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Contents
On girth-biregular graphs
György Kiss, Štefko Miklavič, Tamás Szőnyi . . . . . . . . . . . . . . . . 509
An extension of the Erdős-Ko-Rado theorem to uniform set partitions
Karen Meagher, Mahsa N. Shirazi, Brett Stevens . . . . . . . . . . . . . . 531
Almost simple groups as flag-transitive automorphism groups of symmetric
designs with λ prime
Seyed Hassan Alavi, Ashraf Daneshkhah, Fatemeh Mouseli . . . . . . . . . 553
A classification of connected cubic vertex-transitive bi-Cayley graphs over
semidihedral group
Jianji Cao, Young Soo Kwon, Mimi Zhang . . . . . . . . . . . . . . . . . . 563
Domination and independence numbers of large 2-crossing-critical graphs
Vesna Iršič, Maruša Lekše, Mihael Pačnik, Petra Podlogar,
Martin Praček . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
Enumerating symmetric pyramids in Motzkin paths
Rigoberto Flórez, José L. Ramírez . . . . . . . . . . . . . . . . . . . . . . 591
The core of a vertex-transitive complementary prism
Marko Orel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
Normal Cayley digraphs of dihedral groups with the CI-property
Jin-Hua Xie, Yan-Quan Feng, Jin-Xin Zhou . . . . . . . . . . . . . . . . . 615
On cubic bi-Cayley graphs of p-groups
Na Li, Young Soo Kwon, Jin-Xin Zhou . . . . . . . . . . . . . . . . . . . 631
Using a q-shuffle algebra to describe the basic module V (Λ0) for the
quantized enveloping algebra Uq(ŝl2)
Paul Terwilliger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
Volume 23, Number 4, Fall/Winter 2023, Pages 509–682
xxxi

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P4.01 / 509–530
https://doi.org/10.26493/1855-3974.2935.a7b
(Also available at http://amc-journal.eu)
On girth-biregular graphs
György Kiss *
Department of Geometry and ELKH-ELTE Geometric and Algebraic Combinatorics
Research Group, Eötvös Loránd University, 1117 Budapest, Pázmány s. 1/c, Hungary and
Faculty of Mathematics, Natural Sciences and Information Technologies,
University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia
Štefko Miklavič †
Andrej Marušič Institute, University of Primorska, Muzejski trg 2, 6000 Koper and
Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
Tamás Szőnyi ‡
Department of Computer Science and ELKH-ELTE Geometric and
Algebraic Combinatorics Research Group, Eötvös Loránd University,
1117 Budapest, Pázmány s. 1/c, Hungary and
Faculty of Mathematics, Natural Sciences and Information Technologies,
University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia
Received 25 July 2022, accepted 12 November 2022, published online 13 February 2023
Abstract
Let Γ denote a finite, connected, simple graph. For an edge e of Γ let n(e) denote
the number of girth cycles containing e. For a vertex v of Γ let {e1, e2, . . . , ek} be the
set of edges incident to v ordered such that n(e1) ≤ n(e2) ≤ · · · ≤ n(ek). Then
(n(e1), n(e2), . . . , n(ek)) is called the signature of v. The graph Γ is said to be girth-
biregular if it is bipartite, and all of its vertices belonging to the same bipartition have the
same signature.
*Corresponding author. This research was supported in part by the Hungarian National Research, Development
and Innovation Office OTKA grant no. SNN 132625, by the HAS Slovenian-Hungarian bilateral research project
”Graph Colourings and Finite Geometries” (NKM-95/2019000206), and by the Slovenian Research Agency (re-
search project J1-9110).
†This research was supported in part by the Slovenian Research Agency (research program P1-0285 and
research projects N1-0032, N1-0038, J1-5433, and J1-6720).
‡This research was supported in part by the Hungarian National Research, Development and Innovation Office
OTKA grant no. K 124950, by the HAS Slovenian-Hungarian bilateral research project ”Graph Colourings and
Finite Geometries” (NKM-95/2019000206), and by the Slovenian Research Agency (research project J1-9110).
