
contradiction.
Proof of Theorem 1.1. Suppose G is a saturated simple 2-plane topological graph of n
vertices and e edges. We assume that each special cell contains an isolated vertex.
Claim 2.4. All flags are empty in G.
Proof. Let (u, uv) be a flag. Suppose to the contrary there is at least one crossing on uv.
Let p be the crossing on uv closest to u, with edge xy. Since it is a 2-plane drawing, there
is no crossing on xy between p and one of the endpoints, x say. In this case, we can connect
u to x along up and px. Since the drawing was saturated, u and x are adjacent in G, and
x ̸= v, that contradicts to d(u) = 1.
Remove all empty flags from G. Observe the resulting topological graph G′ is also
saturated. If we can add an edge to G′, then we could have added the same edge to G.
Suppose to the contrary that G′ contains a flag (v, vw). Since G′ is saturated, the flag
is empty by Claim 2.4. In G, vertex v had degree at least 2, so v had some other neighbors,
u1, . . . , um say, in clockwise order. The flags (ui, uiv) were all empty. However, u1 can
be connected to w, which is a contradiction. Therefore, there are no flags in G′. On the
other hand, the graph G′ may contain isolated vertices. Let n′ and e′ denote the number of
vertices and edges of G′. Since n − n′ = e − e′, it suffices to show that e′ ≥ n′ − 1. If
there are no isolated vertices in G′, then e′ ≥ n′ is immediate.
We assign weight 1 to each edge. If G′ has no edge, then it has one vertex and we are
done. We discharge the weights to the vertices so that each vertex gets weight at least 1. If
uv is not a special edge, then it gives weight 1/2 to both endpoints u and v. Suppose now
that uv is a special edge. It bounds the special cell C containing the isolated vertex x. If
d(u) = 2, then uv gives weight 1/2 to u, if d(u) ≥ 3, then it gives weight 1/3 to u. We
similarly distribute the weight to vertex v. We give the remaining weight of uv to x.
We show that each vertex gets weight at least 1. This holds immediately for all vertices
of positive degree. We have to show the statement only for isolated vertices. Let x be an
isolated vertex in a special cell C bounded by e1, e2, . . . , em in clockwise direction. Let
ei = uivi such that the oriented curve −−→uivi has C on its right. See Figure 1 for m = 5.
Let pi be the crossing of ei and ei+1. Indices are understood modulo m. In general, it may
happen that some of the points in { ui, vi | i = 1, . . . ,m } coincide. For each vertex ui
or vi of degree at least 3, the corresponding boundary edge of C has a remainder charge
70 Ars Math. Contemp. 24 (2024) #P1.05 / 67–74
v u
u
v
vu
2 4
2 4
1
3
v
1
=u
3
v
u
v u
u
v
vu
2 4
3
1
2 4
1
3
v u
u
1
2
1
=u
3
2
v
v
3
v
u4
u
4
v
5
5
Figure 1: Case 1, d(v1) ≥ 4, Case 2, d(v1) ≥ 3 and Case 2, u1 = u3.
at least 1/6. We have to prove that (with multiplicity) at least 6 of the vertices ui, vi have
degree at least 3. Consider vertex vi.
Case 1: vi = ui+2. The vertex vi = ui+2 can be connected to ui+1 along the segments
vipi and piui+1, that are crossing-free segments of the corresponding edges. Similarly,
vi = ui+2 can be connected to vi+1 along vipi+1 and pi+1vi+1. Since the drawing was
simple and saturated, ui, ui+1, vi+1, vi+2 are all different and they are already connected
to vi = ui+2, so it has degree at least 4.
Case 2: vi ̸= ui+2. The vertex vi can be connected to ui+1 as before, and to ui+2
along vipi, pipi+1 and pi+1ui+2. Since the drawing was saturated, vi is already adjacent
to ui, ui+1, ui+2. Unless ui = ui+2, vertex vi has degree at least 3. Note that ui+1 ̸= ui
and ui+1 ̸= ui+2, since the drawing was 1-simple.
We can argue analogously for ui. We conclude that vi has degree 2 only if ui = ui+2,
and ui has degree 2 only if vi = vi−2.
Recall that m is the number of bounding edges of the special cell C. For m = 3, it is
impossible that ui = ui+2 or vi = vi−2, therefore, for i = 1, 2, 3 all six vertices ui, vi have
degree at least 3.
Let m > 3, and suppose v1 has degree 2, consequently u1 = u3. In this case, we prove
that um, u1, u2, u3, vm, v2 all have degree at least 3.
We show it for u2, the argument is the same for the other vertices. Let γ be the closed
curve formed by the segments u1p1, p1p2 and p2u3. (We have u1 = u3.) Suppose
d(u2) = 2. By the previous observations, vm = v2. However, vm and v2 lie on dif-
ferent sides of γ, therefore they cannot coincide. Therefore, there are always at least six
vertices ui, vi, with multiplicity, which have degree at least 3, so the isolated vertex x gets
weight at least 1. This concludes the proof.
We recall that s32(n) denotes the minimum number of edges of a saturated n-vertex
3-simple 2-plane drawing.
Proof of Theorem 1.2. We start with the upper bounds. Let
f(n) =
{
3 if n = 3
⌊3n/4⌋ otherwise.
First we construct a saturated 2-plane, 2-simple topological graph with n vertices and
f(n) edges, for every n. Let k ≥ 3. A k-propeller is isomorphic to a star with k edges as an
J. Barát and G. Tóth: Saturated 2-plane drawings with few edges 71
Figure 2: A 3-propeller and a 2-propeller.
abstract graph, drawn as in Figure 2. Clearly it is a saturated 2-plane, 2-simple topological
graph with k + 1 vertices, k edges and the unbounded cell is special.
For n = 1, 2, 3, a complete graph of n vertices satisfies the statement. For n ≥ 4,
n ≡ 0 mod 4, consider n/4 disjoint 3-propellers such that each of them is in the unbounded
cell of the others. For n ≥ 4, n ≡ 1, 2, 3 mod 4, replace one of the propellers by an isolated
vertex, a K2, and a 4-propeller, respectively. This implies the upper bound in (i), that is,
s22(n) ≤ f(n).
Now we construct a saturated 2-plane, 3-simple topological graph with n vertices and
⌊2n/3⌋ edges, for every n. A 2-propeller is isomorphic to a path of 2 edges as an abstract
graph, drawn as in Figure 2. Clearly it is a saturated 2-plane, 3-simple topological graph
with 3 vertices, 2 edges and the unbounded cell is special.
For n ≡ 0 mod 3, take n/3 disjoint 2-propellers such that each of them is in the un-
bounded cell of the others. For n ≡ 1, 2 mod 3, add an isolated vertex or an independent
edge. This implies the upper bound in (ii), s32(n) ≤ ⌊2n/3⌋.
We prove by induction on n that s22(n) ≥ ⌊2n/3⌋ and s32(n) ≥ ⌊2n/3⌋. It is trivial for
n ≤ 4. Let n > 4 and assume that s22(m), s32(m) ≥ ⌊2m/3⌋ for every m < n. Let G
be a saturated 2-plane, 2-simple or 3-simple drawing with n vertices and e edges. We may
assume again that every special cell contains an isolated vertex.
Suppose that (u, uv) is an empty flag. We remove u from G. Analogous to the proof
of Theorem 1.1, the obtained topological graph is saturated, it has n− 1 vertices and e− 1
edges. By the induction hypothesis, e−1 ≥ ⌊2(n−1)/3⌋, which implies that e ≥ ⌊2n/3⌋.
Therefore, we assume for the rest of the proof that G does not contain empty flags.
Claim 2.5. If (u, uv) is a flag, then either d(v) ≥ 3 or u and v are included in a 2-propeller.
Proof. Since G does not contain empty flags, there is a crossing on uv. Let p be the
crossing on uv closest to u, with edge xy. There is no crossing on xy between p and one
of the endpoints, x say, and x ̸= u by the assumptions. We can connect u to x along the
segments up and px. Since the drawing was saturated, u and x are adjacent in G. Since u
has degree 1, x = v. This implies d(v) ≥ 2. We exclude parallel edges, so y ̸= u.
Suppose d(v) = 2. There is a crossing on the segment py of vy, otherwise we could
connect u to y along the segments up and py contradicting the degree assumption on u.
Let q be the crossing of vy and ab. There is no crossing on ab between q and one of the
endpoints, a say. If a and u are on the same side of edge vy (that is, the directed edges
−→
ab
72 Ars Math. Contemp. 24 (2024) #P1.05 / 67–74
and −→uv cross the directed edge −→vy from the same side), then we can connect u to a along
the segments up, pq, qa. Therefore a = v, so either d(v) ≥ 3, or b = u, and edges uv and
vy form a 2-propeller. Note that this case is possible only if G is 3-simple.
So we may assume that a is on the other side. If a = v, then d(v) ≥ 3, so we also
assume that a ̸= v. Consider now the edge uv. If there was no crossing on the segment pv
of uv, then we can connect u to a along up, the segment pv of yv, the segment vp of uv,
pq, and qa. Therefore, there is a crossing on the segment pv of uv. Let r be this crossing
of uv with edge cd, and we can assume there is no crossing on the segment cr. (Here, c or
d might coincide with a.) If c and y are on the same side of uv (that is, the directed edges
−→vy and
−→
dc cross the directed edge −→vu from the same side), then we can connect u to c along
up, px, xr, rc, which means that c = v, so d(v) ≥ 3. If c and y are on opposite sides of
uv, then we can connect c to v, so they are already connected. Therefore, c = y. However,
we assumed that −→vy and
−→
dc cross the directed edge −→vu from the opposite sides, so there is
another crossing of uv and vy. If G is 2-simple, this is impossible and we are done. If G is
3-simple, then this crossing can only be r, so c = y and d = x. Now the edges uv and vy
form a 2-propeller.
In a graph G, a connected component with at least two vertices is an essential compo-
nent. If G has only one essential component, then G is essentially connected.
Claim 2.6. We can assume without loss of generality that G is essentially connected.
Proof. Suppose to the contrary G has at least two essential components. We define a partial
order on the essential components of G: Gi ≺ Gj if and only if Gi lies in a bounded cell
of Gj . Let G1 be a minimal element with respect to ≺ and let G2 be the union of all
other essential components. There is a cell C of G, which is bounded by both G1 and
G2. Let C correspond to cell C1 of G1 and cell C2 of G2. By the definition of G1, C1 is
the unbounded cell of G1. Since G is saturated, at least one of C1 or C2 is a special cell,
otherwise G1 and G2 can be connected.
For i = 1, 2, let Hi be the topological graph Gi together with an isolated vertex in every
special cell. Let ni denote the number of vertices and ei the number of edges in Hi. We
notice e = e1 + e2 and n = n1 + n2 − 1 if exactly one of C1 and C2 is a special cell. Also
n = n1+n2−1 if both of them are special cells, since we can add 1 isolated vertex instead
of 2. By the induction hypothesis, we have ei ≥ ⌊2ni/3⌋, so e ≥ ⌊2n1/3⌋+ ⌊2n2/3⌋, and
it is easy to check, that for any n1, n2 ≥ 2, ⌊2n1/3⌋ + ⌊2n2/3⌋ ≥ ⌊2(n1 + n2 − 1)/3⌋.
Therefore, e ≥ ⌊2n1/3⌋ + ⌊2n2/3⌋ ≥ ⌊2(n1 + n2 − 1)/3⌋ = ⌊2n/3⌋. So, if G is not
essentially connected, then we reduce the problem and proceed by induction.
Assume the 3-simple 2-plane drawing G has a flag (u, uv). If d(v) = 1, then G is
isomorphic to K2 and the theorem holds. If d(v) = 2, then G contains a 2-propeller u, v, w
by Claim 2.5. Since G is essentially connected, but there is an isolated vertex in every
special cell, there is an isolated vertex x in the special cell of the 2-propeller. Therefore,
if d(v) = 2 and d(w) = 1, then G is isomorphic to a 2-propeller plus an isolated vertex
and we are done. If d(v) = 2 and d(w) > 1, then remove vertices u, v, x. We removed 3
vertices and 2 edges, so we can use induction.
In the rest of the proof, we assume that every leaf of G is adjacent to a vertex of degree
at least 3, and there is no 2-propeller subgraph in G. We give weight 3/2 to every edge. We
J. Barát and G. Tóth: Saturated 2-plane drawings with few edges 73
discharge the weights to the vertices and show that either every vertex gets weight at least
1, or we can prove the lower bound on the number of edges by induction.
Let uv be an edge. Vertex u gets 1/d(u) weight and v gets 1/d(v) weight from uv.
Every edge has a non-negative remaining charge.
If uv is a special edge, then it is easy to verify that uv bounds only one special cell,
and the special cell contains an isolated vertex by the assumption, just like in the proof of
Claim 2.3. In this case, edge uv gives the remaining charge to this isolated vertex. After
the discharging step, any vertex x with d(x) > 0 gets charge at least 1.
Now let x be an isolated vertex, its special cell being C. We distinguish several cases.
Case 1: The special cell C has two sides. Let u1v1 and u2v2 be the bounding edges.
They cross twice, in p and q say, so there are no further crossings on u1v1 and u2v2.
The four endpoints are either distinct, or two of them u1 and u2 might coincide, if G was
3-simple. Suppose the order of crossings on the edges is uipqvi, for i = 1, 2. If the vertices
u1 and u2 are distinct, then they can be connected along u1p and pu2. Therefore, u1 and
u2 are either adjacent or coincide in G. Similarly, v1 and v2 are also adjacent. Therefore,
all four endpoints have degree at least 2, and both u1v1 and u2v2 give at most charge 1/2
to its endpoints. Their remaining charges are at least 1/2, so x gets at least charge 1.
For the rest of the proof, suppose C is bounded by e1, e2, . . . , em in clockwise direction,
ei = uivi such that −−→uivi has C on its right.
Case 2: m = 3. If none of the bounding edges is a flag, then we are done since each
of those edges give weight at least 1/2 to x. Suppose that u1 is a leaf. We can connect u1
to v2 along segments of the edges u1v1 and u2v2. Since u1 is a leaf and the drawing was
saturated, u1 and v2 are adjacent, consequently v1 = v2. Similarly, we can connect u1 to
v3, so v1 = v2 = v3.
If u2 is not a leaf, then u1v1 and u3v3 both give at least 1/6 to x, and u2v2 gives at least
2/3, so we have charge at least 1 for x. The same applies if u3 is not a leaf. So assume u1,
u2 and u3 are all leaves. If there are no other edges in G, then we can see from the crossing
pattern that G is a 3-propeller and an isolated vertex. That is, n = 5 and e = 3 and the
required inequality holds.
Suppose there are further edges. By Claim 2.6, G is essentially connected. Since u1,
u2, u3 are leaves, v1 is a cut vertex. Let H1 = G \ {x, u1, u2, u3}. The induced subgraph
H1 has n − 4 vertices and e − 3 edges, and it is saturated. Therefore, by the induction
hypothesis, e− 3 ≥ f(n− 4). Notice that f(n) ≤ f(n− 4) + 3, consequently e ≥ f(n).
Case 3: m > 3. Each edge gives at least 1/6 charge to x by Claim 2.5. If an edge is not
a flag, then it gives at least 1/2 charge to x. If there is at least one non-flag bounding edge,
we are done. Suppose that each edge uivi is a flag (that is, d(ui) or d(vi) is 1). We may
also assume that u1 is a leaf. Now, as in the previous case, we can argue that v3 = v2 = v1.
It implies u2 and u3 are leaves, and by the same argument, v5 = v4 = v3 = v2 = v1.
We can continue and finally we obtain that all vi are identical and all ui are leaves. So
the vertices ui, vi 1 ≤ i ≤ m form a star, and they have the same crossing pattern as an
m-propeller. Therefore, ui, vi 1 ≤ i ≤ m span an m-propeller. We can finish this case
exactly as Case 2. If there are no further edges in G, then the graph is an m-propeller and
an isolated vertex. That is, n = m + 2 and e = m and the inequality holds. If there are
further edges, then v1 is a cut vertex, and we can apply induction. This concludes the proof
of Theorem 1.2.
74 Ars Math. Contemp. 24 (2024) #P1.05 / 67–74
Remarks
• We have established lower and upper bounds on the number of edges of a saturated,
k-simple, 2-plane drawing of a graph. As we mentioned in the introduction, this
problem has many modifications, generalizations. Probably the most natural modifi-
cation is that instead of graphs already drawn, we consider saturated abstract graphs.
A graph G is saturated l-simple k-planar, if it has an l-simple k-planar drawing but
adding any edge, the resulting graph does not have such a drawing. Let tlk(n) be
the minimum number of edges of a saturated l-simple k-planar graph of n vertices.
By definition, slk(n) ≤ tlk(n). We are not aware of any case when the best lower
bound on tlk(n) is better than for s
l
k(n). On the other hand, it seems to be much
harder to establish an upper bound construction for tlk(n) than for s
l
k(n). In fact,
we know nontrivial upper bounds only in two cases, t11(n) ≤ 2.64n + O(1) [3] and
t12(n) ≤ 2.63n+O(1) [1], the latter without a full proof.
It is known that a k-planar graph has at most c
√
kn edges [6], so tlk(n) ≤ c
√
kn, for
some c > 0.
Problem 1. Prove that for every c > 0, tlk(n) ≤ c
√
kn if k, l, n are large enough.
• For any n and k, the best known upper and lower bounds on slk decrease or stay the
same as we increase l. This would suggest that slk ≤ s
l−1
k for any n, k, l, or at least
if n is large enough, however, we cannot prove it.
Problem 2. Is it true, that for any k and l, and n large enough, slk ≤ s
l−1
k ?
ORCID iDs
János Barát https://orcid.org/0000-0002-8474-487X
Géza Tóth https://orcid.org/0000-0003-1751-6911
References
[1] C. Auer, F. J. Brandenburg, A. Gleißner and K. Hanauer, On sparse maximal 2-planar graphs,
in: W. Didimo and M. Patrignani (eds.), Graph Drawing, Springer Berlin Heidelberg, Berlin,
Heidelberg, 2013 pp. 555–556.
[2] J. Barát and G. Tóth, Improvements on the density of maximal 1-planar graphs, J. Graph Theory
88 (2018), 101–109, doi:10.1002/jgt.22187, https://doi.org/10.1002/jgt.22187.
[3] F. J. Brandenburg, D. Eppstein, A. Gleißner, M. T. Goodrich, K. Hanauer and J. Reislhu-
ber, On the density of maximal 1-planar graphs, in: Graph drawing. 20th international sym-
posium, GD 2012, Redmond, WA, USA, Springer, Berlin, pp. 327–338, 2013, doi:10.1007/
978-3-642-36763-2 29, https://doi.org/10.1007/978-3-642-36763-2_29.
[4] S. Chaplick, F. Klute, I. Parada, J. Rollin and T. Ueckerdt, Edge-minimum saturated k-planar
drawings, 2021, arXiv:2012.08631 [math.CO].
[5] F. Klute and I. Parada, Saturated k-plane drawings with few edges, 2021,
arXiv:2012.02281 [math.CO].
[6] J. Pach and G. Tóth, Graphs drawn with few crossings per edge, Combinatorica 17 (1997),
427–439, doi:10.1007/bf01215922, https://doi.org/10.1007/bf01215922.
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.06 / 75–91
https://doi.org/10.26493/1855-3974.2503.f17
(Also available at http://amc-journal.eu)
Classification of thin regular map
representations of hypermaps*
Antonio Breda d’Azevedo , Domenico A. Catalano †
Department of Mathematics, University of Aveiro, Aveiro, Portugal
Received 12 December 2020, accepted 13 January 2023, published online 22 August 2023
Abstract
There are two well known maps representations of hypermaps, namely the Walsh and
the Vince map representations, being dual of each other. They correspond to normal sub-
groups of index two of a free product Γ = (C2 × C2) ∗ C2 which decompose as “elemen-
tary” free product C2 ∗C2 ∗C2. However Γ has three normal subgroups that decompose as
“elementary” free product C2 ∗ C2 ∗ C2, the third of these sbgroups giving the less known
petrie-path map representation. By relaxing the “elementary” free product condition to free
product of rank 3, and under the extra condition “words of smaller length” on the genera-
tors, we prove that the number of map representations of hypermaps increases to 15 (up to
a restrictedly dual), all of which described in this paper.
Keywords: Map representation, hypermaps, maps, regularity, restricted regularity, orientably regular.
Math. Subj. Class. (2020): 05C10, 05C25, 05C65, 05E18, 20F65
1 Introduction
Using maps to describe hypermaps is not new. The well-known Walsh [8] bipartite map
representation uses a bipartite map M to describe a hypermap H by interpreting the two
monochromatic vertices of the map as hypervertices and hyperedges (respectively), and
the faces of M as the hyperfaces of H. The Vince 2-face bipartite map [7], a dual of a
bipartite map, also describes a hypermap by assigning the two monochromatic faces to
hyperedges and hyperfaces respectively, and vertices to hypervertices. These are two, out
*The authors thank the support given by the Portuguese funds through the CIDMA - Center for Research and
Development in Mathematics and Applications and the Portuguese Foundation for Science and Technology (FCT-
Fundação para a Ciência e a Tecnologia) within project UIDB/04106/2020.
†Corresponding author.
E-mail addresses: breda@ua.pt (Antonio Breda d’Azevedo), domenico@ua.pt (Domenico A. Catalano)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
76 Ars Math. Contemp. 24 (2024) #P1.06 / 75–91
of three, Θ-marked map representations realised by an index 2 normal subgroups Θ of the
free product
Γ = ∆(∞, 2,∞) = ⟨R0, R1, R2 | R20, R21, R22, (R0R2)2⟩ = C∞ ∗ (C2 × C2) ,
which are isomorphic to ∆ = ∆(∞,∞,∞) = ⟨S0, S1, S2 | S20 , S21 , S22⟩ (see [1] and
section 3). They are namely, Γ2.4 = ⟨RR01 , R1, R2⟩ and Γ2.1 = ⟨R0, R1, R
R2
1 ⟩. The third
subgroup of Γ of index 2 isomorphic to ∆ is Γ2.5 = ⟨RR01 , R1, R0R2⟩ (see Subsection 4.2).
This induces the third less known representation, succinctly described in [2], given by Γ2.5-
marked maps. In this representation Petrie-path-bipartite maps represent hypermaps by
assigning the two monochromatic Petrie polygons (closed zig-zag paths turning alternately
left and right) to hypervertices and hyperedges, and faces to hyperfaces.
More generally, a regular representation of hypermaps by maps is given by an epimor-
phism ρ from a finite index normal subgroup Θ of Γ to ∆.
This paper is inspired by the work of Lynne James on map representation of topological
categories (see [5]) and is organised as follow: In Section 2 we give an introduction to the
theory of hypermaps and maps focusing on restrictedly marked hypermaps and maps, a
theory developed in [1]. In particular, we focus on Θ-marked maps for normal subgroups
Θ of finite index in Γ. Section 3 is devoted to define the notion of clean and thin Θ-marked
representation of a hypermap by a map. As we will focus on Θ-marked representations for
rank 3 normal subgroups Θ of Γ, in Section 4 we derive a rank formula and classify the
rank 3 normal subgroups of Γ. The rank formula is derived using presentations for NEC
groups (see [3]). Last section is devoted to thin representations (given in Table 2) and its
geometric description by means of an example.
In what follows by “representation” we always mean “regular representation”. Note
that we use right notation, that is, we denote by xf the image of x by the function f .
2 Preliminaries
Hypermaps are 4-tuples H = (F ; r0, r1, r2) where F is a finite set and r0, r1, r2 are invo-
lutory permutations of F (r2i = 1) generating a transitive group on F . The elements of F
are called flags and the transitive group Mon(H) = ⟨r0, r1, r2⟩ is the monodromy group of
H. The orbits of the action of the subgroups of Mon(H) generated by {r0, r1, r2} \ {ri}
for i = 0, 1, 2 are respectively the hypervertices, hyperedges and hyperfaces of the hyper-
map H, called respectively 0-cells, 1-cells and 2-cells of H. The valency of an i−cell is
the length of the orbit of one of its flags by rjrk where {i, j, k} = {0, 1, 2}. If, for some
positive integers k, ℓ, m all hypervertices have valency k, all hyperedges have valency ℓ
and all hyperfaces have valency m, then we say that H is a uniform hypermap (of type
(k, ℓ,m)). In this case, (k, ℓ,m) = (|r1r2|, |r2r0|, |r0r1|), where |g| denotes the order
of g. If r0, r1 and r2 have no fixed point then we say that H has no boundary. Thus, a
uniform hypermap H = (F ; r0, r1, r2) without boundary has V = |F |2|r1r2| hypervertices,
E = |F |2|r2r0| hyperedges and F =
|F |
2|r0r1| hyperfaces.
A morphism or covering from the hypermap H1 = (E; r0, r1, r2) to the hypermap
H2 = (F ; s0, s1, s2) is a function ϕ : E → F satisfying
xriϕ = xϕsi,
for any x ∈ E and any i ∈ {0, 1, 2}. We say that the hypermap H1 covers the hypermap H2
if there is a covering from H1 to H2. It is straightforward to see that any covering is onto
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 77
and uniquely determined by the image of a flag. Injective coverings are therefore called
isomorphisms. An automorphism of H is an isomorphism from H to itself. We will denote
by Aut(H) the set of automorphisms of H, which is obviously a group under composition.
Topologically, a hypermap H can be seen as a triangulation of a compact surface S
with vertices labelled 0, 1 and 2 such that each triangle (a flag of H) has labels 0, 1 and 2
assigned to its vertices; the vertices labelled 0, 1 and 2 are respectively the hypervertices,
hyperedges and hyperfaces. For each x ∈ F the two triangles x and xri share the common
edge e opposite to the vertices labelled i if x ̸= xri; if x = xri, then the edge e is on
the boundary of S and so S is a bordered surface. This triangulation is a topological map
representation of hypermaps whose dual is the James topological map representation of
hypermaps [4]; here the faces are labelled 0 (grey faces), 1 (dotted faces) and 2 (white faces)
(see Figure 3). The hypermap H has (no) boundary if and only if S has (no) boundary. The
characteristic of H is the Euler characteristic of S. In particular, if H = (F ; r0, r1, r2) is a
uniform hypermap without boundary, then the Euler characteristic of H is
χ(H) = |F |
2
(
1
|r1r2|
+
1
|r2r0|
+
1
|r0r1|
− 1
)
.
Alternatively, a hypermap is a cellular embedding of a hypergraph in a compact connected
surface.
The monodromy group Mon(H) of a hypermap H is a quotient of the triangle group
∆. Hence we have an epimorphism π : ∆ → Mon(H) and an action
F ×∆ → F, (x, d) 7→ x(dπ)
of ∆ on the set F of flags of H. The stabiliser H of a flag under this action is a sub-
group of ∆ called a hypermap subgroup of H. As the action of ∆ is transitive, hyper-
map subgroups of H are conjugate. The hypermap H is then isomorphic to the hypermap
(∆/H;H∆R0, H∆R1, H∆R2), where ∆/H denotes the set of right cosets of a hypermap
subgroup H of H in ∆, H∆ is the normal core of H in ∆ and (Hd)H∆Ri = HdRi for
any d ∈ ∆ and any i ∈ {0, 1, 2} (see, for instance [1]).
Let Θ be a normal subgroup of finite index n in ∆ and let H be a hypermap with
hypermap subgroup H . Then Θ acts (as a subgroup of ∆) on the set F = ∆/H of flags
of H partitioning it into at most n orbits, called Θ-orbits; in fact, suppose that H is not a
subgroup of Θ and let b ∈ H \ Θ. Then Hb = H and bΘ ̸= Θ. Therefore the Θ-orbit
{Hbt : t ∈ Θ} is equal to the Θ-orbit {Ht : t ∈ Θ}, forcing the number of Θ-orbits being
at most n. The number of Θ-orbits is n if and only if H < Θ; in this case we say that
H is Θ-conservative. A Θ-conservative hypermap H is Θ-regular if the group AutΘ(H)
of automorphisms preserving Θ-orbits acts transitively on each Θ-orbit, or equivalently, if
H is normal in Θ. However, if H is normal in ∆, then H is a regular hypermap, that is,
∆-regular. We shall say that a hypermap H is restrictedly-marked if it is Θ-conservative for
some normal subgroup Θ of finite index in ∆. Ought to emphasise that not every hypermap
is restrictedly-marked (see [1] for examples).
A hypermap (F ; r0, r1, r2) satisfying (r0r2)2 = 1 is called a map. The hypervertices,
hyperedges and hyperfaces of a map are called vertices, edges and faces, since topologically
a map is a cellular embedding of a graph on a compact surface. The monodromy group of
a map M is then a quotient of the “right” triangle group Γ. This group acts on the set
of flags of M via the canonical projection π : Γ → Mon(M) sending Ri to ri. The
78 Ars Math. Contemp. 24 (2024) #P1.06 / 75–91
stabiliser of a flag under this action will be called a map subgroup of M. Keeping the
same notation as already used for hypermaps, we have that a map M is then isomorphic
to the map (Γ/M ;MΓR0,MΓR1,MΓR2), where M is a map subgroup of M. The theory
of restrictedly-marked maps unfolds in the same way as the theory of restrictedly-marked
hypermaps by taking finite index normal subgroups Θ of Γ instead of ∆. The group Γ is a
free product of C2 = ⟨R1⟩ with D2 = ⟨R0, R2⟩ and by the Kurosh’s Subgroup Theorem,
any normal subgroup Θ of Γ freely decomposes uniquely (up to a permutation of factors)
in a (indecomposable) free product (see [6] page 243 and 245)
C2 ∗ · · · ∗ C2 ∗D2 ∗ · · · ∗D2 ∗ C∞ ∗ · · · ∗ C∞ =
⟨A1⟩ ∗ · · · ∗ ⟨As⟩ ∗ ⟨B1, C1⟩ ∗ ⟨Bt, Ct⟩ ∗ ⟨Z1⟩ ∗ · · · ∗ ⟨Zu⟩
for a certain numbers s, t and u of factors ⟨Ai⟩ = C2, ⟨Bj , Cj⟩ = D2 and ⟨Zu⟩ = C∞
respectively, whereas s, t or u may be zero. Let m = s+ 2t+ u = rank(Θ) and let
{A1, . . . , As, B1, . . . , Bt, C1, . . . , Ct, Z1, . . . , Zu} = {X1, . . . , Xm}.
Then a Θ-conservative map M with map subgroup M can be represented by the Θ-marked
map
Q = (Ω;x1, . . . , xm),
where Ω = Θ/M is the set of right cosets of M in Θ and x1, . . . , xm are permutations of
Ω generating a group G acting transitively on Ω such that the function
X1 7→ x1 , . . . , Xm 7→ xm
extends to an epimorphism from Θ to G.
Any Θ-regular map M covers the regular map
TΘ = (Γ/Θ;ΘR0,ΘR1,ΘR2),
called the Θ-trivial map. As TΘ is a regular map, we have that
• any two vertices of TΘ have same valency, say k,
• any two edges of TΘ have same valency, say l ∈ {1, 2},
• any two faces of TΘ have same valency, say m.
The triple (k, l,m) is called the type of the regular map TΘ. As M is Θ-regular and covers
TΘ, we also have that:
• the vertices of M covering a vertex v of TΘ also have same valency, say kv (which
is a multiple of k),
• the faces of M covering a face f of TΘ also have same valency, say mf (which is a
multiple of m).
Denoting by V , E and F the sets of vertices, edges and faces of TΘ and assuming that M
has no boundary, then this together with Euler formula gives that the characteristic of M is
χ(M) = |Θ : M |
2
∑
v∈V
µv
k
kv
+
∑
e∈E
µe
l
2
+
∑
f∈F
µf
m
mf
− |Γ : Θ|
 , (2.1)
where µv = 1 or 2 according as the vertex v is on the boundary or not and similarly for µe
and µf . For details we refer the reader to [1].
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 79
3 Thin map representations of hypermaps
Let Θ be a finite index normal subgroup of Γ of rank 3 and let {X1, X2, X3} be a set of
generators of Θ. The pair R = (Θ, {X1, X2, X3}) will be called a Θ-marked representa-
tion (of hypermaps by maps) if the function
X1 7→ S0 , X2 7→ S1 , X3 7→ S2
extends to an epimorphism ρ from Θ onto ∆. We call ρ the canonical epimorphism of the
representation R. Two representations (Θ1, {X1, X2, X3}) and (Θ2, {Y1, Y2, Y3}) are to
be considered equal if Θ1 = Θ2 = Θ and their canonical epimorphisms ρ1, ρ2 : Θ → ∆
are such that ρ1 = ιρ2 for some inner automorphism ι of Θ. For example, since S0, S1, S2
are involutions, inverting one or more generators of R give the same representation.
Given a hypermap H with hypermap subgroup H , setting Ω = {(Hρ−1)t : t ∈ Θ} and
xi : Ω → Ω, (Hρ−1)t 7→ (Hρ−1)tXi, i = 1, 2, 3
we get a Θ-marked map (Ω;x1, x2, x3) called a Θ-marked map representation of H.
Remark 3.1. In fact, denoting by N the normal core of Hρ−1 in Θ, the group G =
⟨x1, x2, x3⟩ is isomorphic to Θ/N by an isomorphism φ mapping xi to NXi for any
i ∈ {1, 2, 3}. Hence πφ−1, where π : Θ → Θ/N is the canonical epimorphism, is an
epimorphism from Θ to G extending the function X1 7→ x1, X2 7→ x2, X3 7→ x2. Remark
also that ρ induces a bijection ρ̃ from Ω to {Hd : d ∈ ∆} which sends (Hρ−1)t to H(tρ)
and satisfies xi ρ̃ = ρ̃ ri−1, where ri−1 maps Hd to HdRi−1 for any i ∈ {1, 2, 3}. Thus,
we say that ρ̃ is an isomorphism from the Θ-marked map representation (Ω;x1, x2, x3) of
H to H.
A (Θ-marked) representation R = (Θ, (X1, X2, X3)) will be called clean if Θ is the
free product of the cyclic groups ⟨X1⟩, ⟨X2⟩, ⟨X3⟩, in which case we write
Θ = ⟨X1⟩ ∗ ⟨X2⟩ ∗ ⟨X3⟩ .
A clean representation is called thin if the sum of the lengths of its generators (as words in
the free group over {R0, R1, R2}) is minimal.
The number of rank 3 normal subgroups Θ of Γ is finite, but there are infinitely many
clean representations given by all possible sets {X1, X2, X3} such that Θ = ⟨X1⟩ ∗ ⟨X2⟩ ∗
⟨X3⟩. On the other hand the number of thin representations is finite (see Sections 4 and 5).
4 The rank 3 normal subgroups of Γ
4.1 Rank computation (see also [3])
In order to compute the rank of a normal subgroup Θ of finite index in Γ, we remark that Γ
acts as a group of isometries on the hyperbolic plane H, regarding its generators R0, R1,
R2 as the reflections on the geodesics given in Figure 1 in the Poincaré disk model.
80 Ars Math. Contemp. 24 (2024) #P1.06 / 75–91
R2
R0
R1
Figure 1: The generators of Γ as hyperbolic reflections.
The action of Θ on H gives rise to a quotient orbifold H/Θ which is a punctured
surface (with or without boundary) punctured at the vertices and at the face centers of the
regular map M = (Γ/Θ;ΘR0,ΘR1,ΘR2) with underlying surface S. If ΘR0 = ΘR2,
then the covering H → H/Θ is also branched at the edge centers of M. The group Θ,
being the fundamental group of H/Θ, has a presentation P with p + 2 − χ generators
X1, . . . , Xp, Y1, . . . , Y2−χ, where p is the total number of punctures and branching points
of H/Θ and χ is the characteristic of S. The presentation P has a relator S = X1 · · ·Xp ·
k∏
i=1
Yi · W (Yk+1, . . . , Y2−χ), where
⟨Y1, . . . , Y2−χ | W (Yk+1, . . . , Y2−χ)⟩
is a presentation of the fundamental group of the surface S with k boundary components
(setting
k∏
i=1
Yi = 1 if k = 0), and eventually e relators X21 , . . . , X
2
e if ΘR0 = ΘR2, where
e is the number of edges of M.
Hence rank(Θ) = p+ 2− χ− 1 = p+ 1− χ. More precisely:
• If S has no boundary (k = 0) and is non-orientable, then 2 − χ is the genus g of S,
W (Y1, . . . , Y2−χ) = W (Y1, . . . , Yg) =
g∏
i=1
Y 2i and therefore
S = X1 · · ·Xp ·
g∏
i=1
Y 2i .
• If S has no boundary and is orientable, then 2 − χ is even and the genus g of S is
2−χ
2 . Replacing (Y1, . . . , Y2−χ) by (A1, B1, . . . , Ag, Bg) we have
S = X1 · · ·Xp ·
g∏
i=1
[Ai, Bi] .
In the particular case of χ = 2 (sphere) the word W (Y1, . . . , Y2−χ) is empty and
therefore
S = X1 · · ·Xp.
• If S has boundary, then {R0, R1, R2}∩Θ ̸= ∅. Thus, any triangle of the triangulation
of S given by the flags of M has at least an edge on the boundary, since Θ is normal
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 81
in Γ. This shows that S is a closed disk, that is, a bordered surface on a sphere
with only one boundary component (k = 1). Hence χ = 2 − k = 1 and therefore,
setting Y = Y1 we have that
k∏
i=1
Y1 = Y , W (Yk+1, . . . , Y2−χ) is the empty word
and ⟨Y ⟩ ∼= C∞ is the fundamental group of S. Hence
S = X1 · · ·Xp · Y .
In particular, rank(Θ) = p in this case.
The next proposition relates the rank of Θ with its index n in Γ for n > 4. Relating rank
with all indices will give a clumsy formula which does not give more information about the
index bound for fixed rank.
Proposition 4.1. If Θ is a normal subgroup of finite index n > 4 in Γ, then n is even and
rank(Θ) =

1 + n if H/Θ has boundary and branching points;
1 + n2 if H/Θ has boundary and no branching point;
or H/Θ has no boundary but has branching points;
1 + n4 if H/Θ has no boundary and no branching point.
( in this case n is a multiple of 4).
Proof. Using the above notations and remarks we have the following:
If S has boundary, then ΘR1 ̸= Θ since |Γ/Θ| = n > 4. Hence Γ/Θ = ⟨ΘR1,ΘRj⟩ for
some j ∈ {0, 2}, that is, Γ/Θ is dihedral of even order n. The total number of vertices and
faces of the map M = (Γ/Θ;ΘR0,ΘR1,ΘR2) is then 1 + n2 . This gives
rank(Θ) = p =
{
1 + n2 if H/Θ has no branching points,
1 + n if H/Θ has branching points,
since in the case when H/Θ has branching points, M has n2 edges.
If S has no boundary, then n is a multiple of 4 and from Euler formula we have that
χ =
{
p− n4 if H/Θ has no branching points,
p− n2 if H/Θ has branching points.
Therefore
rank(Θ) = p+ 1− χ =
{
1 + n4 if H/Θ has no branching points,
1 + n2 if H/Θ has branching points.
Corollary 4.2. If rank(Θ) = 3, then the index n is 2, 4 or 8 and Γ/Θ is isomorphic to C2,
C2 × C2, C2 × C2 × C2 or D4.
Proof. Proposition 4.1 guaranties that n ∈ {2, 4, 8} if rank(Θ) = 3. The groups of order
2, 4 and 8 not listed in the statement are not generated by involutions.
82 Ars Math. Contemp. 24 (2024) #P1.06 / 75–91
4.2 The rank 3 normal subgroups of Γ
(1) n = 2: As mentioned in the introduction, there are seven epimorphisms from Γ to C2
having kernels Γ2.1, . . . ,Γ2.7. Only three of them have rank 3, as it is easily checked by
applying the Reidemeister-Schreier rewriting process. In this way, one gets that the rank 3
kernels Θ = ⟨X⟩ ∗ ⟨Y ⟩ ∗ ⟨Z⟩ are
Γ2.1 = ⟨R0⟩ ∗ ⟨R1⟩ ∗ ⟨RR21 ⟩ , Γ2.4 = ⟨R1⟩ ∗ ⟨R2⟩ ∗ ⟨R
R0
1 ⟩ and
Γ2.5 = ⟨R1⟩ ∗ ⟨R0R2⟩ ∗ ⟨RR01 ⟩ .
These three groups are isomorphic to the free product C2∗C2∗C2 and therefore isomorphic
to ∆. The remaining four epimorphisms have kernels
Γ2.2 = ⟨R0, R2⟩ ∗ ⟨RR10 , R
R1
2 ⟩ , Γ2.3 = ⟨R0⟩ ∗ ⟨R1R2⟩ , Γ2.6 = ⟨R2⟩ ∗ ⟨R0R1⟩ and
Γ2.7 = ⟨R0R2⟩ ∗ ⟨R1R2⟩ .
The group Γ2.2 has rank 4 and is isomorphic to the free product D2 ∗D2, while the other
three groups Γ2.3, Γ2.6 and Γ2.7 have rank 2 and are all isomorphic to C2 ∗ C∞.
(2) n = 4: Up to an automorphism of G = C2×C2 there are seven epimorphisms from
Γ to G with kernels Γ4.1, . . . ,Γ4.7. One can check that three of them have rank 3, namely
Γ4.1 = ⟨R0⟩ ∗ ⟨RR10 ⟩ ∗ ⟨(R1R2)2⟩ , Γ4.4 = ⟨R2⟩ ∗ ⟨R
R1
2 ⟩ ∗ ⟨(R0R1)2⟩ and
Γ4.5 = ⟨R0R2⟩ ∗ ⟨(R0R1)2⟩ ∗ ⟨(R0R2)R1⟩ .
These groups are all isomorphic to the free product C2∗C2∗C∞ so that ∆ is an epimorphic
image of each of them.
Remark 4.3. Γ4.1 = Γ2.3 ∩ Γ2.2 = Γ2.3 ∩ Γ2.1 = Γ2.2 ∩ Γ2.1 = Γ2.3 ∩ Γ2.2 ∩ Γ2.1 ,
Γ4.4 = Γ2.4 ∩ Γ2.2 = Γ2.4 ∩ Γ2.6 = Γ2.2 ∩ Γ2.6 = Γ2.4 ∩ Γ2.2 ∩ Γ2.6 ,
Γ4.5 = Γ2.7 ∩ Γ2.5 = Γ2.7 ∩ Γ2.2 = Γ2.5 ∩ Γ2.2 = Γ2.7 ∩ Γ2.5 ∩ Γ2.2 .
(3) n = 8, G = D4: Up to an automorphism of G there are six epimorphism from Γ to
G with kernels Γ8.1, . . . ,Γ8.6. Three of them have rank 3 and are all free groups, namely
Γ8.4 = ⟨R0R1R2R1⟩ ∗ ⟨R1R0R1R2⟩ ∗ ⟨(R0R1)2R0R2⟩ ,
Γ8.5 = ⟨(R1R2)2⟩ ∗ ⟨R2(R1R0)2⟩ ∗ ⟨R2(R0R1)2⟩ and
Γ8.6 = ⟨(R0R1)2⟩ ∗ ⟨R0(R1R2)2⟩ ∗ ⟨R0(R2R1)2⟩ .
Remark 4.4. Γ4.5 is the unique normal subgroup of index 4 containing Γ8.4, while Γ4.4
is the unique normal subgroup of index 4 containing Γ8.5 and Γ4.1 is the unique normal
subgroup of index 4 containing Γ8.6 .
(4) n = 8, G = C2 × C2 × C2: Up to an automorphism of G there is only one
epimorphism from Γ to G with kernel isomorphic to the rank 3 free group C∞ ∗C∞ ∗C∞,
namely
Γ8.7 = ⟨(R0R1)2⟩ ∗ ⟨(R1R2)2⟩ ∗ ⟨R0(R1R2)2R0⟩ .
Remark 4.5. Γ8.7 = Γ4.i ∩ Γ4.j for any distinct i, j ∈ {1, . . . , 7}.
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 83
The following table gives a overall description of Θ and the Θ-trivial map for each
normal subgroup Θ of Γ of index 2, 4, 6 and 8.
Θ index rank Free-Product dec. Type of T
Θ
surface χ fig
Γ2.1 2 3 C2 ∗ C2 ∗ C2 (2,2,1) border 1
Γ2.2 2 4 D2 ∗D2 (2,1,2) border 1
Γ2.3 2 2 C2 ∗ C∞ (1,2,2) border 1
Γ2.4 2 3 C2 ∗ C2 ∗ C2 (1,2,2) border 1
Γ2.5 2 3 C2 ∗ C2 ∗ C2 (2,1,2) border 1
Γ2.6 2 2 C2 ∗ C∞ (2,2,1) border 1
Γ2.7 2 2 C2 ∗ C∞ (1,1,1) orient. 2
Γ4.1 4 3 C2 ∗ C2 ∗ C∞ (2,2,2) border 1
Γ4.2 4 4 C2 ∗ C2 ∗ C2 ∗ C2 (2,2,2) border 1
Γ4.3 4 2 C∞ ∗ C∞ (2,2,1) orient. 2
Γ4.4 4 3 C2 ∗ C2 ∗ C∞ (2,2,2) border 1
Γ4.5 4 3 C2 ∗ C2 ∗ C∞ (2,1,2) orient. 2
Γ4.6 4 2 C∞ ∗ C∞ (1,2,2) orient. 2
Γ4.7 4 2 C∞ ∗ C∞ (2,2,2) nonori. 1
Γ6.1 6 4 C2 ∗ C2 ∗ C2 ∗ C∞ (3,2,2) border 1
Γ6.2 6 4 C2 ∗ C2 ∗ C2 ∗ C∞ (2,2,3) border 1
Γ6.3 6 4 C2 ∗ C2 ∗ C2 ∗ C∞ (3,1,3) orient. 2
Γ8.1 8 5 C2 ∗ C2 ∗ C2 ∗ C2 ∗ C∞ (4,2,2) border 1
Γ8.2 8 5 C2 ∗ C2 ∗ C2 ∗ C2 ∗ C∞ (2,2,4) border 1
Γ8.3 8 5 C2 ∗ C2 ∗ C2 ∗ C∞ ∗ C∞ (4,1,4) orient. 2
Γ8.4 8 3 C∞ ∗ C∞ ∗ C∞ (4,2,4) orient. 0
Γ8.5 8 3 C∞ ∗ C∞ ∗ C∞ (2,2,4) nonori. 1
Γ8.6 8 3 C∞ ∗ C∞ ∗ C∞ (4,2,2) nonori. 1
84 Ars Math. Contemp. 24 (2024) #P1.06 / 75–91
Γ8.7 8 3 C∞ ∗ C∞ ∗ C∞ (2,2,2) orient. 2
Table 1: Normal subgroups of indices 2, 4, 6 and 8 in Γ = C2 ∗D2.
5 Description of the thin map representations
In the previous section we computed all rank 3 normal subgroups Θ = Γi.j together with a
set of generators {X1, X2, X3} such that Γi.j = ⟨X1⟩∗⟨X2⟩∗⟨X3⟩ is a thin representation
Ri.j. Since some Γi.j gives rise to more than one thin representation, we label the corre-
sponding representations by Ri.ja, Ri.jb, etc. Note that the generators of a thin represen-
tation can be read out as fundamental group generators (written as words on {R0, R1, R2})
from the respective trivial map (Section 4). The classification is done up to a restrictedly
dual, that is, the generators of a Θ-marked representation are computed up to the usual map
dual if its restriction to Θ is an automorphism of Θ (see also Remark below). The following
table gives all the thin Θ-marked representations. Generators of Γi.j which are involutions
will be denoted by A, B, C and those which are not will be denoted by X , Y , Z.
Remark 5.1. The assignments
R0 7→ R2 , R1 7→ R1 , R2 7→ R0 and
R0 7→ R0R2 , R1 7→ R1 , R2 7→ R2
extend to automorphisms of Γ and give rise to the map dualities D (the usual map duality)
and P (the Petrie duality). Together they generate the outer automorphism group Out(Γ) =
⟨D,P ⟩ ∼= S3. The following diagram graphically pictures the action of Out(Γ) on the set
of rank 3 normal subgroups of Γ, where lines and dash lines represent the action of D and
P , respectively. Note that D, or P , fixes some Θ and therefore for those Θ’s it is a Θ-
¡2.1 ¡2.5¡2.4
¡4.1 ¡4.5¡4.4
¡8.7¡8.6¡8.4 ¡8.5
Figure 2: The actions of D and P on the Θ’s.
restrictedly duality. The Petrie duality is not a thin-preserving duality except in the case of
Γ8.7; here R8.7a and R8.7b are Petrie duals of each other. The duality D fixes Γ2.5, Γ4.5,
Γ8.4 and Γ8.7. These give rise to the restrictedly-dual representations given in the following
Table, but not listed in Table 2.
To illustrate each thin representation, we exhibit the Θ-marked map representation of
the toroidal regular hypermap H pictured in Figure 3 using the James hypermap represen-
tation [4], where hypervertices, hyperedges and hyperfaces of H are represented by simply
connected regions colored grey, dotted and white, respectively, and flags are the numbered
points. We note that Lynne James hypermap representation is actually the Γ6.1-marked
map representation sending (R2R1)3 to 1. Here Γ6.1 is the normal subgroup of index 6 of
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 85
Γ isomorphic to C2 ∗ C2 ∗ C2 ∗ C∞ generated by R0, RR10 , R
R1R2
0 and (R2R1)
3 (given
in Table 1). This representation is not listed in Table 2 because this restrictedly marked
representation is not thin (it is not even clean).
From Figure 3, H = (F ; r0, r1, r2) with F = {1, . . . , 6} and, up to permutation (coloring),
r0 = (1, 4)(2, 5)(3, 6) , r1 = (1, 2)(3, 4)(5, 6) , r2 = (1, 6)(2, 3)(4, 5) .
The hypermap H has one hypervertex, one hyperedge and one hyperface all of valency 3.
The monodromy group of H is G = ⟨r0, r1, r2⟩ ∼= S3. The Euler characteristic of a map
representation of H is given by (2.1) taking into account the Θ-trivial map given in Table 2
and using the isomorphism ρ̃ given in 3.1.
# Rep. Generators Epim. Θ-slice
1 R2.1 A=R0 B=R1 C=R2R1R2
A→r0
B→r1
C→r2
a
b
c
2 R2.4 A=R0R1R0 B=R1 C=R2
A→r0
B→r1
C→r2 a
b c
3 R2.5 A=R0R1R0 B=R1 C=R0R2
A→r0
B→r1
C→r2
a
b
c
4 R4.1 A=R0 B=R1R0R1 X=R1R2R1R2
A→r0
B→r1
X→r2 a
b c
d
5 R4.4 A=R2 B=R1R2R1 Z=R0R1R0R1
A→r2
B→r1
Z→r0
a
b
c
d
6 R4.5a A=R0R2 B=R1R0R2R1 Z=R0R1R0R1
A→r0
B→r2
Z→r1 a
b c
d
7 R4.5b A=R0R2 B=R1R0R2R1 X=R0R1R2R1
A→r0
B→r2
X→r1 a
b c
d
8 R8.4a X=R0R1R2R1 Y=R1R0R1R2 Z=R0R1R0R1R0R2
X→r0
Y→r1
Z→r2 x
a
b c
d
y
9 R8.4b X=R0R1R2R1 Y=R1R0R1R2 Z=R0R2R1R0R2R1
X→r0
Y→r1
Z→r2 x
a
b c
d
y
10 R8.5a X=R1R2R1R2 Y=R0R1R0R1R2 Z=R0R2R1R0R1
X→r0
Y→r1
Z→r2
a
d c
b
11 R8.5b X=R1R2R1R2 Y=R0R1R0R2R1 Z=R0R2R1R0R1
X→r0
Y→r1
Z→r2 a
d
c
b
e
12 R8.6a X=R0R1R2R1R2 Y=R0R1R0R1 Z=R0R2R1R2R1
X→r1
Y→r0
Z→r2 x
a
b c
d
y
13 R8.6b X=R0R2R1R2R1 Y=R0R1R0R1 Z=R1R0R2R1R2
X→r1
Y→r0
Z→r2 x
a
b c
d
y
14 R8.7a X=R1R2R1R2 Y=R0R1R2R1R2R0 Z=R0R1R0R1
X→r1
Y→r2
Z→r0
a
b
cd
15 R8.7b X=R1R2R1R2 Y=R0R2R1R2R0R1 Z=R0R1R0R1
X→r1
Y→r2
Z→r0
a
b
cd
Table 2: The 15 thin representations.
86 Ars Math. Contemp. 24 (2024) #P1.06 / 75–91
Θ-dual of Rep. Generators
R2.5 A = R2R1R2 B = R1 C = R0R2
R4.5a A = R0R2 B = R1R0R2R1 Z = R2R1R2R1
R4.5b A = R0R2 B = R1R0R2R1 Z = R2R1R0R1
R8.4a X = R2R1R0R1 Y = R1R2R1R0 Z = R2R1R2R1R0R2
R8.7a X = R1R0R1R0 Y = R0R2R1R2R0R1 Z = R2R1R2R1
Table 3: The dual representations.
a
a
b
b
c
c
2 1
6
5
4
3
¡  -Slice6.1 ¡  -Marked map rep. of H6.1 James rep. of H
a
a
b
b
c
c
2 1
6
5
4
3
Figure 3: The toroidal regular hypermap H.
As an example, we give a detailed construction of the thin representation R4.1 of H
following the generic description given in [1]:
The words R0, RR10 and (R1R2)
2, in this order, generate the subgroup Θ = Γ4.1 as a
free product C2 ∗ C2 ∗ C∞ (Table 2). A rooted Θ-slice can be obtained from a Schreier
transversal of Θ in Γ, or alternatively by a cut-opening of the trivial Θ-map (see Ta-
ble 1). The rooted Θ-slice we are taking here is the one given by the Schreier transver-
sal {1, R1, R2, R1R2}. Another Schreier transversal may lead to a different rooted Θ-
a
b c
d
Figure 4: The rooted Γ4.1-slice.
slice, and a choice of another flag as root corresponds to take another Schreier transver-
sal, and both will lead to “similar” Θ-marked maps, in the sense that the underlying map
is the same. The Θ-marked map representation of H is obtained by the isomorphism
ρ : Θ/Hρ−1 → ∆/H given by R0 7→ r0, RR10 7→ r1 and (R1R2)2 7→ r2. So we have
R0 = (1, 4)(2, 5)(3, 6), RR10 = (1, 2)(3, 4)(5, 6) and (R1R2)
2 = (1, 6)(2, 3)(4, 5). Now
we take 6 rooted Θ-slices labelled 1, 2, 3, 4, 5 and 6 and join them through their sides a, b,
c and d accordingly to the action of the words R0, RR10 and (R1R2)
2 on the root flag of the
slices. In this way, the word R0 joins the slices 1 and 4, 2 and 5, and 3 and 6, by their sides
labelled c, while RR10 joins the slices 1 and 2, 3 and 4, and 5 and 6, by their sides b. This
leaves to an incomplete picture:
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 87
b
c
e f
d
a
1
6
2
5
4
3
g
h
i
j
kl
Now (R1R2)2, which is an involution, says that the slices 1 and 6, 2 and 3, and 4 and 5,
are joined together through their sides a and d, that is, in the picture above we have the
following equality between labels: g = a and f = l, h = b and i = c, and d = j and
k = e. This lead to the final picture of R4.1 in Table 5.
In the following tables we illustrate the fifteen map representations Rep of the toroidal
regular hypermap H, we display the general Euler’s characteristic formula for the map
representation Rep of any hypermap, the actual Euler’s characteristic of Rep(H) and the
orientability (up to restricted dual) of Rep(H) - and when possible we record their overall
orientability behaviour in parenthesis.
H =
a
a
b
b
c
c
2
1
6
5
4
3
Euler characteristic of H = 0 oriented
Rep Rep(H) Euler characteristic of Rep(H) orient.
R2.1
a
a
b
b
c
c
2
1
65
4
3 |G|
(
1
2|AB| +
1
2|BC| +
1
2|CA| −
1
2
)
= 0 yes
R2.4
a
a
b
b
c
c
2
1
6
5
4
3 |G|
(
1
2|AB| +
1
2|BC| +
1
2|CA| −
1
2
)
= 0 yes
R2.5
a
a
b b
1
2
3
4
5
6
|G|
(
1
2|BCAC| +
1
2|BA| −
1
2
)
= 1 no
Table 4: The Θ-marked map representations of H for |Γ : Θ| = 2.
We discuss now orientability in more details. The first two thin representations R2.1
and R2.4 (Vince and Walsh representations) are the unique orientation-preserving repre-
sentations, that is, if they are orientable they represent orientable hypermaps and if they
are nonorientable they represent nonorientable hypermaps. However, the maps coming out
from the representations R4.5a, R4.5b, R8.4a, R8.4b, R8.7a and R8.7b are always ori-
entable, since the Θ-trivial maps for Θ ∈ {Γ4.5,Γ8.4,Γ8.7} are orientable. This poses the
question: when they represent non-orientable hypermaps? The same question hang over
the other representations with an additional hitch, both orientable and non-orientable maps
can represent orientable and nonorientable hypermaps. This means that for these represen-
tations we no longer have the clue given by R2.1 and R2.4, and for this reason we need
to make a local teste. In general, a Θ-marked map representation M = (Ω;x1, x2, x3)
88 Ars Math. Contemp. 24 (2024) #P1.06 / 75–91
is a representation of an orientable hypermap if and only if x1x2 and x2x3 act on the set
of Θ-slices with two orbits (Θ-orbits). As a hypermap H is orientable if and only if H
covers the orientably-trivial hypermap T +
H
(Figure 5), a thin map representation Rep(H)
of H represents an orientable hypermap if and only if Rep(H) covers the corresponding
representation Rep(T +
H
) of the orientably-trivial hypermap, call this representation RoriT-
map. In the cases of R2.1 and R2.4, the RoriT-map is spherical and so for any hypermap
2
1
Figure 5: The orientably-trivial hypermap T +
H
.
H the representations R2.1(H) and R2.4(H) are orientable if and only if H is orientable.
For the other cases, and specially the cases in which the representation R is always ori-
entable (R4.5a, R4.5b, R8.4a, R8.4b, R8.7a, R8.7b), the representation M = R(H) is
an orientable hypermap H if M covers the respective RoriT-map. Any RoriT-map has two
flags, so a Θ-marked map representation M = (Ω;x1, x2, x3) is a representation of an
orientable hypermap if and only if the triple (x1, x2, x3) induce a two blocks system on the
set of Θ-slices Ω (the two Θ-orbits) such that each xi permutes the two blocks exactly as
the permutation xi of the two flags of the RoriT-map does. That is, xi exchanges the two
blocks if and only if xi exchanges the two flags.
H =
a
a
b
b
c
c
2
1
6
5
4
3
Euler characteristic of H = 0 oriented
Rep Rep(H) Euler characteristic of Rep(H) orient.
R4.1 b
c
e f
d
a
1
6
2
5
4
3
a
b
c
d
ef
|G|
(
1
2|AB| +
1
2|AXBX| −
1
2
)
= 1 no
R4.4 a
cb b
d dc
2 3
4 5 6
1a |G|
(
1
2|BA| +
1
2|AZBZ| −
1
2
)
= 1 no
R4.5a
1
2 4
35
6
a
b
c
d
a d
b
c
|G|
(
1
|AZB| −
1
2
)
= 0 yes (always)
R4.5b
a
b
c
e
d
a b
d
e c
1
2
4
3
5
6
|G|
(
1
|AX| +
1
|XB| − 1
)
= −2 yes (always)
Table 5: The Θ-marked map representations of H for |Γ : Θ| = 4.
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 89
H =
a
a
b
b
c
c
2
1
6
5
4
3
Euler characteristic of H = 0 oriented
Rep Rep(H) Euler’s charac. form. on Rep(H) orient.
R8.4a
f
af
e
g
g
1 3
2 6
5
4a
b
b
c
c
d
d
e
|G|
(
1
|XZXY | +
1
|Y Z| − 2
)
= −4 yes (always)
R8.4b
1
6
2
3 5
4ab
e i
c
f
g
h
j
ae
c i b
d
f
g
hj
d
|G|
(
1
|XZY | +
1
|ZXY | − 2
)
= −6 yes (always)
R8.5a 1 5
2
3
6
4
aa
c
c
b
b
d d
e
f
f
e
g
g
|G|
(
1
|Y XZ| +
1
|ZY | −
3
2
)
= −4 no
R8.5b
1
4
2
5
3
6
b
c
e
i
a c d
f
g
h
h
a
e
b
f
d
g
i
|G|
(
1
|Y Z| +
1
|Y XZ| −
3
2
)
= −4 no
R8.6a 1
2
6
5
4
3
a
a
b
b
c
e
d
d
i
ci
e
h
ff
g
g hj k
j
k
|G|
(
1
|XZ| +
1
|ZYX| −
3
2
)
= −4 no
R8.6b
1
4
3
6
5
2
ab
e
d
i
c
f g
h
j
ae c
i b
d
f
g hj
|G|
(
1
|XY Z| +
1
|ZX| −
3
2
)
= −4 no
R8.7a
a
bce d fg
a
bc
defg
1
2
5
4
3
6 |G|
(
1
|Y ZX| −
1
2
)
= 0 yes (always)
R8.7b
a
bc
e
d
g
a
bd
e fg
c
f
1
2
5
4
3
6 |G|
(
1
|XY | +
1
|Y Z| − 1
)
= −2 yes (always)
Table 6: The Θ-marked map representations of H for |Γ : Θ| = 8.
Take for example the two map representations (always orientable) given by R4.5a and
R4.5b on the non-orientable hypermap H pictured in Figure 6 left, a non-regular, but uni-
form of type (3, 3, 3), the 6 flags hypermap with monodromy group generated by
r0 = (1, 5)(2, 4)(3, 6) , r1 = (1, 2)(3, 4)(5, 6) , r2 = (1, 6)(2, 3)(4, 5) .
It is simple to see that R4.5a(H) represents a non-orientable hypermap because by
having two vertices of valency 2 the map R4.5a(H) does not cover the uniform (regular)
toroidal map R4.5a(T +
H
) of type {4, 4}.
For the case R4.5b(H), the argument is not so simple as before because this map is uni-
form of type {6, 6} and the trivial oriented map R4.5b(T +
H
) is also uniform of type {2, 2}.
However, the word AXB = R0R2R0R1R0R1 fix the root flag 1 in the map R4.5b(H),
but does not fix any flag on the RoriT-map R4.5b(T +
H
).
90 Ars Math. Contemp. 24 (2024) #P1.06 / 75–91
Rep Rep(T +
H
)
R2.1
12
R2.4 1
2
R2.5
12
R4.1
1
2
R4.4
1
2
R4.5a
1
2
R4.5b 1 2
Rep Rep(T +
H
)
R8.4a a
b c
a
c
1 2
b
R8.4b a
b b
a
c
d
c
d
R8.5a
a
a
c
b
b
c12
R8.5b
b
c
d a
1
2
a
b
c
d
R8.6a a a
bc
1
2
bc
R8.6b
a a
b
c
b
c
d
d
1
2
R8.7a
a
a b
b
1
2
R8.7b
c
a
a
b
b
1
2
c
Table 7: The 15 RoriT-maps (thin representations of the orientably-trivial hypermap).
a
a
b
b
c
c
2 1
6
5
4
3
H
3
4
6
5
1
2
f
e
d
c
c
d
e
f
R4.5a(H)
e
b
c
1
6
4
b
c d
e
2 5
3
a
a
d
R4.5b(H)
Figure 6: A non-orientable hypermap H and its representations R4.5a(H) and R4.5b(H).
Alternatively, following the block system argument described above, painting by red
and blue the possible two blocks, we have for R4.5a(T +
H
)
1(red)
A−→ 5(blue) Z−→ 6(red) B−→ 1(blue)
and for R4.5b(T +
H
)
1(red)
A−→ 5(blue) X−→ 6(red) B−→ 1(blue)
In both cases such two block system does not exist.
A. Breda d’Azevedo et al.: Classification of thin regular map representations of hypermaps 91
ORCID iDs
Antonio Breda d’Azevedo https://orcid.org/0000-0002-7099-4704
Domenico A. Catalano https://orcid.org/0000-0002-1542-9614
References
[1] A. Breda D’Azevedo, A theory of restricted regularity of hypermaps, J. Korean Math. Soc.
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JKMS.2006.43.5.991.
[2] A. Breda D’Azevedo and G. Jones, Double coverings and reflexible abelian hypermaps, Beitr. Al-
gebra Geom. 41 (2000), 371–389, https://www.emis.de/journals/BAG/vol.41/
no.2/7.html.
[3] A. Breda d’Azevedo, A. Mednykh and R. Nedela, Enumeration of maps regardless of genus:
geometric approach, Discrete Math. 310 (2010), 1184–1203, doi:10.1016/j.disc.2009.11.017,
https://doi.org/10.1016/j.disc.2009.11.017.
[4] L. D. James, Operations on hypermaps, and outer automorphisms, Eur. J. Comb. 9
(1988), 551–560, doi:10.1016/S0195-6698(88)80052-0, https://doi.org/10.1016/
S0195-6698(88)80052-0.
[5] L. D. James, Representation of maps, in: Dissertacciones Del Seminario de Matemáticas Fun-
damentales, 1990.
[6] W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Dover Publications, Inc.,
New York, 1976, presentations of groups in terms of generators and relations.
[7] A. Vince, Combinatorial maps, J. Comb. Theory Ser. B 34 (1983), 1–21, doi:10.1016/
0095-8956(83)90002-3, https://doi.org/10.1016/0095-8956(83)90002-3.
[8] T. R. S. Walsh, Hypermaps versus bipartite maps, J. Comb. Theory Ser. B 18
(1975), 155–163, doi:10.1016/0095-8956(75)90042-8, https://doi.org/10.1016/
0095-8956(75)90042-8.

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.07 / 93–105
https://doi.org/10.26493/1855-3974.2677.b7f
(Also available at http://amc-journal.eu)
The covering lemma and q-analogues of
extremal set theory problems
Dániel Gerbner *
Alfréd Rényi Institute of Mathematics, P.O.B. 127, Budapest H-1364, Hungary
Received 8 August 2021, accepted 27 January 2023, published online 22 August 2023
Abstract
We prove a general lemma (inspired by a lemma of Holroyd and Talbot) about the
connection of the largest cardinalities (or weight) of structures satisfying some hereditary
property and substructures satisfying the same hereditary property. We use it to show how
results concerning forbidden subposet problems in the Boolean poset imply analogous re-
sults in the poset of subspaces of a finite vector space. We also study generalized forbidden
subposet problems in the poset of subspaces.
Keywords: Subspace lattice, forbidden subposet, covering, profile polytope.
Math. Subj. Class. (2020): 06A07, 05D05
1 Introduction
One of the most basic questions in extremal finite set theory is the following. Given a
property of families of subsets of an n-element set set, what is the cardinality of the largest
family satisfying it? Sperner [29] showed that for the property that no member of the family
contains another member (in other words: the family is an antichain), the answer is
(
n
⌊n/2⌋
)
.
This cardinality is realized by the family of all the ⌊n/2⌋-element subsets.
Our underlying set is [n] := {1, 2, . . . , n}. We denote the family of all its subsets by
2[n]. This family together with the containment relation forms the Boolean lattice and is
denoted by Bn. The family of all i-element subsets of [n] is called level i and is denoted by(
[n]
i
)
. Let Σ(n, k) denote the cardinality of the largest k levels (i.e. the middle k levels) of
Bn. More precisely, Σ(n, k) =
∑k
i=1
(
n
⌊n−k2 ⌋+i
)
.
*Research supported by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences and the
National Research, Development and Innovation Office – NKFIH under the grants K 116769, KH 130371 and
SNN 129364.
E-mail address: gerbner.daniel@renyi.hu (Dániel Gerbner)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
94 Ars Math. Contemp. 24 (2024) #P1.07 / 93–105
To generalize Sperner’s theorem, Katona and Tarján [24] initiated the study of prop-
erties given by forbidding inclusion patterns. More precisely, let P be a finite poset. We
say that a family F ⊂ 2[n] (weakly) contains P if there is an order-preserving injection
f :P → F , i.e., an injection such that if x <P y, then f(x) ⊂ f(y). Otherwise F is
P -free. Let La(n, P ) denote the size of the largest P -free family F of Bn. We say that a
poset is a chain if its members pairwise contain each other The chain of k elements is said
to have size k and is denoted by Pk. A chain in Bn is called a full chain if it has n + 1
members (thus one from each level).
Let us denote by e(P ) the largest integer m such that for any n, any family F of Bn
that consists of m consecutive levels is P -free. Every result in this area suggests that the
following might hold.
Conjecture 1.1. For any integer n and poset P , we have La(n, P ) = (1+o(1))Σ(n, e(P ))
= (e(P ) + o(1))
(
n
⌊n/2⌋
)
.
This conjecture was first stated by Griggs and Lu [19] and by Bukh [2], although it
was already widely believed in the extremal finite set theory community. For a survey on
forbidden subposet problems see [18].
Another basic type of extremal finite set theory problems is related to intersection pat-
terns. We say that a family F is intersecting if any two members of it share at least one
element. Erdős, Ko and Rado [6] proved that if F ⊂
(
[n]
k
)
is intersecting and n ≥ 2k, then
|F|≤
(
n−1
k−1
)
. For a treatment of several kinds of extremal finite set theory questions, see
[16].
A variant of the basic question arises when we are given a weight function (in addi-
tion to a property) and we want to determine the largest weight of a family satisfying the
property. The most usual version is when the weight of a family is the sum of the weights
of its members, and the weight of a subset of [n] depends only on its size. For example
the celebrated LYM inequality [25, 26, 31] states that for any antichain F ⊂ 2[n], we have∑
F∈F 1/
(
n
|F |
)
≤ 1.
A method to handle together all the weights of the above kind was introduced by
P. L. Erdős, Frankl and Katona [7]. The profile vector of a family F is p(F) = (f0, . . . , fn),
where fi = |F ∩
(
[n]
i
)
|. The weight vector corresponding to a weight function is w =
(w0, . . . , wn), where wi is the weight of an i-element set. Then the weight of F is the
scalar product of the profile vector and the weight vector. For a property T and a posi-
tive integer n, there is a set of profile vectors in the (n + 1)-dimensional Euclidean space
corresponding to the families with property T . It is well-known that the scalar product is
maximized at one of the extreme points of the convex hull of the set of profile vectors,
which is called the profile polytope. The extreme points of the profile polytopes have been
since determined for several properties of families, see [5, 10] for most of them.
We say that a property T of families is hereditary if for any family F with property
T , every subfamily of F has property T . It is easy to see that a property is hereditary
if and only if it can be defined by some forbidden substructures, like all the properties
considered above. We remark that in the case of hereditary properties, we can assume that
all the coordinates of weight functions are non-negative, as we could simply delete the sets
of negative weights anyway. Regarding the extreme points, it means that we can obtain
all the extreme points by changing to zero some coordinates of those extreme points that
maximize the non-negative weight functions.
D. Gerbner: The covering lemma and q-analogues of extremal set theory problems 95
Forbidden subposet problems can be studied in any poset, and intersection problems
can also be studied in structures other than the Boolean poset. A structure where both have
been studied is the lattice of subspaces. Let q be a prime power, Fq be a field of order q
and Fnq be a vector space of dimension n over Fq . Let
[
n
k
]
q
= (q
n−1)(qn−1−1)...(qn−k+1−1)
(qk−1)(qk−1−1)...(q−1)
be the Gaussian (q-nomial) coefficient. It is well-known that
[
n
k
]
q
is the number of k-
dimensional subspaces in Fnq . Let us denote by Ln(q) the lattice of subspaces with the
containment relation. We also say that the k-dimensional subspaces form level k. The
family of all k-dimensional subspaces is called level k of Ln(q).
We are going to consider analogues of extremal finite set theory questions, where i-
element subsets of [n] are replaced by i-dimensional subspaces of Fnq . We say that two
subspaces intersect if their intersection is more than just the zero vector, i.e. there is a
1-dimensional subspace contained in both. Hsieh [22] proved an analogue of the Erdős-
Ko-Rado theorem by showing that an intersecting family of k-dimensional subspaces has
cardinality at most
[
n−1
k−1
]
q
, provided n > 2k. Greene and Kleitman [17] extended it to
the case n = 2k. The analogue of Sperner’s theorem is also well-known (see [5]). Profile
polytopes were studied in this setting in [15].
Recently, other forbidden subposet problems have been examined in Ln(q) [27, 28].
Let Laq(n, P ) denote the largest number of members of a P -free family in Ln(q). Anal-
ogously to the Boolean case, we can define eq(P ) to be the largest integer such that
the union of the middle eq(P ) levels of Ln(q) does not contain P for any n, and let
Σq(n, k) =
∑k
i=1
[
n
⌊n−k2 ⌋+i
]
q
. One might formulate the following.
Conjecture 1.2. For any integer n and poset P , we have Laq(n, P ) = (1 + o(1))
Σq(n, eq(P )).
Observe that for several posets we have eq(P ) = e(P ). Rather than proving results
analogous to those known in the Boolean case, the focus of the papers mentioned above
is to prove “stronger” results. For example, the diamond poset D2 has four elements with
relations a < b < d and a < c < d. It is unknown if Conjecture 1.1 holds for this poset.
The best upper bound is La(n,D2) ≤ (2.20711 + o(1))
(
n
⌊n/2⌋
)
[20]. Sarkis, Shahriari
and students [27] obtained, for the analogous question in the lattice of subspaces, the upper
bound Laq(n,D2) ≤ (2 + 1/q)
[
n
⌊n/2⌋
]
q
.
Let ∨ be the poset on three elements with relations a < b and a < c, and ∧ be the
poset on three elements with relations a < c and b < c. Katona and Tarján [24] determined
La(n, {∨,∧}), where we forbid ∨ and ∧ at the same time. The solution is
(
n
n/2
)
if n is
even, but slightly more than
(
n
⌊n/2⌋
)
if n is odd. Shahriari and Yu [28] showed that in
Ln(q) we have Laq(n, {∨,∧}) =
[
n
⌊n/2⌋
]
q
for every prime power q and n ≥ 2. They also
studied the case we forbid a broom ∧u and a fork ∨v at the same time, where ∧u has u+ 1
elements a1, . . . , au, b and the relations ai < b for any i ≤ u, while ∨v has v + 1 elements
a, b1, . . . , bv and the relations a < bi for every i ≤ v.
The butterfly poset B has four elements and the relations a < c, a < d, b < c and
b < d. De Bonis, Katona and Swanepoel [4] proved La(n,B) = Σ(n, 2). Shahriari and
Yu [28] proved Laq(n,B) = Σq(n, 2).
In this paper we state a simple lemma (Lemma 2.1), that generalizes the so-called per-
mutation method and explore its consequences. It can be applied to other structures, and in
particular for the subspaces it implies the following theorem.
96 Ars Math. Contemp. 24 (2024) #P1.07 / 93–105
Theorem 1.3. Let T be a hereditary property. If any family of Bn with property T has
at most Σ(n, k) members, then any family in Ln(q) with property T has at most Σq(n, k)
members.
This means that the result of De Bonis, Katona and Swanepoel [4] about butterflies
implies the result of Shahriari and Yu [28]. Note that Shahriari and Yu also determined
the extremal families. They also consider the poset Yk on k + 2 elements, defined by
the following relations: ck < ck−1 < · · · < c1 < a and c1 < b. Let Y ′k defined in
the same way but all the relations are reversed. Shahriari and Yu [28] conjectured that
Laq(n, {Yk, Y ′k}) = Σq(n, k); this follows from a result in [13], using Theorem 1.3. We
remark that Xiao and Tompkins [30] independently also found the connection between
La(n, P ) and Laq(n, P ) and used it to prove the conjecture of Shahriari and Yu [28].
The asymptotic version of Theorem 1.3 is also true, giving the following result.
Theorem 1.4. Let T be a hereditary property. If any family of Bn satisfying T has at
most (1 + o(1))Σ(n, k) members, then any family in Ln(q) with property T has at most
(1 + o(1))Σq(n, k) members.
Corollary 1.5. If Conjecture 1.1 holds for P and eq(P ) = e(P ), then Conjecture 1.2 also
holds for P .
To state the Covering Lemma (Lemma 2.1), we need some preparation, hence we post-
pone it to Section 2. We also describe how it relates to several known proofs. In Section 3
we prove Theorems 1.3 and 1.4. In Section 4 we examine how the Covering Lemma can
be modified to apply in the study of profile polytopes and related topics, and we initiate the
study of generalized forbidden subposet problems in Ln(q).
2 The main lemma
Our lemma is motivated by a lemma by Holroyd and Talbot [21]. We say that a family of
subsets of S is a t-covering family of S if every element of S is contained in exactly t sets
of the family. Given a partition of S into S0∪S1∪. . .∪Sn and a vector t = (t0, t1, . . . , tn),
we say that a family of subsets of S is a t-covering family of S if for each 0 ≤ i ≤ n, every
element of Si is contained in exactly ti sets of the family.
In our applications, S will be 2[n] or the family of subspaces of Fnq , and Si will be level
i. Holroyd and Talbot [21] considered coverings of subfamilies F of one level
(
[n]
i
)
. Their
lemma states that if F ⊂
(
[n]
i
)
, Γ is a t-covering family of subfamilies of F , and an element
x has the property that the largest intersecting family in every G ∈ Γ is {G ∈ G : x ∈ G},
then the largest intersecting family in F is {F ∈ F : x ∈ F}. Our main contribution is the
simple observation that we can extend their method to other forbidden configurations and
more levels.
For a weight vector w = (w0, . . . , wn) and a set F ⊂ S, let w(F ) =
∑n
i=0 wi|F ∩Si|.
Let w/t = (w0/t0, . . . , wn/tn). We will always assume that every coordinate of every
weight vector is non-negative. A version of the lemma below has already appeared in my
master’s thesis [9].
Lemma 2.1 (Covering Lemma). Let T be a hereditary property of subsets of S and Γ be a
t-covering family of S. Assume that there exists a real number x such that for every G ∈ Γ,
every subset G′ of G with property T has w/t(G′) ≤ x. Then w(F ) ≤ |Γ|x for every
F ⊂ S with property T .
D. Gerbner: The covering lemma and q-analogues of extremal set theory problems 97
Proof. Let F be a set with property T .
Observe that we have ti|F ∩ Si|=
∑
G∈Γ|G ∩ F ∩ Si|, as every element of F ∩ Si is
counted ti times on both sides. Thus we have
w(F ) =
n∑
i=0
wi|F ∩ Si|=
n∑
i=0
wi
ti
ti|F ∩ Si|=
n∑
i=0
wi
ti
∑
G∈Γ
|G ∩ F ∩ Si|
=
∑
G∈Γ
n∑
i=0
wi
ti
|G ∩ F ∩ Si|=
∑
G∈Γ
w/t(G ∩ F ) ≤
∑
G∈Γ
x = |Γ|x.
Let us describe how one can use this lemma in extremal finite set theory. Let S = 2[n]
and Si =
(
[n]
i
)
. Then the subsets of S are families of Bn, and we will denote them by F
and G instead of F and G.
The prime examples of covering families where the above lemma is useful are given
by the permutation method. Given a permutation α : [n] → [n], and a set F ⊂ [n], let
α(F ) = {α(i) : i ∈ F}. Similarly, for a family F of Bn, let α(F) = {α(F ) : F ∈ F}.
Let G0 be a family of Bn that has at least one i-element set for every 0 ≤ i ≤ n, and let
Γ consist of α(G0) for all permutations α. Let gi = |G0 ∩
(
[n]
i
)
|> 0 and ti = gii! (n− i)!,
then Γ is a t-covering of Bn.
The simplest example is when G0 is a full chain. Consider a Sperner family F of Bn
and let w = t. Then∑
F∈F
|F |! (n− |F |)! = w(F) =
∑
G∈Γ
∑
H∈G∩F
w/t(H) ≤
∑
G∈Γ
1 = |Γ|= n! .
Dividing by n! we obtain the already mentioned LYM-inequality. Another example is when
G0 is the family of intervals in a cyclic ordering of [n], resulting in the cycle method [23].
Any family G0 of Bn can be used to give upper bounds on problems in extremal finite
set theory, but these bounds are unlikely to be sharp. For that, G0 has to be very symmetric
in a sense. We need that for every permutation α, the largest subfamily of α(G0) with
property T has the same size. Other examples for families G0 that sometimes give sharp
bounds are the chain-pairs [10] and double chains [3].
Let us return to Lemma 2.1 and examine a very special case. Assume that Si1∪. . .∪Sik
has property T and for every G ∈ Γ, w/t(G′) = x for G′ = G ∩ (Si1 ∪ . . . ∪ Sik) (in
the case of the permutation method, it means that the union of k full levels has property T ,
and the weight inside α(G0) is maximized by those k levels). This implies that we have
equality in Lemma 2.1.
Now assume that we conjecture that w(F) is maximized by a family that is the union of
k full levels (among families with property T ). Let H0 be the intersection of those k levels
with G0, then H0 has property T . If H0 happens to have the largest weight w/t among
subfamilies of G0 with property T , then it proves the conjecture (here we use the simple
observation that α(H0) would maximize w/t among subfamilies of α(G0)). Thus our goal
would be to find G0 with this property.
For example, in the case of antichains, it is a natural idea to consider a full chain as
G0. Indeed, for every weight, the maximum will be given by a family that consists of one
member, which is a full level on the chain. Moreover, it is one of the levels with the largest
weight, thus we can choose the same level all the time. This implies that for every weight
98 Ars Math. Contemp. 24 (2024) #P1.07 / 93–105
function, the maximum in the Boolean poset is also given by a full level, giving us not
only Sperner’s theorem and the LYM inequality, but all the extreme points of the profile
polytope, reproving a result in [7]. Moreover, we say that a family is k-Sperner if it is
Pk+1-free. The above argument works for k-Sperner families as well, since on any chain,
for any weight, the maximum is given by k full levels. This, again, gives the extreme points
of the profile polytope as well, reproving a result in [8].
Observe that we do not need to have full levels in our conjecture to obtain an exact result
without further computations. Assume that in our conjecture, for every i, the extremal
family H contains γi
(
n
i
)
sets from level i, and H contains a γi fraction of the intersection
of α(G0) and level i. Then the same argument works. For example consider intersecting
families on level k, and use the cycle method [23]. We choose a cyclic ordering of the
elements of [n] and let G0 be the family of k-intervals, i.e. k-sets of consecutive elements.
There are n such k-sets, and k of them contain a fixed element x. Let H be the family of
k-sets containing x, and H0 be its intersection with G0. It is not hard to see that H0 is the
largest intersecting subfamily of G0 (provided k ≤ n/2). Thus, for every α we have that H
contains a k/n fraction of the members of α(G0). As H contains a k/n fraction of all the
sets, we are done.
To finish this section, let us remark that we are mostly interested in the case where
every wi = 1. For that wi/ti = 1/(gii! (n − i)! ) =
(
n
i
)
/(n! gi). In the case where G0
is a full chain, every gi is the same. In the case where G0 is the family of intervals on the
cycle, almost every gi is the same (with the exception of g0 and gn). As multiplying with
the same number does not change the extremal families, we can consider maximizing the
weight function with w′i =
(
n
i
)
instead (assuming we can deal with the empty set and the
full set some other way). If, on the other hand we can deal with the case of constant weight
on the chain or the cycle for a property T , and the optimal family consists of the middle
levels, then we obtain a LYM-type inequality for subfamilies of 2[n] with property T , see
for example the case of butterfly-free families in [4].
3 Subspaces
Let us turn our attention to q-analogues. Similarly to the Boolean case and the permuta-
tion method, it will again simplify our tasks if all G ∈ Γ are isomorphic. Moreover, we
would prefer to use G where proving extremal results is either easy or has already been
done. Therefore, we will use a subfamily G of Ln(q) that is isomorphic to Bn. Choose an
arbitrary basis B = {v1, . . . , vn} of Fnq , and let GB be the family of those subspaces that
are generated by a set of these vectors. Obviously the function that maps H ⊂ [n] to the
subspace ⟨vx : x ∈ H⟩ keeps inclusion and intersection properties. Let Γ be the union over
all bases B of the families GB .
There are f(q, n) = (qn−1)(qn−q)(qn−q2) · · · (qn−qn−1)/n! ways to choose a basis,
as we pick the vectors one by one, and we obtain a basis n! ways. Hence f(q, n) is the car-
dinality of Γ, which is a t-covering of Ln(q) with ti = (q
i−1)···(qi−qi−1)(qn−qi)···(qn−qn−1)
i!(n−i)! .
Indeed, to count how many times an i-dimensional subspace is covered, we have to pick a
basis of the i-dimensional subspace first, and then extend it to a basis of Fnq . We counted
every G ∈ Γ exactly i! (n − i)! times, as we picked the basis in an ordered way. Observe
that we have t0 > t1 > · · · > t⌊n/2⌋ = t⌈n/2⌉ < t⌈n/2⌉+1 < · · · < tn.
Now we are ready to prove Theorem 1.3, which states that if every family F of Bn
satisfying a hereditary property T has cardinality at most Σ(n, k), then families in Ln(q)
D. Gerbner: The covering lemma and q-analogues of extremal set theory problems 99
with property T have cardinality at most Σq(n, k). We note that the actual calculation
could be omitted by the arguments presented in Section 2. We include it here for the sake
of completeness.
Proof of Theorem 1.3. Let F be a family in Ln(q) satisfying T . Consider the t-covering
family Γ defined above and let wi = ti. Then every G ∈ Γ is isomorphic to Bn, thus by
our assumption, the largest weight w/t, i.e. the largest cardinality of a subfamily G′ ⊂ G
satisfying T is Σ(n, k). This implies w(F) ≤ |Γ|Σ(n, k). Now we will maximize |F|
among families that satisfy the above inequality, without requiring property T . To do this,
we need to pick subspaces with the smallest weight, i.e. from the middle levels. We claim
that we can pick exactly the k full middle levels, i.e. w(F0) = |Γ|Σ(n, k) for the family
F0 consisting of k middle levels. (Note that if n+ k is even, we have two options for F0).
This will finish the proof, because more than Σq(n, k) subspaces would have larger weight
than |Γ|Σ(n, k).
We have
w(F0) =
⌊n−k2 ⌋+k∑
i=⌊n−k2 ⌋+1
wi
[
n
i
]
q
=
⌊n−k2 ⌋+k∑
i=⌊n−k2 ⌋+1
(qi − 1) · · · (qi − qi−1)(qn − qi) · · · (qn − qn−1)
i! (n− i)!
[
n
i
]
q
=
⌊n−k2 ⌋+k∑
i=⌊n−k2 ⌋+1
(qi − 1) · · · (qi − qi−1)(qn − qi) · · · (qn − qn−1)
i! (n− i)!
(qn − 1) . . . (qn − qn−1)
(qi − 1) . . . (qi − qi−1)(qn − qi) . . . (qn − qn−1)
=
⌊n−k2 ⌋+k∑
i=⌊n−k2 ⌋+1
f(q, n)n!
i! (n− i)!
=
⌊n−k2 ⌋+k∑
i=⌊n−k2 ⌋+1
|Γ|
(
n
i
)
= |Γ|Σ(n, k).
Another way to see that w(F0) = |Γ|Σ(n, k) is by observing that the left hand side
counts the number of pairs (S,B), where S is an i-dimensional subspace and B is a ba-
sis for S (organized by subspaces), while the right hand side counts the same thing, but
organized by the basis.
Note that there are several statements similar to Theorems 1.3 and 1.4 that we could
prove. We chose to state this one because it immediately gives the exact value of Laq(n,B).
Observe that the Boolean result actually gives a weighted result in the case of subspaces,
that is stronger than Theorem 1.3. In the case of the butterfly poset, one can prove an even
stronger result. If F is a butterfly-free family of Bn, then we have the LYM-type inequality∑
F∈F 1/
(
n
|F |
)
≤ 2 by [4]. This and the same calculation as in the proof of Theorem 1.3
imply that for a butterfly-free family G in Ln(q), we have
∑
G∈G 1/
[
n
dim(G)
]
≤ 2.
Let us prove now Theorem 1.4, which is the asymptotic version of Theorem 1.3.
Proof of Theorem 1.4. We follow the proof of Theorem 1.3. Using its notation, we obtain
w(F) ≤ (1+o(1))|Γ|Σ(n, k). Again, to maximize |F| among those families satisfying the
100 Ars Math. Contemp. 24 (2024) #P1.07 / 93–105
above inequality, we need to pick subspaces with the smallest weight, i.e. from the middle
levels. This time we claim that we can pick the subspaces in F0, and o(|F0|) additional
subspaces. This will finish the proof similarly to the proof of Theorem 1.3.
We have proved w(F0) = |Γ|Σ(n, k), thus the remaining subspaces have total weight
o(|Γ|Σ(n, k)) = o(w(F0)). As each of those has weight not smaller than any weight
of a subspace in F0, more than ε|F0| of them would have weight more than εw(F0), a
contradiction that finishes the proof.
4 Profile polytopes, chain profile polytopes, generalized forbidden sub-
poset problems
In the previous sections we considered arbitrary weights. This means our method can
potentially determine the extreme points of the profile polytope for a hereditary property T .
If every extreme point in the Boolean case is the union of full levels, and the corresponding
union of full levels has property T in the case of subspaces, then this is the situation.
Unfortunately, we are only aware of one particular property with this situation. For k-
Sperner families, the Boolean result was proved in [8]. We note that instead of using the
substructure isomorphic to Bn with Lemma 2.1, one could use a simpler substructure: a
full chain with Lemma 2.1, to obtain the same result, i.e. to determine the extreme points.
Moreover, it also easily follows from the LYM-inequality, which is known to hold in Ln(q).
In fact, one can analogously define the profile vectors and polytopes for any graded poset
and show for a large class of posets (those with the so-called Sperner property) that the
extreme points of k-Sperner families are the profiles of the unions of at most k full levels.
Gerbner and Patkós [14] introduced l-chain profile vectors. Given a family F of Bn, its
l-chain profile vector is an element of the
(
n+1
l
)
-dimensional Euclidean space. A coordi-
nate corresponds to a set {i1, . . . , il} with i1 < i2 < · · · < il. The value of that coordinate
is the number of chains of size l in F with one element from level ij for every 1 ≤ j ≤ l.
They determined the extreme points of the l-chain profile polytopes of intersecting families
and of k-Sperner families of Bn.
They mentioned in [15], after determining the extreme points of the profile polytope of
intersecting families in Ln(q), that with the same method, one can determine the extreme
points of the l-chain profile polytope of intersecting families in Ln(q) as well. Here we
show that similarly, the extreme points of the l-chain profile polytope of k-Sperner families
in Ln(q) can be determined. We will state a modified version of Lemma 2.1 that counts
copies of a poset Q in a family instead of counting the members of that family.
Let Q be an arbitrary poset with elements a1, . . . , al. Consider the r = (n + 1)l
functions that map every aj to an Si. Let us fix an ordering of these functions and let βi be
the ith of them. We will consider ordered l-sets, i.e. l-sequences (s1, . . . , sl) of the base
set S. For each 1 ≤ i ≤ r, let Si be an arbitrary family of l-sequences with sj ∈ βi(aj)
for every 1 ≤ j ≤ l. In the applications, where Si is a level, we will let Si consist of those
l-sequences, where the elements form a copy of Q. In particular, if for an embedding βi
and for some j, j′ with aj < aj′ we have that βi(aj) is a higher level than βi(aj′), then Si
is empty. Let us consider only those r′ ≤ r functions βi, where Si is not empty. We can
assume without loss of generality that these functions are β1, . . . , βr′ .
Let t = (t1, . . . , tr′) be a vector. We say that a family Γ of subsets of S is an (l, t)-
covering if for each 1 ≤ i ≤ r′, and each l-sequence in Si, there are exactly ti members of
Γ containing all the elements of that l-sequence (i.e. a particular copy of Q). Let us consider
D. Gerbner: The covering lemma and q-analogues of extremal set theory problems 101
a weight vector w = (w1, . . . , wr′). For a set F ⊂ S, let fi denote the number of l-sets
in Si with every element in F . Let w(F ) =
∑r′
i=1 wifi. Let w/t = (w1/t1, . . . , wr′/tr′).
We will assume that every weight is non-negative (as T is hereditary, elements of S with
negative weight could simply be deleted anyway from any subset of S with property T ).
Lemma 4.1. Let T be a hereditary property of subsets of S and Γ be an (l, t)-covering
family of S. Assume that there exists a real number x such that for every G ∈ Γ, every
subset G′ of G with property T has w/t(G′) ≤ x. Then w(F ) ≤ |Γ|x for every F ⊂ S
with property T .
Proof. Observe that we have tifi =
∑
G∈Γ hi, where hi denotes the number of l-sequences
in Si with each element of it in F ∩G. Indeed, the l-sequences in Si with each element in
F are counted ti times on both sides. Thus we have
w(F ) =
r′∑
i=1
wifi =
r′∑
i=1
wi
ti
tifi =
r′∑
i=1
wi
ti
∑
G∈Γ
hi =
∑
G∈Γ
r′∑
i=1
wi
ti
hi
=
∑
G∈Γ
w/t(G ∩ F ) ≤
∑
G∈Γ
x = |Γ|x.
We have equality here if for every G ∈ Γ, there is a G′ ⊂ G satisfying T with
w/t(G′) = x, and G′ = G ∩ F . This holds in the following situation. Let T be the
k-Sperner property, S be Ln(q) with the usual partition into levels, and Si be the set of
those l-sets that form a chain. Let Γ consist of copies of the Boolean poset, as described in
Section 3 (note that we could use instead the chains given by a basis and its ordering). Let
us assume levels j1, . . . , jk have the maximum weight w/t in the Boolean poset, and let F
consist of the subspaces on levels j1, . . . , jk. Then by the above, F has the largest weight
w(F ) = |Γ|x among k-Sperner families. We obtained that for every non-negative weight
the union of k levels has the largest weight, which implies the following result.
Corollary 4.2. The extreme points of the l-chain profile polytope of k-Sperner families of
subspaces of Fnq are the unions of at most k levels.
We mentioned the l-chain polytopes here because the above result gives the first in-
stance of a generalized forbidden subposet problem in Ln(q). The generalized forbidden
subposet problem seeks to find La(n, P,Q), the largest number of copies of the poset Q
in a P -free subfamily of Bn. Its study was initiated by Gerbner, Keszegh and Patkós [11],
analogously to the graph case [1] that has recently attracted a lot of attention. Further
results on La(n, P, Pl) can be found in [12].
We propose to study generalized forbidden subposet problems in Ln(q). Let
Laq(n, P,Q) denote the largest number of copies of the poset Q in a P -free family in
Ln(q). Corollary 4.2 implies that Laq(n, Pk, Pl) is given by k full levels (it is not hard
to see that the best way to choose the k levels i1, . . . , ik is when the values i1, i2 − i1,
i3 − i2, . . . , ik − ik−1, n− ik differ by at most one). For other pairs of posets, a weighted
version in the Boolean case could give bounds on Laq(n, P,Q).
Let us mention that even though Lemma 4.1 immediately implied a generalized forbid-
den subposet result in Ln(q), it may be the only particular problem where we can use it to
obtain a sharp bound. Lemma 4.1 requires studying a weighted version of a generalized
102 Ars Math. Contemp. 24 (2024) #P1.07 / 93–105
forbidden subposet problem in the smaller structure G, similarly to Lemma 2.1. Observe
that in the case of counting the members of a family of Bn, we had the useful property
that wi/ti is the largest in the middle, exactly where the (conjectured) extremal families
are. Therefore, an unweighted result on the cycle gave a weighted result of Bn that implied
the unweighted result. And similarly, an unweighted result of Bn immediately implied the
analogous bound in Ln(q). However, this is not the case with the more complicated weight
functions and more diverse extremal families that we deal with in generalized forbidden
subposet problems.
To finish the paper, we present some simple results for Laq(n, P,Q). They are unre-
lated to the earlier parts of the paper, but we would like to present some results concerning
this function, since we initiate the study of this topic in this paper. Let the generalized dia-
mond poset Dr have r+2 elements a, b1, . . . , br, c and relations a < bi < c for 1 ≤ i ≤ r.
Proposition 4.3. (i) Laq(n,∨,∧r) = Laq(n,∧,∨r) =
([ n⌊n/2⌋]q
r
)
.
(ii) Laq(n,B,Dr) =
([ n⌊n/2⌋]q
r
)
.
(iii) Laq(n, P3,∧r) = max0≤k≤n
[
n
k
]
q
([ k⌊k/2⌋]q
r
)
.
The Boolean analogues of the above statements were proved in [11], and the proofs
of them also work in our case. We include them for the sake of completeness. We will
use the canonical partition of k-Sperner families F ; it is a partition of F into k antichains
F1, . . . ,Fk, where Fi is the set of minimal elements of F \ ∪i−1j=1Fj .
Proof. The lower bounds for (i) and (ii) are given by the families consisting of all the
⌊n/2⌋-dimensional subspaces together with the zero-dimensional and/or the n-dimensional
subspace. For (iii) consider all the k-dimensional and ⌊k/2⌋-dimensional subspaces for
every k.
For the upper bound in (i), the first equality is trivial by symmetry. Let us consider now
the canonical partition F1 ∪ F2 of a ∨-free family F in Ln(q). Observe that every copy
of ∧r consists of a member of F2, and r members of F1 contained in it. Every member of
F1 is contained in at most one member of F2 by the ∨-free property, thus for every set of
r members of F1, at most one member of F2 forms a copy of ∧r with them. This implies
Laq(n,∨,∧r) ≤
(|F1|
r
)
. As F1 is an antichain, it has at most
[
n
⌊n/2⌋
]
q
members, finishing
the proof of (i).
To prove the upper bound in (ii), let F be a B-free family in Ln(q) and M = {M ∈
F : ∃F ′, F ′′ ∈ F such thatF ′ ⊂ M ⊂ F ′′}. As F is P4-free, M is an antichain. Observe
that for an M ∈ M there is exactly one F ′ ∈ F with F ′ ⊂ M and there is exactly one
F ′′ ∈ F with M ⊂ F ′′. Thus, for every r-tuple from M there is at most one copy of Dr
in F , and there are at most
([ n⌊n/2⌋]q
r
)
such r-tuples.
To prove the upper bound in (iii), let F be a P3-free family in Ln(q) and consider its
canonical partition F1 ∪F2. Every copy of ∧r consists of a member of F2 and r members
of F1. For a member F of F2 with dimension k, we have to pick r subspaces of it that
are in F1. Those members of F1 that can be picked form an antichain of subspaces of a
k-dimensional space, thus there are at most
[
i
⌊k/2⌋
]
q
of them, and there are
[
n
k
]
q
([ k⌊k/2⌋]q
r
)
ways to pick r of them. It means that a k-dimensional member of F2 is in at most w(k) :=([ k⌊k/2⌋]q
r
)
copies of ∧r. Hence the total number of copies of ∧r is at most the total weight
D. Gerbner: The covering lemma and q-analogues of extremal set theory problems 103
of F2, i.e. w(F2). As F2 is an antichain, this is maximized by a level (for a number of
reasons mentioned earlier, for example Corollary 4.2 implies this). The weight of level k is[
n
k
]
q
([ k⌊k/2⌋]q
r
)
, finishing the proof.
ORCID iDs
Dániel Gerbner https://orcid.org/0000-0001-7080-2883
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.08 / 107–126
https://doi.org/10.26493/1855-3974.2976.f76
(Also available at http://amc-journal.eu)
On the beta distribution, the nonlinear Fourier
transform and a combinatorial problem
Pavle Saksida *
Faculty of Mathematics and Physics, University of Ljubljana,
Jadranska ulica 21, Ljubljana, Slovenia
Received 19 October 2022, accepted 31 January 2023, published online 22 August 2023
Abstract
The paper describes some probabilistic and combinatorial aspects of the nonlinear
Fourier transform associated with the AKNS-ZS problems. We show that the volumes
of a family of polytopes that appear in a power expansion of the nonlinear Fourier trans-
form are distributed according to the beta probability distribution. We establish this result
by studying an Euler-type discretization of the nonlinear Fourier transform. This approach
leads to the combinatorial problem of finding the number of alternating ordered partitions
of an integer into a fixed number of distinct parts. We find the explicit formula for these
numbers and show that they are essentially distributed according to a novel discretization
of the beta distribution for a suitable choice of the shape parameters. We also find the
generating functions of the numbers of alternating sums. These functions are expressed in
terms of the our discrete nonlinear Fourier transform.
Keywords: Beta distribution, nonlinear Fourier transform, discretisation.
Math. Subj. Class. (2020): 37K15, 42A99, 60E05, 05A17
1 Introduction
As announced in the title, this paper investigates relations between three topics from differ-
ent parts of mathematics: probability distributions, combinatorics and the theory of nonlin-
ear partial differential equations, more concretely, the nonlinear Fourier transform. Despite
the apparent heterogeneity of the topics, the relations between them are rather natural.
*I am grateful to Matjaž Konvalinka for a very fruitful discussion. I am also grateful to both anonymous
reviewers for their comments and suggestions. The research for this paper was supported in part by the research
program Analysis and Geometry, P1- 0291, funded by the Slovenian Research Agency.
E-mail address: pavle.saksida@fmf.uni-lj.si (Pavle Saksida)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
108 Ars Math. Contemp. 24 (2024) #P1.08 / 107–126
The construction and the study of various versions of the nonlinear Fourier transform
stem from the theory of integrable nonlinear partial differential equations. The most fa-
mous examples of such equations include the Korteweg-de Vries, nonlinear Schrödinger,
and sine-Gordon equations, Heisenberg ferromagnet model, Toda lattices and many oth-
ers. The role of the nonlinear Fourier transform in the theory of integrable equations
is roughly analogous to the role of the linear Fourier transform and, more generally, the
Sturm-Liouville expansions in the theory of linear partial differential equations.
The transformation F used in this text can be thought of as a non-linearization of the
usual Fourier transformation. Let u : [0, 1] → R be a function. The nonlinear Fourier
transform F of u that we shall consider in this paper is of the form
F [u](n) = I +
(
0 F [u](n)
−F [u](−n) 0
)
+
∞∑
d=2
Ad[u](n),
where F is the linear Fourier transform (Fourier series) and u 7→ Ad[u] are the suitable
matrix-valued nonlinear operators.
The beta distribution is one of the oldest and most important probability distributions
with a broad spectrum of applications in different areas of probability and statistics, par-
ticularly in Bayesian statistical inference. It has been recently mentioned in virtually every
book on machine learning and related topics. The beta distribution Beta(x; a, b) with shape
parameters a and b is given by the probability density function
pβ(x; a, b) =
1
B(a+ 1, b+ 1)
xa(1− x)b, x ∈ [0, 1].
In this paper, we shall establish a link between the nonlinear Fourier transform and the
beta distribution. Let uc(x) ≡ u be a constant function. The transformation F is related to
a two-parameter family of polytopes D̂d(λ), where d ∈ N and λ ∈ [0, 1], given by
D̂d(λ) = {(x1, x2, . . . , xd); 1 ≥ x1 ≥ x2 . . . ≥ xd ≥ 0,
d∑
i=1
(−1)i−1xi = λ}
and their projections Dd(λ) in the hyperplane {(x1, x2, . . . , x(d−1), 0)} ⊂ Rd. For the
nonlinear Fourier transform F [uc](n) of the constant function uc ≡ u on [0, 1], we have
F [uc](n) = I +
∞∑
d=1
ud
∫ 1
0
Vol(Dd(λ))
(
e−2πiλn 0
0 e2πiλn
)
·
(
0 1
−1 0
)
dλ.
This formula is proved in Proposition 2.1 on page 113. We shall see that for every fixed d0,
the volumes of the family {Dd0(λ);λ ∈ [0, 1]} are given by the formula for the probability
density function of the beta distribution. Theorem 4.3 on page 122 gives the formula
Vol(Dd(λ)) =
1
d!
{
pβ(λ;
d
2 ,
d
2 + 1); d even
pβ(λ;
d+1
2 ,
d+1
2 ); d odd.
(1.1)
The probabilistic contents of the above formula will be described below.
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 109
The statement and the proof of Theorem 4.3 are obtained by considering a suitable
discretization FN of the nonlinear Fourier transform F . In the expression for FN [uc], the
role of the volumes of the polytopes Dd(λ) is assumed by the numbers
AQN (L, d) = ♯{(l1, l2, . . . , ld) ∈ N; l1 − l2 + l3 − . . .+ (−1)(d−1)ld = L},
where N − 1 ≥ l1 > l2 > . . . > ld ≥ 0. So, AQN (L, d) is the number of ordered
alternating partitions of L into d distinct parts not grater than N − 1.
The central result of the paper is the explicit formula for the numbers AQN (L, d). It is
given in Theorem 3.3 on page 116. We show that
AQN (L, d) =
{( L−1
⌊ d−12 ⌋
)(
N−L
⌊ d2 ⌋
)
; d even(
L
⌊ d−12 ⌋
)(
N−L−1
⌊ d2 ⌋
)
; d odd.
(1.2)
The relationship between the numbers AQN (L, d) and the nonlinear Fourier transform is
best described by the fact that the generating functions for the numbers AQN (L, d) are in
a natural way expressed in terms of the discrete nonlinear Fourier transform FN . This is
proved in Proposition 3.2 on page 115. We actually get separate generating functions for
odd and for even values of d. Understanding the structure of the numbers AQN (L, d) was
important in the construction of the inverse of FN in our recent paper [14].
Results (1.1) and (1.2) can be recast into probabilistic terms. Let our sample space
consist of all strictly decreasing d-tuples of integers
∆Dd (N) = {(l1, l2, . . . , ld); N − 1 ≥ l1 > l2 > . . . , ld ≥ 0)},
Let all the samples (l1, l2, . . . , ld) be equally probable and let
XAS [N, d] : ∆
D
d (N) → N
be the random variable which assigns to a randomly chosen point in ∆Dd (N) the alternating
sum,
XAS [N, d](l1, l2, . . . , ld) = l1 − l2 + l3 − . . .+ (−1)(d−1)ld.
We want to compute the probability P (XAS [N, d] = L) of the event that a randomly
chosen d-tuple has the alternating sum equal to L. We shall show that
AS[N, d](L) = P (XAS [N, d] = L) =

(
L−1
⌊ d−1
2
⌋)(
N−L
⌊ d
2
⌋ )
(Nd)
; d even
( L⌊ d−1
2
⌋)(
N−L−1
⌊ d
2
⌋ )
(Nd)
; d odd.
(1.3)
The random variable XAS is distributed according to the probability mass function AS[N, d]
defined by the right hand side of the above formula.
The question arises: Does this distribution have a sensible limit as N goes to infinity?
One possibility is to proceed as follows. Let λ ∈ [0, 1] be arbitrary. Let us choose a
sequence {LN}N∈N such that LN < N and limN→∞ LN/N = λ. We shall see that
lim
N→∞
P (XAS [N, d] = LN )N =
{
pβ(λ;
d
2 ,
d
2 + 1) ; d even
pβ(λ;
d+1
2 ,
d+1
2 ) ; d odd,
(1.4)
110 Ars Math. Contemp. 24 (2024) #P1.08 / 107–126
where pβ(λ; a, b) is the beta distribution with shape parameters a and b.
Let our sample space now be the ordered simplex ∆d ⊂ Rd of the dimension d, given
by
∆d = {(x1, x2, . . . , xd); 1 ≥ x1 ≥ x2 . . . ≥ xd ≥ 0},
and let all the samples (x1, x2, . . . , xd) be equally probable. This means that we assigned
on ∆d the uniform distribution v : ∆d → R given by v(x1, x2, . . . , xd) ≡ d!. Let the
random variable Xas[d] defined on ∆d be given by
Xas[d](x1, x2, . . . , xd) = x1 − x2 + x3 − . . .+ (−1)(d−1)xd.
Formula (1.4) shows that the cumulative probability distribution
Fas[d] : [0, 1] −→ R, λ → Fas[d](λ)
of the random variable Xas is given by
Fas[d](λ) = P (Xas[d] ≤ λ) =
{∫ λ
0
pβ(µ;
d
2 ,
d
2 + 1) dµ ; d even∫ λ
0
pβ(µ;
d+1
2 ,
d+1
2 ) dµ ; d odd.
This result can be recast in geometric terms. Taking into account that the d-dimensional
volume of the simplex ∆d is equal to 1d! , we see from the above that the (d−1)-dimensional
volume of the polytope Dd(λ) is indeed given by formula (1.1) explained in Theorem 4.3.
The equality (1.4) suggests a natural generalisation of the probability mass function of
XAS [N, d]. It can be defined by
PN (L; a, b) =
(
L−1
a
)(
N−L
b
)(
N
a+b+1
) ,
where L ∈ {1, 2, . . . , N} and are integers such that a + b < N . In Proposition 4.2 we
show that
lim
N→∞
PN (
LN
N
, a, b)N = pβ(λ; a, b) =
1
β(a+ 1, b+ 1)
λa(λ− 1)b.
So, the probability mass function PN (a, b) is a natural discretization of the continuous beta
distribution for arbitrary values of shape parameters. But, at the moment, a convincing
combinatorial or geometric description of PN (a, b) remains a task for the future.
Above, we have described a way how the beta distribution emerges as an appropriate
limit from a discrete and finite probability distribution. This result is reminiscent to the
relation between the Pólya-Eggenberger urn and the beta distribution. Pólya-Eggenberger
urn is an urn model with replacement and is tenable - one can draw the balls from the urn
infinitely many times. The limit of the quotient Wnn , where Wn is the number of white balls
drawn in n draws, is given by
lim
n→∞
P (
Wn
n
< λ) =
∫ λ
0
pβ(µ;
W0
s
,
B0
s
) dµ,
where W0 and B0 are the initial numbers of white and black balls in the urn and s is the
number of the balls added to the urn after drawing and returning a ball of the same colour.
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 111
Let s = 1. After a finite number n of draws, the probability of Wn = w and Bn = b is
equal to
P (Wn = w,Bn = b) =
(
w−1
W0−1
)(
b−1
B0−1
)(
n+τ0−1
τ0−1
) , (1.5)
where τ0 = W0 + B0. The proofs can be found in the comprehensive treatment of Pólya
urn models [9] by H. M. Mahmoud. The values in the (1.5) are related by w + b = W0 +
B0 + n = τ0 + n. Our formula (1.3) could therefore be tentatively understood as an
outcome of Pólya-Eggenberger process after roughly N + d steps. But the number of steps
in constructing an alternating sum l1−l2+. . .+(−1)d−1ld is d. That d is indeed the correct
number of steps in our process will become even clearer in the proofs of Theorem 3.3 and
Corollary 3.4. These proofs are different from the usual proof of formula (1.5). While
the number of steps in the limit of the Pólya-Eggenberger process is infinite, the number
of steps in our process remains d even after performing the limit. This comes naturally
from the source of our construction which is the nonlinear Fourier transform. The core
of our limiting construction is the replacement of the alternating sum of integers: first, by
alternating sums of rational numbers and then, in the limit, by the alternating sum of real
numbers. This also leads to the geometric expression of our results in terms of the volumes
of the polytopes Dd(λ), mentioned above.
There are other discretization of the beta distribution with useful properties. One pos-
sible approach was studied by A. Punzo in [11]. But, as far as the author is aware, this
discretization does not come from some combinatorial source and is given by a very differ-
ent formula.
The plan of the paper is the following. In section 2, we recall the AKNS-ZS type
of the nonlinear Fourier transform and prepare the necessary formulas. We establish the
connection between F and the family of polytopes {Dd(λ)}. We also introduce the dis-
cretization FN of F . In section 3, we describe and prove the main facts about our central
combinatorial problem: the evaluation of the numbers AQN (L, d), and the derivation of
the generating functions of the numbers AQN (L, d) in terms of FN . In section 4, we prove
proposition 4.2 and theorem 4.3 stated above. Section 5 contains graphs illustrating the re-
lation between the beta distribution and our discrete approximation. We conclude the paper
by mentioning some problems for further research.
2 Nonlinear Fourier transforms and its discretization
We review the definition of the nonlinear Fourier transform F which first appeared in the
work of M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur in [1] and [2] and more
or less simultaneously in the work of V. Zakharov and A. Shabat in [19]. They studied
and solved a certain class of integrable partial differential equations which are now called
AKNS-ZS equations. The acronym is also used to denote the nonlinear Fourier transform
which figures in the AKNS-ZS theory. In this section, we shall also introduce the Euler-
type discretization FN of F .
2.1 Nonlinear Fourier transform of AKNS-ZS type
We shall consider the nonlinear Fourier transform F which appears in the study of the
periodic AKNS-ZS problems. To every well-behaved function u(x) : [0, 1] → C it as-
signs the doubly infinite sequence {F [u](n)}n∈Z of SU(2) matrices, given by F [u](n) =
112 Ars Math. Contemp. 24 (2024) #P1.08 / 107–126
(−1)nΦ(x = 1, n), where Φ(x, n) is the solution of the linear initial value problem
Φx(x, n) = L(x, n) · Φ(x, n), Φ(0, n) = I. (2.1)
The coefficient matrix L(x, n) is given by
L(x, n) =
(
πi n u(x)
−u(x) −πi n
)
.
As we mentioned in the Introduction, we will see that F is of the form
F [u](n) = I +
(
0 F [u](n)
−F [u](−n] 0
)
+
∞∑
d=2
Ad[u](n).
The amount of literature on various aspects of the inverse scattering method is truly vast, so
we shall only mention a few works in which the Fourier analysis aspect is more pronounced.
The foundational work was done by Gardner, Greene, Kruskal and Miura in [8] and [7].
The transform, used in this paper was first constructed by Ablowitz, Kaup, Newell and
Segur, in [1, 2], and simultaneously by Zakharov and Shabat in [19]. Nonlinear Fourier
transforms of functions, defined on R and R+, were studied by I. Gelfan’d, A. Fokas and
B. Pelloni in [5, 6, 10], and in their other works. A version of transformation, closely
related to the one studied in this paper is described by T. Tao and C. Thiele in [15]. Some
aspects of the transformation, defined above, were studied in my papers [12, 13] and [14].
Definition of F , given above is the one that is usually found in the texts which study
the integrable ANKS-ZS equations. We shall rather represent F in a different gauge. Let
G(x, n) = diag(e−πinx, eπinx) be the (diagonal) matrix of our gauge transformation. In
the new gauge Φ is replaced by ΦG = G · Φ and ΦG is the solution of the initial-value
problem
ΦGx (x, n) = L
G(x, z) · ΦG(x, n), ΦG(0, n) = I. (2.2)
The transformed coefficient matrix is then LG(x, n) = Gx ·G−1(x, n)+G(x, n) ·L(x, n) ·
G−1(x, n). Its explicit expression is
LG(x, n) =
(
0 e−2πinxu(x)
−e2πinxu(x) 0
)
. (2.3)
In the new gauge we set FG[u](n) = ΦG(x = 1, n). Since n ∈ Z, the equation
ΦG(1, n) = G(1, n) · Φ(1, n) gives F [u](n) = FG[u](n). The solution to the problem
(2.2) can be given in the form of the Dyson series.
ΦG(x, n) = I +
∞∑
d=1
∫
∆d(x)
LG(x1, n) · LG(x2, n) · · ·LG(xd, n) dx⃗, (2.4)
where ∆d(x) is the ordered simplex of dimension d with the edge length equal to x,
∆d(x) = {(x1, x2, . . . , xd) ∈ Rd;x ≥ x1 ≥ x2 ≥ . . . ≥ xd ≥ 0}.
Let us denote
E(x, n) =
(
eπixn 0
0 e−πixn
)
, J =
(
0 1
−1 0
)
, (2.5)
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 113
and let u(x) be real valued. Then we have LG(x, n) = u(x)E(−2x, n) · J. Matrices
E(x, n) and J do not commute. Instead, they obey the relation
E(x, n) · J = J · E(−x, n). (2.6)
Recall that D̂d(λ) denotes the polytope given by
D̂d(λ) = {(x1, x2, . . . xd) ∈ ∆d(1);
d∑
j=1
(−1)j−1xj = λ},
and Dd(λ) is its projection on the hyperplane xd = 0. These are the polytopes, mentioned
in the introduction. Denote
U(x1, x2, . . . , xd−1;λ) = u(x1) · · ·u(xd−1)u((−1)d−1(λ−
d−1∑
j=1
(−1)j−1xj)),
and let dλx⃗ be the measure on D̂d(λ) ⊂ Rd, inherited from the Euclidean measure on Rd.
Using (2.6) in the Dyson series and evaluating at x = 1 gives
F [u](n) = I +
∞∑
d=1
∫
∆d(1)
u(x1)u(x2) · · ·u(xd)E
(
−2(
d∑
j=1
(−1)j−1xj), n
)
· Jd dx⃗
which, upon setting x1 − x2 + . . .+ (−1)d−1xd = λ, can be rewritten as
F [u](n) = I +
∞∑
d=1
∫ 1
0
E(−2λ, n)
(∫
D̂d(λ)
u(x1)u(x2) · · ·u(xd) dλx⃗
)
· Jd 1√
d
dλ
= I +
∞∑
d=1
∫ 1
0
E(−2λ, n)
(∫
Dd(λ)
U(x1, . . . , xd−1;λ) dx1 · · · dxd−1
)
Jd dλ,
Inserting the constant function uc(x) ≡ u we immediately get the following proposition.
Proposition 2.1. In the case where uc(x) ≡ u is a constant function, we get
F [uc](n) = I +
∞∑
d=1
ud
∫ 1
0
Vol(Dd(λ))E(−2λ, n) · Jd dλ. (2.7)
2.2 Euler-type discretization of F
Many authors studied various discretizations of transformations similar to F , but usually
acting on the functions defined on R or R+, see e.g. [16, 17, 18]. Important are the dis-
cretizations that preserve the integrability of the AKNS-ZS systems. These are constructed
in well known works of M. Ablowitz and J. Ladik and also L. Faddeev and A. Yu Volkov,
see [3, 4]. A discrete nonlinear Fourier transform, similar to the one studied below, was
considered by Tao and Thiele in [15]. In the author’s paper [14] an algorithm for evaluat-
ing the inverse of the nonlinear Fourier transform, defined below, is constructed. (In [14]
a nonlinear Fourier transform of distributions of the form u(x) =
∑N
n=1 un δxn(x) is also
constructed, together with its inverse.)
114 Ars Math. Contemp. 24 (2024) #P1.08 / 107–126
We have obtained the nonlinear Fourier transform from an initial value problem for
a particular first-order linear differential equation. An obvious approach to construct a
discretization is to replace the differential equation with a suitable difference equation. Let
u⃗ = (u0, u1, . . . , uN−1) ∈ RN be a vector which plays a role of a function of a discrete
variable. Let the L-matrix be given by
LN (k, n) =
(
0 e−2πi
kn
N uk
−e2πi knN uk 0
)
.
Definition 2.2. Let k, n ∈ {0, 1, . . . , N − 1}. Discrete nonlinear Fourier transform FN [u⃗]
of u⃗ is defined by FN [u⃗](n) = ΦN (k = N − 1, n), where ΦN is the solution of the
difference initial value problem
ΦN (k + 1, n)− ΦN (k, n)
1
N
= LN (k, n) · ΦN (k, n), ΦN (0, n) = I.
Solving the above initial value problem and evaluating at k = N − 1 gives
FN [u⃗](n) =
0∏
k=N−1
(
I +
1
N
LN (k, n)
)
,
and this can be expanded into
FN [u⃗](n) = I +
N∑
d=1
1
Nd
∑
N−1≥l1>l2>...>ld≥0
LN (l1, n) · LN (l2, n) · · ·LN (ld, n). (2.8)
This expression is a discrete analogue of Dyson’s expansion (2.4).
Let us introduce the notation
Eδ(l, n) = E(l,
n
N
) =
(
eπil
n
N 0
0 e−πil
n
N
)
l, n ∈ {0, 1, . . . , N − 1}
where E is given by (2.5), and the subscript δ refers to the use in the discretized context.
The coefficient matrix LN can be written in the form
LN (l, n) = ul Eδ(−2l, n) · J,
with J defined in (2.5). By means of relation (2.6), we can collect all the copies of J in
(2.8) on the right. Let u⃗c = (u, . . . , u) be a constant vector. We get
FN [u⃗c](n) = I+
N∑
d=1
(
u
N
)d
∑
N−1≥l1>l2>...>ld≥0
Eδ
(
−2(l1− l2+ . . .+(−1)d−1ld), n
)
·Jd.
If we denote L = l1 − l2 + . . .+ (−1)d−1ld, we can finally write
FN [u⃗c](n) = I +
N−1∑
d=1
(
u
N
)d
N−1∑
L=0
Eδ(−2L, n)
∑
(l1,...,ld)∈D̂discd,N (L)
Jd, (2.9)
where
D̂discd,N (L) = {(l1, . . . , ld) ∈ Nd; N − 1 ≥ l1 > . . . > ld ≥ 0,
d∑
j=1
(−1)j−1lj = L}.
(2.10)
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 115
3 Ordered alternating partitions with distinct parts
In this section we introduce the central combinatorial object of the paper, namely the num-
bers AQN (L, d). We establish the connection between the family {AQN (L, d)} and the
discrete nonlinear Fourier transform FN . The transformation FN yields the generating
functions for {AQN (L, d)} separately for even and odd values of d. The main result of the
section is the statement and proof of the explicit formula for the numbers AQN (L, d) and
the evaluation of the probability distribution of these numbers.
Definition 3.1. Let
∆DN,d = {(l1, l2, . . . , ld) ∈ Nd;N − 1 ≥ l1 > l2 > . . . > ld ≥ 0}
be the discrete ordered simplex. Denote by AQN (L, d) the numbers which count the or-
dered alternating partitions of L ∈ N into d distinct parts not greater than N − 1,
AQN (L, d) = ♯{(l1, l2, . . . , ld) ∈ ∆DN,d; l1 − l2 + l3 − . . . (−1)d−1ld = L}. (3.1)
In other words, AQN (L, d) is the number of solutions of the equation
l1 − l2 + l3 − . . .+ (−1)d−1ld = L
where (l1, l2, . . . , ld) is an element of the simplex ∆DN,d.
The next proposition shows that FN [uc](n) can, roughly speaking, be understood as
the discrete linear Fourier transform of the generating polynomial of the finite sequence
{AQN (L, d)}Nd=1.
Let us denote by Fev[uc](n) the upper left entry and by Fodd[uc](n) the upper right
entry of the 2× 2 matrix FN [uc](n).
Proposition 3.2. The power series expansion of FN [uc] around u = 0 is given by
FN [uc](n) = I +
N∑
d=1
(
u
N
)d
N−1∑
L=0
AQN (L, d)Eδ(−2L, n) · Jd. (3.2)
For every L ∈ {0, 1, . . . , N − 1}, the generating polynomials of the numbers
{AQN (L, 2k)}k=1,...,⌊N2 ⌋ and {AQN (L, 2k − 1)}k=1,...,⌊N+12 ⌋
are given by the equations
⌊N2 ⌋∑
k=1
(−1)k( u
N
)2kAQN (L, 2k) =
N−1∑
n=0
e2πi
Ln
N · Fev[uc](n) (3.3)
⌊N+12 ⌋∑
k=1
(−1)k+1( u
N
)2k−1AQN (L, 2k − 1) =
N−1∑
n=0
e−2πi
Ln
N · Fodd[uc](n). (3.4)
116 Ars Math. Contemp. 24 (2024) #P1.08 / 107–126
Proof. Recall formula (2.9)
FN [u⃗c](n) = I +
N−1∑
d=1
(
u
N
)d
N−1∑
L=0
Eδ(−2L, n)
∑
(l1,...,ld)∈D̂discd,N (L)
Jd.
The last sum in the formula yields the constant matrix Jd multiplied by the integer
♯D̂discd,N (L). The number AQN (L, d) is by its definition the number of elements in D̂
disc
d,N (L),
so we have ∑
(l1,...,ld)∈D̂discd,N (L)
Jd = AQN (L, d) · Jd.
Let us now take into account
J2k = (−1)k · I =
(
(−1)k 0
0 (−1)k
)
and
J2k−1 = (−1)k+1 · J =
(
0 (−1)2k−1
−(−1)2k−1 0
)
,
and consider the diagonal and anti-diagonal parts of FN separately. From 3.2, we get two
equations, one for each parity of k:
Fev[uc](n) =
⌊N2 ⌋∑
k=1
(−1)k( u
N
)2k
N−1∑
L=0
e−2πi
Ln
N ·AQN (L, 2k)
Fodd[uc](n) =
⌊N+12 ⌋∑
k=1
(−1)k+1( u
N
)2k−1
N−1∑
L=0
e2πi
Ln
N AQN (L, 2k − 1).
Now, we perform the inverse discrete linear Fourier transforms on both of the above equa-
tions and get the expressions (3.3) and (3.4).
We now state and prove the explicit formula for the function AQN (L, d).
Theorem 3.3. For any N ∈ N, d ≤ N and L ∈ {0, . . . , N − 1}, we have
AQN (L, d) =
{( L−1
⌊ d−12 ⌋
)(
N−L
⌊ d2 ⌋
)
; d even(
L
⌊ d−12 ⌋
)(
N−L−1
⌊ d2 ⌋
)
; d odd.
(3.5)
Above we use the definition of the binomial symbol for which
(
a
b
)
= 0 for negative a.
Proof. Let us define
ÂQN (L, d) = ♯{(l1, . . . , ld);N ≥ l1 > . . . > ld ≥ 1, and
d∑
j=1
(−1)j−1lj = L}.
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 117




                         

                         d                       l1

                                                                  

                      d-1

                                                    l2

                      d-2

                                                                   l3

                                                                                                     

                                                         l4



                                                                    l5                       



                          3                                 l6



                          2                                   l7 

  

                          1                                  l8

                                   



                                    a1    a2   a3      L      b4       b3     b2         b1      N    
Figure 1: Zigzag path interpretation of an element of ÂQN (L, d) with d = 8.
We claim that for ÂQN (L, d) we have
ÂQN (L, d) =
(
L− 1
⌊d−12 ⌋
)(
N − L
⌊d2⌋
)
. (3.6)
Suppose that d = 2k is even. Let us consider the partial sums of the alternating sum
ÂQN (L, d) = (l1 − l2) + (l3 − l4) + . . .+ (ld−1 − ld) = L, (3.7)
namely:
a1 = (l1 − l2)
a2 = (l1 − l2) + (l3 − l4)
...
...
ak−1 = (l1 − l2) + (l3 − l4) + (l5 − l6) + . . .+ (ld−3 − ld−2).
Let us also introduce the integers bm, given by
b1 = (N − l1)
b2 = (N − l1) + (l2 − l3)
...
...
bk = (N − l1) + (l2 − l3) + (l4 − l5) + . . .+ (ld−2 − ld−1)
From the above definitions we see that
l1 = N − b1
l2m = (N − bm)− am
l2m−1 = (N − bm)− am−1.
118 Ars Math. Contemp. 24 (2024) #P1.08 / 107–126
We shall now turn the situation around. Let
1 ≤ α1 < α2 < . . . , < αk−1 ≤ L− 1 (3.8)
be an arbitrary ordered subset of {1, 2, 3 . . . , L− 1} and let
0 ≤ β1 < β2 < . . . < βk ≤ N − L− 1 (3.9)
be an arbitrary ordered subset of {0, 1, 2, . . . , N − L− 1}. Let us define
λ1 = N − β1
λ2m = (N − βm)− αm, m = 1, 2, . . . , k − 1
λ2m−1 = (N − βm)− αm−1, m = 2, 3, . . . , k
From (3.8) and (3.9) we see that
N ≥ λ1 > λ2 > λ3 > . . . > λd−1 > 1
and
λ1 − λ2 + λ3 − . . .+ λd−1 ≥ L+ 1.
Therefore there exists precisely one λd such that
(λ1 − λ2 + λ3 − . . .+ λd−1)− λd = L
From the construction we also see that λd < λd−1.
We have shown that for every choice of a pair (3.8) and (3.9) of subsets of
{1, 2, 3 . . . , L− 1} and {0, 1, 2, . . . , N − L− 1},
respectively, there exists precisely one solution {λ1, λ2, . . . , λd} of the equation (3.7).
Since the number of such pairs is equal to(
L− 1
k − 1
)(
N − L
k
)
=
(
L− 1
⌊d−12 ⌋
)(
N − L
⌊d2⌋
)
,
our proposition is proved for even d. The proof for odd d is only a slight variation of the
above and we shall omit it.
Proof by induction. Our formula can be proved by induction on N . For N = 2, formula
(3.6) can be checked by hand. If we divide the alternating sums from ÂQN (L, d) into
those, for which l1 = N and those for which l1 < N , we get the recursion relation
ÂQN (L, d) = ÂQN−1(L, d) + ÂQN−1(N − L, d− 1).
By the induction hypothesis, the above equation becomes
ÂQN (L, d) =
(
L− 1
⌊d−12 ⌋
)(
N − L− 1
⌊d2⌋
)
+
(
N − L− 1
⌊d−22 ⌋
)(
L− 1
⌊d−12 ⌋
)
=
(
L− 1
⌊d−12 ⌋
)(
N − L
⌊d2⌋
)
,
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 119
and this proves (3.6). The second equality above comes from the recurrence relation of the
Pascal triangle.
Finally, we observe that
AQN (L, d) = ÂQN (L, d), for d even, and AQN (L, d) = ÂQN (L+1, d), for d odd.
These relations, together with formula (3.6), prove the proposition.
Two of the central results of this paper are corollaries of the above theorem.
Corollary 3.4. Let the random variable
XAS [N, d] : ∆
D
d (N) −→ R
defined on the discrete ordered simplex
∆Dd (N) = {(l1, l2, . . . , ld); N − 1 ≥ l1 > l2 > . . . > ld ≥ 0}
be given by
XAS [N, d](l1, l2, . . . , ld) = l1 − l2 + l3 − . . .+ (−1)(d−1)ld.
Then its probability mass function is
P (XAS [N, d] = L) =

(
L−1
⌊ d−1
2
⌋)(
N−L
⌊ d
2
⌋ )
(Nd)
; d even
( L⌊ d−1
2
⌋)(
N−L−1
⌊ d
2
⌋ )
(Nd)
; d odd.
Proof. The number of the favourable events is given by Theorem 3.3, proved above. To
evaluate the number of all outcomes it helps to consider Figure 1. We see that the number
of all outcomes is equal to the number of the subsets which are composed of all the integer
points ai, all the points bi and the point L. These are precisely all the subsets with d
elements in the set {1, 2, . . . , N}. Their number is of course
(
N
d
)
. This proves our corollary.
Inserting the formula (3.5) in the expressions (3.3) and (3.4) yields the following corol-
lary:
Corollary 3.5. The power series of FN [uc](n) around u = 0 is given by
FN [uc](n) = I +
⌊N2 ⌋∑
k=1
(−1)k( u
N
)2k
N−1∑
L=0
(
L− 1
k − 1
)(
N − L
k
)
· Eδ(−2L, n)
+
⌊N+12 ⌋∑
k=0
(−1)k( u
N
)2k+1
N−1∑
L=0
(
L
k
)(
N − L− 1
k
)
· Eδ(−2L, n) · J.
120 Ars Math. Contemp. 24 (2024) #P1.08 / 107–126
4 Beta distribution and polytopes Dd(λ)
In this section we prove our second theorem which is the expression of the volumes of
polytopes Dd(λ) in terms of the probability density function of the beta distribution. We
obtain this result by taking a suitable limit of the probability mass functions of the random
variables XAS [N, d].
4.1 Discretization of beta distribution
The subset D̂discd,N (L), given by (2.10) of the discrete ordered simplex
∆discd,N = {(l1, l2, . . . , ld) ∈ (N ∪ {0})d;N − 1 ≥ l1 > l2 > . . . > ld ≥ 0}
with the edge of size N is given by one equation. The size AQN (L, d) of D̂discd,N (L) is
therefore of the order Nd−1.
Lemma 4.1. Let λ be a real number in [0, 1] and let {LN}N∈N be a sequence of positive
integers such that LN < N and limN→∞ LNN = λ. Then we have
lim
N→∞
AQN (LN , d)(
N
d
) N = {pβ(λ; d2 , d2 + 1) ; d even
pβ(λ;
d+1
2 ,
d+1
2 ) ; d odd.
(4.1)
where
pβ(λ; a, b) =
1
B(a+ 1, b+ 1)
λa(1− λ)b
is the probability density function of the beta distribution Beta(λ; a, b).
Proof. We shall prove the formula only for even d. The proof for odd d is essentially
the same. Consider first the numerator of the quotient under the limit. For d = 2m,
formula (3.5) gives
AQN (LN , d) =
(
LN − 1
m− 1
)(
N − LN
m
)
.
This expression can be expanded into
AQN (LN , d) =
1
(m− 1)!m!
m−2∏
k=0
((LN − 1)− k)
m−1∏
k=0
((N − LN )− k). (4.2)
Consider the first product above. It is a polynomial of degree m− 1 in the variable (LN −
1) = (N LNN − 1). Expanding this polynomial gives
(N
LN
N
− 1)m−1 +
m−2∑
k=1
(N
LN
N
− 1)k · n(k) = (N LN
N
− 1)m−1 +O(N LN
N
)m−2
For large N we can replace LNN by λ. Taking into account also the second product, (4.2)
gives
AQN (LN , d) =
1
(m− 1)!m!
(
(N
LN
N
− 1)m−1 +R1
)(
(N −N LN
N
)m +R2
)
=
1
(m− 1)!m!
(
Nm+m−1(
LN
N
)m−1(1− LN
N
)m +R3
)
(4.3)
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 121
where
R1 = R3 = O(
1
(Nλ)m−2
) and R2 = O(
1
(Nλ)m−1
).
For the denominator
(
N
d
)
we have(
N
d
)
=
1
d!
(
N(N − 1) · · · (N − (d− 1))
)
=
Nd
d!
+O( 1
N (d−2)
). (4.4)
Because d− 1 = m+ (m− 1) and limN→∞ LNN = λ formulas (4.3) and (4.4) yield
lim
N→∞
AQN (LN , d)(
N
d
) N = d!
(m− 1)!m!
λm−1λm.
The definition of the Euler beta function for positive integers gives d!(m−1)!m! =
1
B(m,m+1) ,
and this proves formula (4.1) for even d.
The above calculation suggests the definition of a discrete version BetaN (a, b) of beta
distribution for arbitrary choice of the shape parameters. Let a, b and N be integers. Let
the probability mass function of BetaN (a, b) be defined by
PN (L; a, b) =
(
L−1
a
)(
N−L
b
)(
N
a+b+1
)
for L ∈ {1, 2, . . . , N}.
Proposition 4.2. Let λ be an arbitrary real number in the unit interval [0, 1] and let
{LN}N∈N be a sequence, such that LN < N and limN→∞ LNN = λ. Then
lim
N→∞
PN (
LN
N
, a, b)N = pβ(λ; a, b) =
1
β(a+ 1, b+ 1)
λa(λ− 1)b
Proof. The proof is an obvious adaptation of the proof of Lemma 4.1. We only have to
replace the particular values m − 1 and m of the shape parameters by an arbitrary pair a
and b of positive integers. Then the same calculations as those performed in the proof of
Lemma 4.1 yield the proof of the proposition.
4.2 Volumes of polytopes Dd(λ)
Recall formula (2.7):
F [uc](n) = I +
∞∑
d=1
ud
∫ 1
0
Vol(Dd(λ))
(
e−2πiλn 0
0 −e2πiλn
)
·
(
0 1
−1 0
)d
dλ.
We have the following theorem.
122 Ars Math. Contemp. 24 (2024) #P1.08 / 107–126
Theorem 4.3. For every dimension d, the volumes of polytopes Dd(λ) are essentially dis-
tributed according to the beta distribution with the shape parameters (d2 ,
d
2 + 1), if d is
even, and (d+12 ,
d+1
2 ), if d is odd. More concretely, we have the following expression:
Vol(Dd(λ)) =
1
d!

1
B( d2 ,
d
2+1)
λ
d
2−1(1− λ) d2 = pβ(λ; d2 ,
d
2 + 1); d even
1
B( d+12 ,
d+1
2 )
λ
d−1
2 (1− λ) d−12 = pβ(λ; d+12 ,
d+1
2 ); d odd,
(4.5)
where pβ(λ; a, b) is the probability density function of the distribution with shape parame-
ters a and b.
Proof. Recall the set D̂discd,N (L), given by (2.10). Rescaling it by the factor 1/N gives the
set
D̂discd (
L
N
) = {( l1
N
,
l2
N
. . . ,
ld
N
);
N − 1
N
≥ l1
N
> . . . >
ld
N
≥ 0,
d∑
j=1
(−1)j−1lj = L}
which contains the same number of points as D̂discd,N (L), but lies in the polytope D̂d(
L
N ).
Let Ddiscd (
L
N ) denote the orthogonal projection of D̂
disc
d (
L
N ) on the hyperplane
{(x1, . . . , xd−1, 0)} ⊂ Rd.
The number ♯Ddiscd (
L
N ) of points in D
disc
d (
L
N ) is clearly equal to the number of points in
D̂discd (
L
N ). The value
1
Nd−1
♯Ddiscd (
L
N ) is approximately equal to the volume Vol(Dd(
L
N ))
of the projection Dd( LN ) of D̂d(
L
N ) on the hyperplane xd = 0 in R
d. So, on the one hand,
the number ♯Ddiscd (
L
N ) is equal to AQN (L, d), while on the other, the value
1
Nd−1
♯Ddiscd (
L
N )
is an approximation of Vol(Dd(L)). Let now {λN}N∈N be a sequence of rationals LNN
converging to λ ∈ [0, 1]. We have
lim
N→∞
1
Nd−1
AQN (NLN , d) = Vol(Dd(λ)).
In the proof of Lemma 4.1 we have seen that(
N
d
)
=
Nd
d!
+O( 1
N (d−2)
),
so (
N
d
)
N
=
N (d−1)
d!
+O( 1
N (d−3)
).
Therefore
lim
N→∞
1
Nd−1
AQN (NLN , d) = lim
N→∞
AQN (LN , d)(
N
d
) N.
This equality, together with Lemma 4.1, proves our theorem.
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 123
5 Quantitative comparisons
In this section we shall investigate by experimental means the comparison between the
probability density function of Beta(l; a, b) distribution and its approximations, given by
the probability mass functions PN (l; a, b). For the sake of brevity we shall concentrate
on the shape parameters (a, b) = (m − 1,m) which appear in connection with the non-
linear Fourier transform. It is now clear that absolute value of the difference pβ(l; a, b) −
PN (l; a, b) decreases for every l = LNN as N increases. But the quality of the approx-
imation also depends crucially on the choice of the shape parameters. We shall see that,
roughly speaking, the value |pβ(l; a, b)−PN (l; a, b)| at a fixed N , increases with increasing
of a+ b. Explicit formula for this difference can be deduced from formulas (4.1) and (4.2),
but it is quite complicated. The images will provide a better illustration of the relations
between pβ(l; a, b) and PN (l; a, b).
The two images in Figure 2 show the comparison between pβ(l; 21, 22) and PN (l; 21, 22)
for N = 200 and N = 1000.
0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
6
(a) N = 200, a = 21, b = 22
0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
(b) N = 1000, a = 21, b = 22
Figure 2: Comparison of graphs.
Figure 3 shows that for any choice of the shape parameters the difference PN (l; a, b)−
pβ(l; a, b)) has three local extrema. For the cases, related to the number of alternating
partitions of integers where a = b − 1 or a = b, the maximum is located roughly at the
center of the interval [0, 1].
0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.2
0.4
0.6
0.8
(a) N = 200, a = 21, b = 22
0.2 0.4 0.6 0.8 1.0
-0.1
0.1
0.2
(b) N = 1000, a = 21, b = 22
Figure 3: The shape of the difference.
124 Ars Math. Contemp. 24 (2024) #P1.08 / 107–126
The two images in Figure 4 illustrate the dependence of the difference pβ(l; a, b) −
PN (l; a, b) on the size of the shape parameters. Again we consider (a, b) = (a, a+ 1). We
see, that at a fixed N the difference increases with increasing of the shape parameter a.
(a) N = 100, a ∈ [3, 20] (b) N = 1000, a ∈ [3, 20]
Figure 4: Dependence of the difference on the size of the shape parameter.
Even if the shape parameters a and b are very different, the corresponding graphs are of
similar shapes to the above. The only difference is that, in case where the shape parameters
a and b are significantly different the peaks of the graphs are shifted away from the center.
This is clear from the following fact. Suppose that a is considerably larger than b. Then
the left zero of limit function pβ(l; a, b) = 1B(a+1,b+1) l
a(1 − l)b is of higher degree than
the right one. The function is therefore flatter and closer to zero in the vicinity of 0 and
the peak of the graph is pushed towards the right. Qualitatively the shape of the difference
does not change.
6 Conclusions and outlook
In the paper we arrived at the construction of a discrete probability distribution with proba-
bility mass function PN (l; a, b) which converges to the probability density function
pβ(l; a, b) as N → ∞. The result is precisely stated in Proposition 4.2. Crucial in the
construction is the connection of PN (l; a, a) and PN (l; a−1, a) to the following combina-
torial problem: find the number AQN (L, d) of alternating ordered partitions of the positive
integer L < N into d distinct parts, not greater than N − 1. The number AQN (L, d) can
also be represented by the number of the zig-zag paths, drawn in Figure (1). This combina-
torial problem naturally appeared in the context of the discretization FN of the nonlinear
Fourier transform F , described in Section 2. The essential connection between the num-
bers AQN (L, d) and FN is given by Proposition 3.2 where we show that the inverse linear
Fourier transform of the entries of FN yields the generating polynomials of the numbers
AQN (L, d).
The formula for distribution PN (l; a, b) can also be interpreted as the distribution de-
scribing the Pólya-Eggenberger urn, but this interpretation is different from ours. We have
the connection of PN (l; a, b) to the combinatorial problem and the nonlinear Fourier trans-
form only for the shape parameters of the form (a, b) = (a, a) or (a, b) = (a − 1, a).
The natural question arises: can we find a combinatorial problem whose relation with
PN (l; a, b) for an arbitrary choice of a and b would be analogous to the relation be-
P. Saksida: On the beta distribution, the nonlinear Fourier transform and . . . 125
tween PN (l; a − 1, a) and PN (l; a, a) and the problem of alternating ordered partitions
AQN (L, d)? Does there exist a meaningful generalisation Fa,b of the nonlinear Fourier
transform F , whose relation with pβ(x; a, b) would be analogous to the relation between
F and pβ(x; a, a) and pβ(x; a − 1, a), described in theorem 4.3. These are the natural
problems for further investigation, based on this paper. Finding answers to these questions
would importantly improve understanding the nonlinear Fourier transform and its structure.
In this paper, we considered the nonlinear Fourier transform F [u] evaluated on the
simplest of functions, namely, the constant function u ≡ c. An obvious direction of further
research is to try to extend the approach used in this paper, to the context, where F [u] is
evaluated on some more interesting class of functions u.
ORCID iDs
Pavle Saksida https://orcid.org/0000-0003-3093-9863
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.09 / 127–135
https://doi.org/10.26493/1855-3974.2975.1b2
(Also available at http://amc-journal.eu)
Products of subgroups, subnormality,
and relative orders of elements
Luca Sabatini *
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
Reáltanoda utca 13-15, H-1053, Budapest, Hungary
Received 6 October 2022, accepted 22 February 2023, published online 31 August 2023
Abstract
Let G be a group. We give an explicit description of the set of elements x ∈ G such
that x|G:H| ∈ H for every subgroup of finite index H ⩽ G. This is related to the following
problem: given two subgroups H and K, with H of finite index, when does |HK : H|
divide |G : H|?
Keywords: Relative order, product of subgroups, subnormal subgroup.
Math. Subj. Class. (2020): 20D40, 20D25, 20F99.
1 Introduction
Let G be an arbitrary group, and let us write H ⩽f G to say that H is a subgroup of G
of finite index. Let x ∈ G and H ⩽f G. If H is a normal subgroup of G, then it is
easy to see that x|G:H| ∈ H . The same is not true in general: fixed H ⩽f G, the set
{x ∈ G : x|G:H| ∈ H} may not even be closed under multiplication (take G = Sym(3)
and H = ⟨(1 2)⟩). The goal of this paper is to understand this phenomenom and its
implications. As far as we can see, this has not been dealt with before in the literature.
Definition 1.1. Let x ∈ G and H ⩽ G. The relative order of x with respect to H is
oH(x) := |⟨x⟩ : ⟨x⟩ ∩H|.
The following result is proved in Section 2.
Lemma 1.2. Let n ≥ 1. Then xn ∈ H if and only if oH(x) is finite and divides n.
*The author thanks Bob Guralnick and Orazio Puglisi for useful conversations.
E-mail address: sabatini.math@gmail.com (Luca Sabatini)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
128 Ars Math. Contemp. 24 (2024) #P1.09 / 127–135
Given H,K ⩽ G, |HK : H| is the cardinality of the set of all cosets of H which are
intersected by K (we refer to Section 2 for more details). Since oH(x) = |H⟨x⟩ : H|, we
obtain
Corollary 1.3. x|G:H| ∈ H if and only if |H⟨x⟩ : H| divides |G : H|.
If H,K ⩽f G, then |HK : H| divides |G : H| if and only if |HK : K| divides
|G : K|. If G is finite, both are equivalent to |HK| dividing |G|. In Section 3, we prove
the following two results:
Proposition 1.4. Let H ◁◁G. Then |HK : K| divides |G : K| for every K ⩽f G.
Theorem 1.5. Let H ⩽f G. Then H ◁ ◁ G if and only if |HK : H| divides |G : H| for
every K ⩽ G.
The converse of Proposition 1.4 is not true in general (see Example 5.11). In particular,
some attention is needed with subgroups of infinite index. During the preparation of this
manuscript, the author found out that the finite version of Theorem 1.5 already appeared in
[5, Theorem 2]. In Section 4, we study the following class of subgroups
Definition 1.6. A subgroup H ⩽f G is exponential if x|G:H| ∈ H for every x ∈ G.
This is a generalization of subnormality, and we prove that it is equivalent to normal-
ity in some cases, namely for the Hall subgroups of a finite group and for the maximal
subgroups of a solvable group. From the dual point of view, in Section 5 we study the set
S(G) := {x ∈ G : x|G:H| ∈ H for every H ⩽f G}.
At first glance S(G) is quite elusive, and indeed working directly with the definition is
not easy. Using the results of Section 3, we give an elementary proof of the next theorem.
Given N ◁ G, let FN (G) be the preimage of F (G/N), where F (G) denotes the Fitting
subgroup of G.
Theorem 1.7. If G is any group, then S(G) = ∩N◁fGFN (G).
In particular, S(G) = F (G) when G is finite (Proposition 5.1). Of course, Theorem 1.7
implies that S(G) is closed under multiplication, a fact which is not immediately clear from
the definition.
2 Preliminaries
We start with the proof of the key Lemma 1.2.
Proof of Lemma 1.2. Let ordH(x) := min{n ≥ 1 : xn ∈ H}. We first notice that
oH(x) = ordH(x). Indeed, from the definitions we have oH(x) = oH∩⟨x⟩(x) and
ordH(x) = ordH∩⟨x⟩(x). The fact that oH∩⟨x⟩(x) = ordH∩⟨x⟩(x) is a simple exercise.
Now, the “if” part of the statement is trivial. On the other hand, if xn ∈ H for some n ≥ 1,
then clearly ordH(x) < ∞. Let n = q · ordH(x) + r with r, q ≥ 0 and r < ordH(x).
Since H is a subgroup, the fact that xn = xq·ordH(x)xr ∈ H implies that xr ∈ H , which
in turn means r = 0.
L. Sabatini: Products of subgroups, subnormality, and relative orders of elements 129
The bulk of this paper is about finite groups. We summarize here the basic tools and
notation that are used with regard to general non-finite groups. Let G be an arbitrary group
and H,K ⩽ G. If H and K have finite index, then so has H ∩ K, and |G : H ∩ K| =
|G : H||H : H ∩ K|. As we have said in the introduction, we write |HK : H| for the
cardinality of the set of all cosets of H which are intersected by K. This is not accidental,
because the product set HK = {hk : h ∈ H, k ∈ K} is a union of cosets of H . It is not
relevant to distinguish between left-cosets and right-cosets, since k ∈ Hx if and only if
k−1 ∈ x−1H . We also observe that |HK : H| = |K : H ∩K| = |KH : H|.
The finite residual R(G) is the intersection of the subgroups of G of finite index. If R(G) =
1, then G is said to be residually finite. It is easy to check that G/R(G) is always residually
finite. Finally, the Fitting subgroup F (G) is defined as the subgroup generated by the
nilpotent normal subgroups, and coincides with the set of the elements x ∈ G such that the
normal closure ⟨x⟩G is nilpotent [1]. In general, this is a stronger condition than ⟨x⟩ being
subnormal in G. If G is finite, then F (G) itself is nilpotent, i.e. it is the largest nilpotent
normal subgroup.
3 Products of subgroups
The proof of Proposition 1.4 follows immediately from the following
Lemma 3.1. Let H ◁M ⩽ G, and let K ⩽f G. Then |HK : K| divides |MK : K|.
Proof. We have to prove that the ratio
|MK : K|
|HK : K|
=
|M : M ∩K|
|H : H ∩K|
is an integer. Now H ◁ M implies that H(M ∩ K) is a subgroup of M , and so we can
write
|M : M ∩K| = |M : H(M ∩K)||H(M ∩K) : M ∩K|
= |M : H(M ∩K)||H : H ∩K|.
In particular, the original ratio equals |M : H(M ∩K)|.
We continue with the easiest direction of Theorem 1.5.
Lemma 3.2. Let H ⩽f M ⩽f G, and let K ⩽ G. Then |HK:H||MK:M | = |M ∩K : H ∩K|.
Proof. We have
|HK : H|
|MK : M |
=
|K : H ∩K|
|K : M ∩K|
=
|K : M ∩K||M ∩K : H ∩K|
|K : M ∩K|
= |M ∩K : H ∩K|.
We prove the claim of Theorem 1.5 by induction on the subnormal defect of H , so let
H ◁f M ◁◁f G, and K ⩽ G. Using Lemma 3.2, we have
|G : H|
|HK : H|
=
|G : M ||M : H|
|MK : M ||M ∩K : H ∩K|
.
130 Ars Math. Contemp. 24 (2024) #P1.09 / 127–135
By induction, it is sufficient to prove that |M :H||M∩K:H∩K| is an integer. Now H ◁M implies
that H(M ∩K) is a subgroup of M , and so we can write
|M : H| = |M : H(M ∩K)||H(M ∩K) : H|
= |M : H(M ∩K)||M ∩K : H ∩K|.
This concludes the proof of the “only if” part.
3.1 The Kegel-Wielandt-Kleidman theorem, revisited
Definition 3.3. Let G be a finite group, H ⩽ G, and let p be a prime. Then H is p-
subnormal in G if H ∩ P is a p-Sylow of H for every p-Sylow P of G.
We characterize p-subnormality with the following
Lemma 3.4. A subgroup H is p-subnormal if and only if |HP | divides |G| for every p-
Sylow P ⩽ G.
Proof. We have that H ∩ P is a p-Sylow of H if and only if |H : H ∩ P | = |HP : P | is
not divisible by p. Since |H : H ∩ P | is a divisor of |G|, the last condition is equivalent to
|HP : P | dividing |G : P |, i.e. |HP | | |G|.
The famous Kegel-Wielandt conjecture [3, 7], proved by Kleidman [4] using the clas-
sification of the finite simple groups, says that H ◁ ◁ G whenever H is p-subnormal for
every p.
Theorem 3.5 (Kegel-Wielandt conjecture). If |HP | divides |G| for every Sylow subgroup
P ⩽ G, then H ◁◁G.
See [2] for some consequences of p-subnormality for a single p. The “if” part of Theo-
rem 1.5 follows easily. Let H ⩽f G, and assume that |HK : H| divides |G : H| for every
K ⩽ G. Let N ◁f G be the normal core of H , and let N ⩽ K ⩽ G be any intermediate
subgroup. Working with G/N and K/N , Theorem 3.5 gives H/N◁◁G/N , i.e. H◁◁G.
We point out that Kegel [3] did not use the classification to prove Theorem 3.5 when H
is solvable. We give a very short proof in the case where H is nilpotent, which is enough
for the characterization of S(G) we will present in Section 5.
Lemma 3.6 (Kegel-Wielandt for nilpotent subgroups). Let H ⩽ G be a nilpotent subgroup
of the finite group G. If |HP | divides |G| for every Sylow subgroup P ⩽ G, then H◁◁G.
Proof. Suppose that H is not subnormal, and in particular H ⩽̸ F (G). So there exists a
p-element x such that x ∈ H \ F (G). Since x /∈ Op(G), there exists a p-Sylow P of G
such that x /∈ P . By hypothesis H ∩P is a p-Sylow of H and, since H is nilpotent, H ∩P
contains all the p-elements of H . This contradicts the fact that x /∈ P .
Levy [5] proves the same result when H is a p-subgroup of G. Another consequence
of Theorem 1.5 is that p-subnormality for every p implies that |HK| divides |G| for every
K ⩽ G. We provide an elementary proof of this fact.
Lemma 3.7. Let G be a finite group and H ⩽ G. If |HP | divides |G| for every Sylow
P ⩽ G, then |HK| divides |G| for every K ⩽ G.
L. Sabatini: Products of subgroups, subnormality, and relative orders of elements 131
Proof. Let K ⩽ G. We have to show that |HK : K| = |H : H ∩K| divides |G : K|. Let
pα be a prime power that divides |H : H ∩K|. Since pα is arbitrary, it is sufficient to prove
that pα | |G : K|. Let P0 ⩽ K be a p-Sylow of K, and let P ⩽ G be a p-Sylow of G such
that P ∩K = P0. Of course, pα | |H : H ∩ P0|. By hypothesis |H : H ∩ P | = |HP : P |
divides |G : P |, and so is not divisible by p. Therefore, pα | |H ∩ P : H ∩ P0|. Now
|H∩P : H∩P0| = |(H∩P )P0 : P0|, and this divides |P : P0| because P is a p-group. So
pα | |P : P0|, and then of course pα | |G : P0|. Since p ∤ |K : P0|, we obtain pα | |G : K|
as desired.
4 Exponential subgroups
We write H ⩽exp G if x|G:H| ∈ H for all x ∈ G. We observe immediately that exponen-
tiality is preserved by quotients.
Lemma 4.1. Let N ◁ G, and N ⩽ H ⩽ G. Then H ⩽exp G if and only if H/N ⩽exp
G/N .
Proof. Let x ∈ G and H ⩽exp G. Then (Nx)|G/N :H/N | = Nx|G:H| ∈ H/N and so
H/N ⩽exp G/N . If H/N ⩽exp G/N , then Nx|G:H| = (Nx)|G/N :H/N | ∈ H , and so
x|G:H| ∈ H .
Since exponential subgroups have finite index, we can apply Lemma 4.1 with the nor-
mal core, and work with a finite group. Let G be a finite group and H ⩽ G. From
Corollary 1.3 and Theorem 1.5, we have
• H ◁◁G if and only if |HK| divides |G| for every K ⩽ G;
• H ⩽exp G if and only if |HC| divides |G| for every cyclic C ⩽ G.
We stress that H ⩽exp G whenever |G : H| is a multiple of the exponent exp(G).
Remark 4.2. Every finite group of order other than a prime has a non-trivial exponential
subgroup: if exp(G) < |G|, then it is sufficient to take any subgroup whose order divides
|G|/ exp(G). Otherwise, all the Sylow subgroups of G are cyclic, and it is well known
that G is solvable. In particular, G has a non-trivial normal subgroup, which is certainly
exponential.
We notice a difference with the stronger condition that HK is a subgroup for every K i.e.
H is a permutable subgroup. Indeed, it is easy to prove that if HC is a subgroup for every
cyclic C ⩽ G, then HK is a subgroup for every K ⩽ G.
For every n ≥ 1, let Gn := ⟨{xn : x ∈ G}⟩. The exponential subgroups of G of index
n are in correspondence with the subgroups of G/Gn of index n. Since Gn is characteristic,
the property of being exponential is preserved by automorphisms. Moreover, we have the
following
Lemma 4.3. Let H ⩽ G have a trivial characteristic core. Then H ⩽exp G if and only if
|G : H| is a multiple of the exponent of G.
Proof. Let n = |G : H|. By the exponentiality of H we have Gn ⩽ H . Since Gn is a
characteristic subgroup of G contained in H , we obtain Gn = 1. But this means exactly
that n is a multiple of exp(G). The converse is trivial.
132 Ars Math. Contemp. 24 (2024) #P1.09 / 127–135
In general, there exist non-subnormal exponential subgroups whose index is not a mul-
tiple of the exponent. A simple example is G = C4 × Sym(3) and H ∼= C2 × C2. The
following corollaries of Lemma 4.3 are obtained with the same strategy.
Corollary 4.4. Let H ⩽ G be a Hall subgroup. If H ⩽exp G, then H ◁G.
Proof. Suppose that H is not normal, and let N ◁G be the normal core of H . Since H/N
is a Hall subgroup of G/N , by induction and Lemma 4.1, we can assume that H is core-
free. Now exp(G) captures every prime dividing |G|, and so the contradiction is given by
Lemma 4.3.
Corollary 4.5. Let M ⩽ G be a maximal subgroup of the solvable group G. If M ⩽exp G,
then M ◁G.
Proof. Suppose that M is not normal, and let N◁G be the normal core of M . Since M/N
is a maximal subgroup of G/N , by induction and Lemma 4.1, we can assume that M is
core-free. Now |G : M | = qα for some prime power qα. If G is a q-group we are done.
Otherwise, the contradiction is given by Lemma 4.3.
We cannot drop the hypothesis of solvability in Corollary 4.5: the alternating group
G = Alt(10) has a conjugacy class of maximal subgroups M of size 720. Since exp(G) =
2520 = |G : M |, it appears that M is an exponential maximal subgroup which is not
normal.
We conclude this section with the hereditary properties of exponential subgroups.
Lemma 4.6. The following are true:
• If H ⩽exp M ⩽exp G, then H ⩽exp G;
• The intersection of exponential subgroups is exponential.
Proof. Let x ∈ G. Since M ⩽exp G, we have m = x|G:M | ∈ M . Then x|G:H| =
m|M :H| ∈ H . To prove the second statement, it is sufficient to notice that |G : H ∩K| is
a multiple of both |G : H| and |G : K|.
Other important properties of the lattice of the subnormal subgroups are not true for
exponential subgroups, and the dihedral group G = D12 is a good source of counterexam-
ples. Every subgroup of G whose order is 2 is exponential in G, since exp(G) = 6. Let H
be any non-central subgroup of order 2. Now
• The subgroup H1 = ⟨H,Z(G)⟩ ∼= C2 × C2 provides a counterexample to the state-
ment that two exponential subgroups generate an exponential subgroup: choosing
any involution x ∈ G \H1 we get x|G:H1| = x /∈ H1.
• The subgroup H2 which satisfies H < H2 ∼= Sym(3) provides a counterexample to
the statement that the intersection of an exponential subgroup of G with any subgroup
of G is exponential in that subgroup: choosing any involution x ∈ H2 \ H , we get
that H is not exponential in H2 although it is exponential in G.
L. Sabatini: Products of subgroups, subnormality, and relative orders of elements 133
5 The set S(G)
Let us recall the definition of S(G) given in the introduction:
S(G) := {x ∈ G : x|G:H| ∈ H for every H ⩽f G}.
From Corollary 1.3, we have
S(G) = {x ∈ G : |H⟨x⟩ : H| divides |G : H| for every H ⩽f G}.
The results of Section 3 allow to settle the finite case easily:
Proposition 5.1. If G is finite, then S(G) = F (G).
Proof. Let x ∈ G. Then x ∈ S(G) if and only if |H⟨x⟩| divides |G| for every H ⩽ G.
From Proposition 1.4 and Lemma 3.6, this is equivalent to ⟨x⟩◁◁G, i.e. x ∈ F (G).
5.1 A top-down approach
Let G be an arbitrary group and let R(G) = ∩H⩽fGH be its finite residual. The condition
in the definition of S(G) is empty on R(G), and so R(G) ⊆ S(G). In fact, S(G) is the
preimage of S(G/R(G)) under the projection G ↠ G/R(G).
Lemma 5.2. Let N ◁G. Then S(G/N) = {Nx : x|G:H| ∈ H for every N ⩽ H ⩽f G}.
In particular, S(G/R(G)) = S(G)/R(G).
Proof. Let x ∈ G and N ⩽ H ⩽f G. The equality (Nx)|G:H| = Nx|G:H| implies that
Nx ∈ H/N if and only if x|G:H| ∈ H , and the first part follows because H is arbitrary.
The second part follows because R(G) contains all the subgroups of G of finite index.
As a consequence of Lemma 5.2, we can assume that G is residually finite. Given
N ◁G, let FN (G) be the preimage of F (G/N).
Proof of Theorem 1.7. We have to prove that S(G) = ∩N◁fGFN (G). Let x ∈ S(G) and
N ◁f G. From Lemma 5.2 and Proposition 5.1 we have Nx ∈ S(G/N) = F (G/N), i.e.
x ∈ FN (G).
On the other hand, let x ∈ ∩N◁fGFN (G) and H ⩽f G. If N ◁f G is the normal core of
H , then in particular x ∈ FN (G). From Proposition 5.1 we have
Nx ∈ FN (G)
N
= F (G/N) = S(G/N),
and so Lemma 5.2 provides x|G:H| ∈ H . The proof follows because H is arbitrary.
The following observation deletes a bunch of terms from ∩N◁fGFN (G).
Lemma 5.3. Let G be a finite group and N ◁G. Then F (G) ⩽ FN (G).
Proof. We have that NF (G)/N ∼= F (G)/(N ∩ F (G)) is a nilpotent normal subgroup of
G/N . Then NF (G)/N ⩽ F (G/N) = FN (G)/N , and so NF (G) ⩽ FN (G).
Corollary 5.4. If N,K ◁f G and K ⩽ N , then FK(G) ⩽ FN (G).
As a particular case of Theorem 1.7, we have
134 Ars Math. Contemp. 24 (2024) #P1.09 / 127–135
Proposition 5.5. Let G be a group. The following are equivalent:
(A) G = S(G);
(B) every subgroup of finite index of G is exponential;
(C) every finite quotient of G is nilpotent;
(D) every subgroup of finite index of G is subnormal.
Proof. This follows easily from Theorem 1.7.
We say that a group G is S-free if S(G) = 1.
Lemma 5.6. Let G be a group which is residually S-free. Then S(G) = 1.
Proof. Let 1 ̸= x ∈ G. By definition, there exists N ◁G such that x /∈ N and S(G/N) =
1. In particular Nx /∈ S(G/N), and so from Lemma 5.2 we obtain x /∈ S(G). Since x is
arbitrary, it follows that S(G) = 1.
Corollary 5.7. If F is a finitely generated free group, then S(F ) = 1.
5.2 Baer groups and S-groups
Following a different approach, now we study S(G) starting from the subgroups of G. This
will provide a counterexample to the converse of Proposition 1.4.
Let B(G) := {x ∈ G : ⟨x⟩ ◁ ◁ G} be the Baer radical of G. It is clear that B(G) is
a characteristic subgroup. Moreover, B(G) coincides with F (G) if G is finite, but it can
be much larger in general (see [1, Example 85]). A group which equals its Baer radical
is called a Baer group. The same argument in the proof of Proposition 5.1 shows that
B(G) ⊆ S(G). We say that a group is an S-group if it satisfies the equivalent conditions of
Proposition 5.5. It is easy to see that the class of S-groups is closed by subgroups of finite
index and quotients. Of course, every Baer group is an S-group.
Proposition 5.8 (Theorem 73 in [1]). A group is a Baer group if and only if every its finitely
generated subgroup is subnormal and nilpotent. In particular, every finitely generated Baer
group is nilpotent.
By Propositions 5.5 and 5.8, every finitely generated non-nilpotent p-group is an S-
group which is not Baer. The next theorem of Wilson [8] provides many groups with trivial
Baer radical. We recall that an infinite group is just-infinite if every its proper quotient is
finite.
Theorem 5.9 (Theorem 2 in [8]). Let G be a just-infinite group. If B(G) ̸= 1, then B(G)
is a free abelian group of finite rank, which coincides with its own centralizer in G.
Lemma 5.10. Let G be a just-infinite p-group. Then S(G) = G, but B(G) = 1.
Proof. The fact that G = S(G) follows from Proposition 5.5 and the fact that finite p-
groups are nilpotent. If B(G) ̸= 1, then B(G) is a free abelian group by Theorem 5.9,
which contraddicts that G is a p-group.
L. Sabatini: Products of subgroups, subnormality, and relative orders of elements 135
Example 5.11 (No converse to Proposition 1.4). Let G be a just-infinite p-group, and let
K ⩽ G be any nilpotent subgroup. Since every subgroup of finite index of G is subnormal,
from Theorem 1.5 we have that |HK : H| divides |G : H| for every H ⩽f G. On the
other hand, K is not subnormal in G, because B(G) = 1.
Finally, it is worth to mention the following theorem of Robinson [6]. Given a group
property P , a group is hyper-P if every its non-trivial homomorphic image has some non-
trivial normal subgroup with the property P .
Theorem 5.12 (Theorem 1 in [6]). Let G be a finitely generated hyperabelian or hyperfinite
group. If G is an S-group, then G is nilpotent.
ORCID iDs
Luca Sabatini https://orcid.org/0000-0002-4781-5579
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.10 / 137–153
https://doi.org/10.26493/1855-3974.2826.3dc
(Also available at http://amc-journal.eu)
Coincident-point rigidity in normed planes
Sean Dewar *
School of Mathematics, University of Bristol, Bristol BS8 1UG, U.K
John Hewetson
Dept. Math. Stats., Lancaster University, Lancaster LA1 4YF, U.K
Anthony Nixon †
Dept. Math. Stats., Lancaster University, Lancaster LA1 4YF, U.K
Received 11 February 2022, accepted 28 March 2023, published online 7 September 2023
Abstract
A bar-joint framework (G, p) is the combination of a graph G and a map p assigning
positions, in some space, to the vertices of G. The framework is rigid if every edge-length-
preserving continuous motion of the vertices arises from an isometry of the space. We
will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated
vertices are mapped to the same position. This non-genericity assumption leads us to a
count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that
independence in this matroid is equivalent to independence as a suitably regular bar-joint
framework in a non-Euclidean normed plane with two coincident points; this characterises
when a regular non-Euclidean normed plane coincident-point framework is rigid and allows
us to deduce a delete-contract characterisation. We then apply this result to show that
an important construction operation (generalised vertex splitting) preserves the stronger
property of global rigidity in non-Euclidean normed planes and use this to construct rich
families of globally rigid graphs when the non-Euclidean normed plane is analytic.
Keywords: Bar-joint framework, global rigidity, non-Euclidean framework, count matroid, recursive
construction, normed spaces, analytic norm.
Math. Subj. Class. (2020): 52C25, 05C10, 52B40, 46B20
*Corresponding author. Supported in part by the Austrian Science Fund (FWF): P31888 and in part by the
Heilbronn Institute for Mathematical Research.
†Supported in part by the Heilbronn Institute for Mathematical Research and in part by EPSRC grant number
EP/W019698/1.
E-mail addresses: sean.dewar@bristol.ac.uk (Sean Dewar), john.hewetson02@gmail.com (John Hewetson),
a.nixon@lancaster.ac.uk (Anthony Nixon)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
138 Ars Math. Contemp. 24 (2024) #P1.10 / 137–153
1 Introduction
A bar-joint framework (G, p) is the combination of a graph G = (V,E) and a map p : V →
Rd assigning positions to the vertices of G (and hence lengths to the edges). Note that in
this article graphs are taken to be finite and simple. Intuitively, the framework is rigid if
every edge-length-preserving continuous motion of the vertices arises from an isometry of
Rd. More strongly, (G, p) is globally rigid if every framework in Rd, on the same graph,
with the same edge lengths actually has the same distance between every pair of vertices.
The rigidity and global rigidity of bar-joint frameworks in Euclidean spaces has been
intensely studied in recent years (e.g. [2, 3, 13, 16, 21, 23]) and has a rich history going
as far back as classical work of Euler and Cauchy on Euclidean polyhedra. In the last
decade, work on rigidity has been generalised to various non-Euclidean normed spaces
(e.g. [6, 9, 10, 19, 20]). All of these results concern characterising the combinatorial nature
of the ‘generic’ behaviour of frameworks. This article extends this to frameworks with two
points lying in the same location. The difficulty that already arises in this context shows
how necessary the genericity assumption in those papers really was. Frameworks with
coincident points have been considered in the Euclidean context [12, 14] and applied to
global rigidity there [4], as well as for frameworks on surfaces [17].
Beyond the natural extension towards non-generic frameworks (and thus nearer to be-
ing of potential use in applications), we are motivated by the study of global rigidity in
non-Euclidean normed planes. The first and third author recently instigated research in this
direction [10] proving global rigidity for an infinite class of graphs in non-Euclidean ana-
lytic normed planes. In this paper we use our analysis of frameworks with two coincident
points to improve this result by creating a substantially richer class of globally rigid graphs.
We now give a short outline of what follows. After introducing the necessary back-
ground on the theory of rigid frameworks in normed planes, coincident-point frameworks
and the relevant notion of graph sparsity, in Section 2, the majority of the paper is con-
tained in Section 3. Here we provide a detailed geometric analysis of the effect of certain
graph operations on the rigidity of a coincident-point framework in a normed plane. In
Section 4 we combine these geometric results with combinatorial results of [17] to es-
tablish a purely combinatorial characterisation of independence in the ‘coincident-point
normed plane rigidity matroid’ and we deduce from this a delete-contract characterisation
of coincident-point rigidity in any strictly convex non-Euclidean normed plane. By delete-
contract characterisation we mean that we characterise the coincident-point rigidity of a
graph G in terms of the rigidity of two graphs related to G; the graphs obtained from G
by deleting the edge between the coincident vertices and the graph obtained by contracting
the two coincident vertices. In Section 5 we provide our other main results; these concern
global rigidity. We deduce from our delete-contract characterisation that another graph
operation preserves global rigidity, and we use this result alongside the results of [10] to
establish global rigidity in the special case of non-Euclidean analytic normed planes for a
rich family of graphs.
We conclude the introduction with a brief comparison with the more familiar Euclidean
case to give context for the reader. Both our characterisation of independence in the
coincident-point normed plane rigidity matroid and our delete-contract characterisation are
precise analogues of results obtained by Fekete, Jordán and Kaszanitzky for the Euclidean
case [12]. Furthermore, in the Euclidean case generic global rigidity in the plane is com-
pletely characterised [16]. Our results provide a key step towards establishing an analogue
of that result in non-Euclidean analytic normed planes. It is worth noting though that the
S. Dewar et al.: Coincident-point rigidity in normed planes 139
results of [12] came later than the Euclidean plane characterisation and, to our knowledge,
have not been used to provide an alternative proof of the global rigidity characterisation in
the Euclidean plane. The non-Euclidean normed plane case requires both subtly different
combinatorics and geometry which motivated our deployment of this technique. We would
expect that our application to global rigidity through ‘generalised vertex splitting’ (defined
in Section 5) could be adapted to the Euclidean case.
2 Rigidity and uv-coincident frameworks in normed spaces
2.1 Rigidity in normed spaces
Let X be a real finite-dimensional normed space with norm ∥ · ∥. We define a support
functional of z ∈ X to be a linear functional f : X → R such that f(z) = ∥z∥2 and
sup∥x∥=1 f(x) = ∥z∥. It follows from the Hahn-Banach theorem that every point has a
support functional and every linear functional of X is the support functional of a point in
X . A non-zero point in X is said to be smooth if it has exactly one support functional,
and we shall denote the unique support functional of a smooth point z by φz . We say X is
smooth if every non-zero point in X is smooth, and strictly convex if every linear functional
of X is the support functional of at most one, and hence exactly one, point in X .1 We note
that for normed planes (2-dimensional normed spaces), strict convexity is equivalent to the
property that any two linearly independent smooth points have linearly independent support
functionals.
Now let (G, p) be a framework in X; that is the combination of a graph G = (V,E)
and a map p : V → X (called a placement of G). A finite flex of (G, p) is a continuous path
α : [0, 1] → XV where α(0) = p and ∥αx(t)−αy(t)∥ = ∥px − py∥ for each edge xy ∈ E
and every t ∈ [0, 1]. If every framework (G,α(t)) is congruent to (G, p), i.e. there exists
an isometry ft : X → X so that αx(t) = ft(px) for every x ∈ V , then we say α is trivial.
We now define (G, p) to be (continuously) rigid if every finite flex of (G, p) is trivial.
Since determining whether a framework is rigid is computationally challenging [1],
we follow the literature and linearise the problem. First, let (G, p) be a well-positioned
framework, i.e. the point px − py is smooth for each edge xy ∈ E. An infinitesimal
flex of (G, p) is a map u : V → X where φpx−py (ux − uy) = 0 for all xy ∈ E. An
infinitesimal flex is trivial if there exists a linear map T : X → X and a point z0 ∈ X
so that ux = T (px) + z0 for every vertex x ∈ V , and the map T is tangent to the linear
isometry group of X at the identity map. Importantly, when X has finitely many linear
isometries – for example, when X is a non-Euclidean normed plane [25, page 83] – the
only trivial infinitesimal flexes are those that stem from translations, i.e., infinitesimal flexes
u = (ux)x∈V where there exists z ∈ X such that ux = z for all x ∈ V . We now say that
a well-positioned framework (G, p) is infinitesimally rigid if every infinitesimal flex of
(G, p) is trivial.
For a d-dimensional normed space X , a well-positioned framework (G, p) in X , and a
fixed basis b1, . . . , bd of X , we can define the rigidity matrix to be the |E| × d|V | matrix
1Here we have opted to use a more relevant – but still equivalent – definition for strict convexity. The more
conventional definition for the property is as follows: a normed space is said to be strictly convex if ∥tx + (1 −
t)y∥ < 1 for all x, y ∈ X with ∥x∥ = ∥y∥ = 1 and each 0 < t < 1. To see the equivalence, note that if
sup∥z∥=1 f(z) = 1 and ∥x∥ = ∥y∥ = 1, then f(x) = f(y) = 1 if and only if ∥tx+(1− t)y∥ = 1 for all 0 <
t < 1: this latter fact stems from the inequality tf(x)+(1−t)f(y) ≤ f(tx+(1−t)y) ≤ ∥tx+(1−t)y∥ ≤ 1.
140 Ars Math. Contemp. 24 (2024) #P1.10 / 137–153
R(G, p), where for every e ∈ E, x ∈ V and i ∈ {1, . . . , d} we have
R(G, p)e,(x,i) =
{
φpx−py (bi) if e = xy,
0 otherwise.
The choice of basis used to define R(G, p) can be arbitrary as we are only interested in
the sets of linearly independent rows of the matrix. We say a well-positioned framework
is independent if rankR(G, p) = |E|, minimally (infinitesimally) rigid if it is both in-
dependent and infinitesimally rigid, and regular if rankR(G, p) ≥ rankR(G, q) for all
other well-positioned frameworks (G, q). It is immediate that all independent and/or in-
finitesimally rigid frameworks are regular. Given k is the dimension of the linear space of
trivial infinitesimal flexes of (G, p), it can be shown that so long as the affine span of the
set {px : x ∈ V } is X , the framework (G, p) will be infinitesimally rigid if and only if
rankR(G, p) = d|V | − k; see [6, Proposition 3.13]. Consequently any well-positioned
framework where the affine span of the set {px : x ∈ V } is X , is minimally rigid if and
only if |E| = rankR(G, p) = d|V | − k.
We can link infinitesimal rigidity to rigidity with the following result.
Theorem 2.1. Let (G, p) be a well-positioned framework in a normed space X .
(i) [7, Theorem 3.7] If (G, p) is infinitesimally rigid, then it is rigid.
(ii) [6, Theorem 1.1 & Lemma 4.4] If (G, p) is regular and rigid, and the set of smooth
points in X is open, then (G, p) is infinitesimally rigid.
We shall make use of the following perturbation result throughout the paper. It will be
convenient to refer to properties of placements rather than frameworks. To this end we say
that a placement p of G has property P if the framework (G, p) has property P .
Lemma 2.2 ([6, Lemmas 4.1 and 4.4]). For any graph G and any normed space X , the
set of well-positioned placements of G in X is a conull (i.e. the complement of a set with
Lebesgue measure zero) subset of XV , and the set of regular placements of G in X is a
non-empty open subset of the set of well-positioned placements.
We say that a graph is rigid (respectively, independent, minimally rigid) if it has an
infinitesimally rigid (respectively, independent, minimally rigid) placement.
Whether a graph G = (V,E) is rigid/independent in a normed plane can be determined
by simple sparsity counting conditions. For ∅ ̸= U ⊆ V , iG(U) will denote the number
of edges in the subgraph, G[U ], of G induced by U . For non-negative integers k, ℓ, we
say G is (k, ℓ)-sparse if iG(U) ≤ k|U | − ℓ for every ∅ ̸= U ⊆ V with |U | ≥ k; if G is
(k, ℓ)-sparse and |E| = k|V | − ℓ, then we say G is (k, ℓ)-tight.
Theorem 2.3 ([5]). A graph G is minimally rigid in a non-Euclidean normed plane X if
and only if G is (2, 2)-tight.
For a family S = {S1, S2, . . . , Sk} of subsets Si ⊆ V , 1 ≤ i ≤ k, we say that S is
a cover of F ⊆ E if F ⊆ {xy : {x, y} ⊆ Si for some 1 ≤ i ≤ k}. We can combine
Theorem 2.3 with [17, Section 3.1] (which simply applies a classical result of Edmonds
[11] on matroids induced by submodular functions) to obtain the following result.
S. Dewar et al.: Coincident-point rigidity in normed planes 141
Corollary 2.4. Let (G, p) be a well-positioned framework in a non-Euclidean normed
plane X . Let S be the set of all covers X := {X1, . . . , Xk}. Given s : N → {0, 1} is the
map with s(x) = 1 if x = 2 and s(x) = 0 otherwise, we have
rankR(G, p) ≤ min
X∈S
k∑
i=1
(2|Xi| − 2− s(|Xi|)) ,
with equality if and only if (G, p) is regular. Moreover it suffices to minimise over all covers
Y := {Y1, . . . , Yk} of the edge set E where |Yi| ≥ 2 for each i and |Yi ∩ Yj | ≤ 1 for all
i ̸= j, with equality only if min{|Yi|, |Yj |} = 2.
2.2 uv-coincident rigidity and uv-sparse graphs
Let G = (V,E) be a graph with vertices u, v ∈ V , and let X be a normed space. A
framework (G, p) in X is uv-coincident if pu = pv; if the framework (G− uv, p) is well-
positioned, then we say that (G, p) is a well-positioned uv-coincident framework. Since
pu = pv , we consider G − uv so as to maintain smoothness of the support functionals
associated with the framework; otherwise, no uv-coincident framework with uv as an edge
would be well-positioned.
A well-positioned uv-coincident framework (G, p) is infinitesimally rigid if (G−uv, p)
is infinitesimally rigid in X . Given the linear space XV /uv := {q ∈ XV : qu = qv},
we say that a well-positioned uv-coincident framework (G, p) is regular if rankR(G −
uv, p) ≥ rankR(G − uv, q) for all q ∈ XV /uv, and independent if uv /∈ E and (G, p)
is independent in X . A well-positioned uv-coincident framework (G, p) is minimally (in-
finitesimally) rigid if it is both infinitesimally rigid and independent. We say a graph G is
uv-rigid (respectively, uv-independent, minimally uv-rigid) if there exists a uv-coincident
framework (G, p) that is infinitesimally rigid (respectively, independent, minimally rigid).
By applying the same methods used to prove Lemma 2.2, we can obtain the natural
analogue for uv-coincident frameworks. The two main observations for proving the result
are: (i) the set of smooth points of a normed space form a conull subset (i.e. the complement
of a set with Lebesgue measure zero) of X and (ii) the map p 7→ R(G − uv, p) is lower
semi-continuous.
Lemma 2.5. For any graph G and any normed space X , the set of well-positioned uv-
coincident placements of G in X is a conull subset of XV /uv, and the set of regular uv-
coincident placements of G in X is a non-empty open subset of the set of well-positioned
uv-coincident placements.
As will be shown in Section 4, uv-rigidity in non-Euclidean normed planes is closely
related to the following sparsity property of graphs.
Let G = (V,E) be a graph and let u, v be two distinct vertices of G. Let X =
{X1, X2, ..., Xk} be a family with Xi ⊆ V , 1 ≤ i ≤ k. We say that X is uv-compatible if
u, v ∈ Xi and |Xi| ≥ 3 hold for all 1 ≤ i ≤ k. We define the value of non-empty subsets
of V and of uv-compatible families, denoted val(·), as follows. For ∅ ≠ U ⊆ V , we let
val(U) = 2|U | − tU ,
where tU = 4 if U = {u, v}, tU = 3 if U ̸= {u, v} and |U | ∈ {2, 3}, and tU = 2
142 Ars Math. Contemp. 24 (2024) #P1.10 / 137–153
otherwise. For a uv-compatible family X = {X1, X2, . . . , Xk} we let
val(X ) =
(
k∑
i=1
val(Xi)
)
− 2(k − 1) = 2 +
k∑
i=1
(2|Xi| − tXi − 2).
Note that if X = {U} is a uv-compatible family containing only one set then the two
definitions agree, i.e. val(X ) = val(U) holds.
We say that G is uv-sparse if for all U ⊆ V with |U | ≥ 2 we have iG(U) ≤ val(U)
and for all uv-compatible families X we have iG(X ) :=
∣∣∣⋃ki=1 E(G[Xi])∣∣∣ ≤ val(X ). A
graph G is uv-tight if it is uv-sparse and |E| = 2|V | − 2. Note that if G is uv-sparse then
uv /∈ E. It was shown in [17] that if G = (V,E) is a graph and u, v ∈ V are distinct
vertices of G then I = {F : F ⊆ E and (V, F ) is uv-sparse} is the family of independent
sets of a matroid Muv on E.
It is straightforward to construct (2, 2)-sparse graphs which are not uv-sparse. Perhaps
the simplest way is to notice that the complete bipartite graph K2,3, with the part of size
two comprising of u and v, is clearly (2, 2)-sparse but fails to be uv-sparse. To see this
let v1, v2, v3 be the vertices in the part of size three and consider the uv-compatible family
X = {X1, X2, X3} where X1 = {u, v, v1}, X2 = {u, v, v2} and X3 = {u, v, v3}. Then
iG(X ) = 2+2+2 = 6 and val(X ) = (2 · 3− 3)+ (2 · 3− 3)+ (2 · 3− 3)− 2(3− 1) = 5.
3 Recursive operations
Let G = (V,E) be a graph. The 0-extension operation (on a pair of distinct vertices
a, b ∈ V ) adds a new vertex z and two edges za, zb to G. The 1-extension operation
(on edge ab ∈ E and vertex c ∈ V \ {a, b}) deletes the edge ab, adds a new vertex z
and edges za, zb, zc. The vertex-to-H move adds a copy of a (2, 2)-tight graph H with
V (H) ∩ V = {w}, along with an arbitrary replacement of each edge xw by an edge of
the form xy with y ∈ V (H). A vertex-to-4-cycle move takes a vertex w with neighbours
v1, v2, . . . , vk for any k ≥ 2, splits w into two new vertices w,w′ with w′ /∈ V , adds edges
wv1, w
′v1, wv2, w
′v2 and then arbitrarily replaces edges xw with edges of the form xy
where x ∈ {v3, . . . , vk} and y ∈ {w,w′}. All (2, 2)-tight graphs can be constructed from
a single vertex by a sequence of 0- and 1-extensions, vertex-to-4-cycle and vertex-to-K4
operations; see [22, Theorem 3.1] for more details. The operations we use are illustrated in
Figures 1 and 2.
Figure 1: 0-extension and 1-extension.
S. Dewar et al.: Coincident-point rigidity in normed planes 143
Figure 2: The vertex-to-H (with H being the complete graph on 4 vertices) and vertex-to-
4-cycle operations.
We shall need the following specialized versions. First, suppose that |V ∩ {u, v}| =
1. The 0-extension that adds u (respectively, 0-extension that adds v) operation is a 0-
extension where z = u and v ∈ V \ {a, b} (respectively, with z = v and u ∈ V \ {a, b}).
The vertex-to-4-cycle move that adds u (respectively, vertex-to-4-cycle move that adds v)
is a vertex-to-4-cycle move where w = v and {w,w′} = {u, v} (respectively, w = u and
{w,w′} = {u, v}). The vertex-to-H move that adds u (respectively, vertex-to-H move that
adds v) is a vertex-to-H move where w = v and u ∈ V (H) \ V (respectively, w = u and
v ∈ V (H) \ V ), and the graph H is uv-tight.
Now suppose u, v ∈ V are two distinct vertices. The uv-0-extension operation is a 0-
extension on a pair a, b with {a, b} ≠ {u, v}. The uv-1-extension operation is a 1-extension
on some edge ab and vertex c for which {u, v} is not a subset of {a, b, c}. The uv-vertex-
to-4-cycle and uv-vertex-to-H moves are simply any vertex-to-4-cycle and vertex-to-H
moves applied to a graph containing both u and v.
We can immediately obtain the following result using the proof technique of [5, Lem-
mas 5.1 and 5.2]. In particular, since the uv-0- and uv-1-extensions are local operations
that relate to at most one of u and v, their coincidence does not have any effect on the
proofs presented in [5].
Lemma 3.1. Let G be a graph that contains both u and v, and let G′ be formed from G by
either a uv-0-extension or a uv-1-extension. If G is uv-independent in a normed plane X ,
then G′ is uv-independent in X .
The next lemma shows 0-extensions that add either u or v preserve independence. It
should be noted that our proof technique requires strict convexity.
Lemma 3.2. Let G = (V,E) be a graph that contains u but not v, and let X be a strictly
convex normed plane. Suppose G′ is formed from G by a 0-extension that adds v. Then G′
is uv-independent if and only if G is independent.
Proof. We note that as G′ contains G as a subgraph, if G′ is uv-independent then G will
be independent. Suppose there is an independent placement p of G in X . By applying
translations, we may suppose that pu = 0. Let v1, v2 be the two neighbours of v in G′.
We may also assume that pv1 and pv2 are linearly independent and smooth; indeed if this
144 Ars Math. Contemp. 24 (2024) #P1.10 / 137–153
was not true, we could apply Lemma 2.2 to (G, p) to find a placement of G where it is
true. Define p′ to be the well-positioned placement of G′ with p′x = px for all x ∈ V and
p′v = pu. From our choice of placement of G
′, we see that
R(G′, p′) =
 R(G, p) 0|E|×2A −φpv1
B −φpv2

for some 1×2|V | matrices A and B. Hence (G′, p′) is independent if and only if φpv1 , φpv2
are linearly independent. Since pv1 , pv2 are linearly independent and X is strictly convex,
the pair φpv1 , φpv2 are linearly independent as required.
For the vertex-to-4-cycle move we will use the technique of [17, Lemma 11] to show
that a vertex-to-4-cycle move which creates two coincident vertices preserves indepen-
dence. Similarly to the previous result, we will require that the normed plane in question is
strictly convex.
Lemma 3.3. Let G = (V,E) and G′ = (V ′, E′) be graphs and let X be a strictly convex
normed plane.
(i) If G is independent in X and G′ is formed from G by a vertex-to-4-cycle move that
adds either u or v, then G′ is uv-independent in X .
(ii) If G is uv-independent in X and G′ is formed from G by a uv-vertex-to-4-cycle
move, then G′ is uv-independent in X .
Proof. Suppose that G is uv-independent (respectively, independent). Using Lemma 2.5
(respectively, Lemma 2.2), choose a uv-independent (respectively, independent) placement
p of G in X so that pw, pv1 , pv2 are not collinear. By applying translations to p, we shall
assume that pw = 0. Now define p′ to be the placement of G′ with p′x = px for all x ∈ V
and p′w′ = pw. The pair (G
′, p′) form a well-positioned uv-coincident framework due to
our choice of p′. Since X is strictly convex, the pair φpv1 , φpv2 are linearly independent.
Define G′′ to be the graph formed from G′ by replacing each edge w′vi for 3 ≤ i ≤ k with
the edge wvi. Then
R(G′′, p′) =
 R(G, p) 0|E|×2A φp′
w′−p
′
v1
B φp′
w′−p
′
v2
 =
 R(G, p) 0|E|×2A −φpv1
B −φpv2
 ,
for some 1 × 2|V | matrices A and B. Since pv1 , pv2 are linearly independent and X is
strictly convex, the pair φpv1 , φpv2 are linearly independent. Hence R(G
′′, p′) has linearly
independent rows. To prove that G′ is uv-independent in X we will describe a series of
rank-preserving row operations that will form R(G′, p′) from R(G′′, p′).
As φpv1 and φpv2 are linearly independent, there exist for each 3 ≤ i ≤ k a unique pair
of values αi and βi such that
αiφpv1 + βiφpv2 = φpvi = φp′vi−p
′
z
,
where z ∈ {w,w′} is chosen so that viz ∈ E′. For 1 ≤ i ≤ k, let (wvi) denote the row of
R(G′′, p′) corresponding to the edge wvi, and similarly let (w′v1) and (w′v2) denote the
S. Dewar et al.: Coincident-point rigidity in normed planes 145
rows of R(G′′, p′) corresponding to edges w′v1 and w′v2 respectively. For vi ∈ NG′(w′),
let [w′vi] denote the row of R(G′, p′) corresponding to the edge w′vi. Now, for all vi ∈
NG′(w
′)\{v1, v2}, we have
[w′vi] = (wvi)− αi(wv1)− βi(wv2) + αi(w′v1) + βi(w′v2).
These row operations, when applied R(G′′, p′), preserve linear independence and form the
matrix R(G′, p′). Therefore the rows of R(G′, p′) are linearly independent.
We now prove that vertex-to-H operations that add either u or v and uv-vertex-to-H
operations will preserve uv-independence.
Lemma 3.4. Let G = (V,E) and G′ = (V ′, E′) be graphs and let X be any non-
Euclidean normed plane.
(i) Suppose G is independent in X and G′ is formed from G by a vertex-to-H move that
adds either u or v. If H is minimally uv-rigid in X , then G′ is uv-independent in X .
(ii) Suppose G is uv-independent in X and G′ is formed from G by a uv-vertex-to-H
move. If H is minimally rigid in X , then G′ is uv-independent in X .
Proof. If (i) holds, let (H, q) be a minimally rigid uv-coincident framework in X and
(G, p) be an independent framework in X , while if (ii) holds, let (H, q) be a minimally
rigid framework in X and (G, p) be an independent uv-coincident framework in X . By
applying translations we may assume qw = pw = 0. For any matrix A with columns
corresponding to a vertex subset of V ∪V (H), define Aw to be the matrix where we delete
all columns corresponding to the vertex w. Given the fixed basis b1, b2 ∈ X used to define
our rigidity matrices in X , we define the matrix
M :=
[
R(H, q)w 0|E(H)|×(2|V |−2)
A R(G, p)w
]
where A is the |E| × (2|V (H)| − 2) matrix with entries
Ae,(y,i) =
{
φpy−pw(bi) if e = xw,
0 otherwise.
By our choices of p and q, the matrix M has linearly independent rows.
For each n ∈ N, choose a well-positioned uv-coincident framework (G′, pn) where
pnx = qx/n for each x ∈ V (H) and ∥pnx − px∥ < 1/n for each x ∈ V (this framework can
be seen to exist from Lemma 2.5). Define Mn to be the matrix formed from multiplying
each row of R(G′, pn)w corresponding to an edge of H by n. As the map x 7→ φx is
continuous on the set of smooth points of X (see [24, Theorem 25.5]), the sequence of
matrices (Mn)n∈N will converge to M . Hence for sufficiently large N ∈ N, the matrix
Mn0 (and hence R(G
′, pn0)w) will have linearly independent rows. By setting p′ = pn0 ,
we obtain our desired independent uv-coincident framework (G′, p′).
146 Ars Math. Contemp. 24 (2024) #P1.10 / 137–153
4 Characterising coincident-point independence
With the geometric results of the previous section in hand, we can use the combinatorics
of [17] to prove the difficult sufficiency direction of our main result on coincident frame-
works. We begin with the following result which can be extracted from the proof of [17,
Theorem 4].
Proposition 4.1 ([17]). Any uv-tight graph on at least five vertices can be constructed
from either a (2, 2)-tight graph with at least four vertices that contains exactly one of u
and v, or from the graph consisting of two copies of K4 intersecting in a single vertex
x /∈ {u, v} where u and v are in different copies of K4 (see Figure 3), by a sequence of
0-extensions that add u or v, vertex-to-4-cycle and vertex-to-H moves that add u or v,
uv-0- and uv-1-extensions, and uv-vertex-to-4-cycle and uv-vertex-to-H moves.
Sketch of proof. Since the proof of [17, Theorem 4] is long and technical we provide a
sketch of the proposition to orient the interested reader into how it can be extracted from
that theorem. It is easy to see that every graph generated as described in the statement
is uv-tight. For the converse, firstly [17, Theorem 4] is stated for independence in Muv ,
i.e. for uv-sparse graphs, but we can extend to a base E of Muv which induces a graph
G = (V,E) that, since |V | ≥ 5, necessarily has 2|V | − 2 edges and hence is uv-tight.
Suppose G has a vertex, w, of degree 2. If w ∈ {u, v} then G − w is (2, 2)-tight. If
w /∈ {u, v} then an easy argument shows that G− w is uv-tight.
Therefore we may assume that the minimum degree is exactly 3, however it is much
harder to reduce degree 3 vertices. [17, Theorem 4] deals firstly with three straightforward
special cases. Firstly, if there is a 4-cycle in G containing u and v then uv is not an edge of
this 4-cycle and we see that G is obtained from a (2, 2)-tight graph by a vertex-to-4-cycle
operation that split u into u and v. Secondly, if G contains a uv-tight subgraph H such that
V (H) ⊊ V then we may assume H is a maximal such subgraph (that is there is no vertex
in V \ V (H) with more than one neighbour in V (H)). Then G/H (the graph obtained
from G by contracting all vertices of H to a single vertex) is (2, 2)-tight and G is obtained
from G/H by a vertex-to-H move that expands u into a subgraph H that contains u and
v. Thirdly, if G contains a degree 3 vertex contained in a subgraph of G isomorphic to K4
and there is a vertex x ∈ V \ V (H) such that |V (H) ∩ N(x)| = 2 (and since we may
assume the second special case does not occur {u, v} ̸⊂ V (H) ∪ {x}), then we may apply
a uv-vertex-to-H move to a uv-tight graph G/(H ∪ {x}) to obtain G.
The proof is then completed by applying the arguments in Cases 5 and 6 of [17, The-
orem 4], which use the fact we do not have the special structures we just dealt with, to
analyse all possibilities for reducing a vertex of degree 3. Note that it is still not true that
1-extensions and uv-1-extensions suffice, however it is true that using precisely the opera-
tions listed in the proposition is sufficient.
We will also require the following lemmas.
Lemma 4.2. Let G = (V,E) be a graph with at most 4 vertices that contains both u and
v, and let X be a strictly convex non-Euclidean normed plane. Then G is uv-sparse if and
only if it is uv-independent in X .
Proof. The only graphs on 4 or fewer vertices that are not uv-sparse are those which con-
tain the edge uv, and if G contains the edge uv then it is not uv-independent. Suppose
uv /∈ E. We note that G must be a subgraph of K4 − uv, so it is sufficient to consider the
S. Dewar et al.: Coincident-point rigidity in normed planes 147
vu
x
Figure 3: A uv-tight graph that is one of the base graphs of the construction described in
Proposition 4.1.
case G = K4 − uv. As G can be formed from G − u by a 0-extension that adds u, G is
uv-independent by Theorem 2.3 and Lemma 3.2.
Lemma 4.3. Let G = (V,E) be the graph consisting of two copies of K4 intersecting in a
single vertex x /∈ {u, v}, where u and v are in different copies of K4 (see Figure 3). Then
G is minimally uv-rigid in any non-Euclidean normed plane X .
Proof. Let Vu = {x, u, au, bu} and Vv = {x, v, av, bv} be the vertex sets of the two copies
of K4 in G. As can be seen in Figure 3, Vu ∩ Vv = {x}. By Theorem 2.3, there exists
a placement pu : Vu → X so that the framework (KVu , pu), where KVu is the complete
graph with vertex set Vu, is minimally rigid in X . Define the placement p : V → X by
setting pav = p
u
au , pbv = p
u
bu
, pv = puu, and py = p
u
y for all y ∈ Vu. We now note that
(G, p) is a minimally rigid uv-coincident framework; this follows from the fact that joining
two minimally rigid frameworks in a non-Euclidean normed plane produces a minimally
rigid framework, since the trivial infinitesimal flexes of a non-Euclidean normed plane
correspond only to translations. Hence G is minimally uv-rigid as required.
Theorem 4.4. A graph is uv-independent in a strictly convex non-Euclidean normed plane
X if and only if it is uv-sparse.
Proof. First suppose G is uv-independent in X . Let G/uv denote the graph obtained from
G by contracting the vertex pair u, v into a new vertex which we denote as z2. Let (G, p) be
a regular (and hence independent) uv-coincident framework in X . We obtain a framework
(G/uv, puv) in X by putting puvz = pu = pv and p
uv
x = px for all x ∈ V \ {u, v}. For any
U ⊆ V , the (possibly uv-coincident) induced subframework (G[U ], p|U ) is independent.
Hence, if {u, v} ̸⊆ U , then iG(U) ≤ val(U) by Theorem 2.3. Since the case when
U = {u, v} is trivial, it now remains to show that iG(X ) ≤ val(X ) for all uv-compatible
families X in G. (Note that the case when U ⊆ V and {u, v} ⊆ U will be included by
taking X = {U}.)
Let X = {X1, . . . , Xk} be a uv-compatible family and consider the subgraph H =
(U,F ) of G, where U =
⋃k
i=1 Xi and F =
⋃k
i=1 E(G[Xi]). By contracting the vertex
pair u, v in H , we obtain the graph H/uv. Define q to be the restriction of p to the vertex
set U and quv to be the restriction of puv to the vertex set U − {u, v} + z. We have
Xuv = {X1/uv, . . . ,Xk/uv} is a cover of E(H/uv) where Xi/uv denotes the set that we
2For us, a contraction will always be the more general vertex-contraction (which does not require u and v be
adjacent) not the stricter edge-contraction (which does require u and v be adjacent).
148 Ars Math. Contemp. 24 (2024) #P1.10 / 137–153
get from Xi by identifying u and v. By Corollary 2.4, we have
rankR(H/uv, quv) ≤
k∑
i=1
(2|Xi/uv| − 2− s(|Xi/uv|)
=
k∑
i=1
(2|Xi| − 2− tXi)
= val(X )− 2.
Every vector µuv in the kernel of R(H/uv, quv) determines a unique vector µ in the kernel
of R(H, q) with µu = µv = µuvz and µx = µ
uv
x for all for all x ∈ U \ {u, v}. Hence
dimkerR(H, q) ≥ dimkerR(H/uv, quv). The rigidity matrix R(H, q) has linearly inde-
pendent rows since R(G, p) has linearly independent rows, hence we have
iG(X ) = rankR(H, q) ≤ rankR(H/uv, quv) + 2 ≤ val(X ).
Thus G is uv-sparse.
We prove the sufficiency by induction on |V |. Suppose that G is uv-sparse. If |V | ≤ 4,
then G is uv-independent in X by Lemma 4.2. So we may suppose that |V | ≥ 5. By
adding additional edges, if necessary, we may assume G is uv-tight3. By Proposition 4.1,
G can be constructed from either a (2, 2)-tight graph containing exactly one of u and v, or
the graph pictured in Figure 3, by the operations defined in Section 3. Furthermore, as X
is strictly convex, the corresponding geometric operations preserve minimal rigidity in X
(see Section 3). The result now follows from Theorem 2.3 (i.e., every (2, 2)-tight graph is
independent in X) and Lemma 4.3.
We next use this result to prove the following delete-contract characterisation of uv-
rigidity.
Theorem 4.5. Let G be a graph with distinct vertices u, v, and let X be a strictly convex
non-Euclidean normed plane. Then G is uv-rigid in X if and only if G−uv and G/uv are
both rigid in X .
Proof. Suppose that G is uv-rigid. It is immediate from the definition that G−uv must be
rigid. Choose a regular uv-coincident placement p of G, and define puv to be the placement
of G/uv where puvx = px for all x ∈ V − {u, v} and (given that z is the vertex obtained
from u and v during the contraction) puvz = pu = pv . Given an infinitesimal flex µ
uv
of (G/uv, puv), we can form an infinitesimal flex µ of (G, p) by setting µx = µuvx for
all x ∈ V − {u, v} and µu = µv = µuvz . Since (G, p) is infinitesimally rigid as a uv-
coincident framework, we must have that µ = (λ)x∈V (and hence µuv = (λ)x∈V (G/uv))
for some vector λ ∈ X . Thus (G/uv, puv) is infinitesimally rigid and G/uv is rigid.
The converse follows from Theorem 4.4 as in the proof of [17, Theorem 1].
We conjecture that the last two results apply in arbitrary non-Euclidean normed planes.
Conjecture 4.6. Let G = (V,E) be a graph and let u, v ∈ V be distinct vertices. Then G
is uv-independent in a non-Euclidean normed plane X if and only if G is uv-sparse.
3Recall that uv-sparse graphs are the independent sets of a matroid and, when |V | ≥ 5, the bases of this
matroid have rank 2|V | − 2.
S. Dewar et al.: Coincident-point rigidity in normed planes 149
Indeed extending the proof of Theorem 4.4 to the non-convex case requires only im-
provements to Lemmas 3.2 and 3.3. For the first of these, the issue is that 0-extensions
that add v require us to precisely place v on top of the placement of u. However in the
not strictly convex case, the proof of [5, Lemmas 5.1] requires one to choose the position
of v carefully so that the support functionals of the edges incident to v guarantee linear
independence. For the latter case, both the vertex-to-4-cycle move that adds v and the
uv-vertex-to-4-cycle move have similar complications that would need to be resolved.
5 Global rigidity in analytic normed planes
A framework (G, p) in a normed space X is said to be globally rigid if every other frame-
work (G, q) in X with ∥pv − pw∥ = ∥qv − qw∥ for every edge vw ∈ E is congruent to
(G, p). A graph is then said to be globally rigid in X if the set{
p ∈ XV : (G, p) is globally rigid
}
has a non-empty interior as a subset of the linear space XV with the product topology
inherited from X . It can be quickly seen that any globally rigid framework/graph will also
be rigid.
Although much is known about global rigidity in Euclidean spaces, very little is known
about the property for non-Euclidean normed spaces. The results that are known are only
for analytic normed spaces, i.e., normed spaces where the norm restricted to the non-zero
points is a real analytic function. As well as being strictly convex ([10, Lemma 3.1]),
analytic normed spaces have many useful properties, including the following.
Lemma 5.1. Let G be a graph with distinct vertices u, v and let X be a non-Euclidean
analytic normed space.
(i) The set of all p ∈ XV where (G, p) is a regular framework is an open conull subset
of XV .
(ii) The set of all p ∈ XV /uv where (G, p) is a regular uv-coincident framework is an
open conull subset of XV /uv.
Proof. If dimX = 1 then the result follows immediately from noticing that all well-
positioned frameworks and uv-coincident frameworks are regular. Suppose dimX ≥ 2.
It was shown in [10, Proposition 3.2] that the set of well-positioned but non-regular place-
ments of G are exactly the zero set of a non-constant analytic function defined on the con-
nected open conull set of well-positioned placements. This gives (i). For (ii) we can use
the same technique to show that the set of well-positioned but non-regular uv-coincident
placements of G are exactly the zero set of a non-constant analytic function defined on the
connected open conull set of well-positioned uv-coincident placements. The result now
holds as the zero set of a non-constant analytic function with connected domain is always
a closed null subset (see [10, Proposition 2.3]).
Importantly, we can define a large class of globally rigid graphs in any non-Euclidean
analytic normed plane.
Proposition 5.2 ([10]). Let X be a non-Euclidean analytic normed plane. Then the graphs
K5−e and H , depicted in Figure 4, are globally rigid in X . Moreover any graph obtained
from either of these by a sequence of degree 3 vertex additions (i.e., add a vertex and join
it to three other vertices) and edge additions is globally rigid.
150 Ars Math. Contemp. 24 (2024) #P1.10 / 137–153
Figure 4: The graphs K−5 (left) and H (right).
We next increase this class of graphs with the following construction operation intro-
duced in [18]. A generalised vertex split, is defined as follows. Choose z ∈ V and a parti-
tion Nu, Nv of the neighbours of z. Next, delete z from G and add two new vertices u, v
joined to Nu, Nv , respectively. Finally add two new edges uv, uw for some w ∈ V \Nu.
See Figure 5 for an illustration of the operation.
z u v
w w
Figure 5: Generalised vertex split.
As the name suggests, this operation generalises the usual vertex splitting operation,
see [26], which is the special case when w is chosen to be a neighbour of v. Note also that
the special case when u has degree 3 (and v = z) is the well known 1-extension operation.
Previously it was not known whether the 1-extension operation or a suitably restricted
version of the vertex splitting operation preserves global rigidity in any non-Euclidean
normed plane X .
As an application of our main result we will deduce that global rigidity can, under cer-
tain conditions, be preserved for generalised vertex splits. We will first need the following
result which can be seen to follow from adapting the methods in [10, Section 3.2] to allow
frameworks with zero-length edges4.
Lemma 5.3. Let (G, p) be a uv-coincident framework in a smooth normed space X with
finitely many linear isometries. If (G, p) is globally rigid and infinitesimally rigid, then
there exists an open neighbourhood U ⊂ XV of p where for each q ∈ U the framework
(G, q) is globally rigid.
Theorem 5.4. Let G be a globally rigid graph in a non-Euclidean analytic normed plane
X . Let G′ be a generalised vertex split of G at the vertex z with new vertices u, v and
suppose that G′ − uv is rigid in X . Then G′ is globally rigid in X .
4Although it is a prerequisite in [10, Section 3.2] that the frameworks are well-positioned, the proof technique
only requires that the squared edge-length map is differentiable. Since the map x 7→ ∥x∥2 is always differentiable
at the point 0, we can refine the result so that it holds for frameworks with zero-length edges.
S. Dewar et al.: Coincident-point rigidity in normed planes 151
Proof. Since G′/uv = G is globally rigid in X it is also rigid in X by Theorem 2.1.
As G′ − uv is also rigid in X , Theorem 4.5 implies that G′ is uv-rigid in X . Hence by
Lemma 5.1, we may choose an infinitesimally and globally rigid framework (G, p) so that
if we define (G′, p′) to be the uv-coincident framework with p′x = px for all x ∈ V and
p′u = p
′
v = pz , then (G
′, p′) will be infinitesimally rigid also. Furthermore, (G′, p′) will
also be globally rigid as (G, p) is globally rigid. We can now use Lemma 5.3 to deduce that
(G′, q) is globally rigid in X for all q sufficiently close to p′. Hence G′ is globally rigid in
X also.
We can now improve upon Proposition 5.2. Here a graph G = (V,E) is redundantly
rigid in X if G− e is rigid in X for any edge e ∈ E.
Corollary 5.5. Let G be a graph obtained from K−5 or H by a sequence of generalised
vertex splits that preserve redundant rigidity, edge additions and degree at least 3 vertex
additions. Then G is globally rigid in any non-Euclidean analytic normed plane.
Proof. Follows immediately from Proposition 5.2 and Theorem 5.4.
Since minimally rigid graphs in X have 2|V | − 2 edges by Theorem 2.3, it is natural
to expect that if G = (V,E) is globally rigid then |E| ≥ 2|V | − 1. The graphs K−5 and
H both achieve equality, but the inequality is strict for every graph in the infinite family
obtained from these as in Proposition 5.2. To illustrate the power of Corollary 5.5 we note
that we now have infinitely many globally rigid graphs for which equality holds and that
this still holds if we restrict generalised vertex splitting to just one of vertex splitting or
1-extension. Two examples are depicted in Figure 6. The graph on the left is obtained from
H by a vertex split and the graph on the right is obtained from H by a 1-extension. Both
are globally rigid in X by Corollary 5.5.
Figure 6: Examples of globally rigid graphs.
6 Concluding remarks
1. Following submission of this article we were able to improve upon Corollary 5.5. Specif-
ically in [8], using the results of this article in a crucial way, we obtained a complete
combinatorial description of graphs that are globally rigid in any non-Euclidean analytic
normed plane. It turns out that we needed just 1 additional operation to those used in Corol-
lary 5.5: this operation deletes an edge xy and adds two new vertices z, w and 5 new edges
xz, xw, yz, yw, zw. In different language, the characterisation of [8] shows that a graph is
globally rigid in any non-Euclidean analytic normed plane if and only if it is 2-connected
and redundantly rigid (which means that it is still rigid after deleting any edge).
152 Ars Math. Contemp. 24 (2024) #P1.10 / 137–153
2. Theorem 4.4 and Theorem 4.5 provide a detailed combinatorial understanding of coin-
cident point rigidity for frameworks in strictly convex non-Euclidean normed planes. As
noted in the introduction, similar results exist for the Euclidean plane [12] and for frame-
works supported on a cylinder in R3 [17]. Given the applicability of coincident point
rigidity to analysing global rigidity (e.g. [4]) it would be interesting to develop analogues
of Theorem 4.4 and Theorem 4.5 in other natural settings in rigidity theory. It may also be
interesting to explore rigidity for frameworks with larger (or multiple) sets of coincident
points. This line of investigation has begun in the case of the Euclidean plane [15].
ORCID iDs
Sean Dewar https://orcid.org/0000-0003-2220-4576
John Hewetson https://orcid.org/0000-0001-9369-7895
Anthony Nixon https://orcid.org/0000-0003-0639-1295
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