The author would like to thank his coauthors Gabriela, Alejandra and Robert for the paper published in 2022. The
discussions leading to mentioned paper considerably influenced the present paper. The author author would like
to thank the Rényi Institute of Mathematics for the hospitality during this research.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
510 Ars Math. Contemp. 23 (2023) #P4.01 / 509–530
Let Γ be a girth-biregular graph with girth g = 2d and signatures (a1, a2, . . . , ak1) and
(b1, b2, . . . , bk2), and assume without loss of generality that k1 ≥ k2. Our first result is that
{a1, a2, . . . , ak1} = {b1, b2, . . . , bk2}. Our next result is the upper bound ak1 ≤ M , where
M = (k1 − 1)⌊g/4⌋(k2 − 1)⌈g/4⌉. We describe the graphs attaining equality. For d = 3
or d ≥ 4 even they are incidence graphs of Steiner systems and generalized polygons,
respectively. Finally, we show that when d is even and ak1 = M −ε for some non-negative
integer ε < k2 − 1, then ε = 0. Similar result is valid for d = 3, ε ≤ 1 and k2 ̸ | k1.
Keywords: Girth cycle, girth-biregular graph, Steiner system, generalized polygons.
Math. Subj. Class. (2020): 05C35, 51E20
1 Introduction
In extremal graph theory one often considers problems of the following type: we fix some
graph parameter or some graph property and want to deduce the extremal number of another
parameter (in many cases the number of points or edges). Typical questions are Turán type
problems, see e.g. the survey of Füredi and Simonovits [7]. The problem considered in
our paper is motivated by the cage problem (and the degree/diameter problem), see [4, 12].
The cage problem was extended recently by several authors to bipartite graphs which are
biregular in the sense that vertices in the same bipartition set have the same degree, see
Jajcay, Ramos-Rivera and their coauthors [1, 6].
The paper by Jajcay, Kiss and Miklavič [8] defined a new type of regularity: a graph
is called edge-girth regular if the number of cycles of length g (the girth) containing an
edge is independent of the edge. This definition was weakened by Potočnik and Vidali
[14] and in [9] it was extended to a stability theorem. One can introduce the signature
(a1, . . . ak) of a point as the ordered sequence of the number of girth cycles containing the
edges emanating from the point (see Definition 2.1). A graph is called girth-regular if all
of its points have the same signature. For such graphs with valency k ≥ 3, it was shown
in [14] that ak ≤ (k − 1)2d, where d = ⌊g/2⌋. In [9], the upper bound was improved
for g = 2d in the sense that it is either (k − 1)2d or at most (k − 1)2d − (k − 1). In the
former case the graph has to be the incidence graph of a thick generalized d-gon of order
(k − 1, k − 1). In particular, we must have d = 2, 3, 4, 6.
The aim of the present paper is to extend some of the results of [9] to the bipartite
biregular case. If the valencies in the bipartition classes are k1 > k2 > 2, then we prove that
the maximum number of girth-cycles containg an edge is at most M = (k1 − 1)⌊g/4⌋(k2 −
1)⌈g/4⌉, see Theorem 2.6. For g = 4, we show that when the graph is girth regular and the
largest element of the signature of a point is equal to M − ε, with ε ≤ k2 − 1, then ε = 0,
and the graph is the complete bipartite graph Kk1,k2 . In Section 3, we prove an analogous
result for g = 2d ≥ 8, d even, relating the ε = 0 case to incidence graphs of a finite thick
generalized d-gon, see Theorem 3.4(vi). For q = 2d, d odd, we have partial results. In
particular, similarly to the results of [1, 6], when g = 6, we could find a connection of
ak = M and block designs. For particular k1 and k2, the connection is with affine planes,
see Corollary 6.3.
E-mail addresses: gyorgy.kiss@ttk.elte.hu (György Kiss), stefko.miklavic@upr.si (Štefko Miklavič),
tamas.szonyi@ttk.elte.hu (Tamás Szőnyi)
Gy. Kiss et al.: On girth-biregular graphs 511
2 Definitions and basic properties
In this section we collect basic notation and terminology. First, for the sake of complete-
ness, we recall some definitions from design theory and finite geometries. In the second
subsection we define girth-biregular graphs and present some simple, important properties
of them.
2.1 Block designs, Steiner systems, generalized polygons
Here we give only the most necessary definitions. A detailed introduction to block de-
signs and Steiner systems we refer the reader to [2] and [3], while the concepts from finite
geometries we use can be found for example in [10] and [11].
Definition 2.1. Let v ≥ k ≥ t ≥ 2 and λ ≥ 1 be integers. A t-(v, k, λ) design is
a collection of k-subsets (blocks) of a v-set S (points) such that every t-subset of S is
contained in exactly λ of the blocks.
A t-(v, k, 1) design is called a Steiner system. In particular, the blocks of a Steiner
system with t = 2 are often called lines.
A parallelism of a design is a partition of its blocks into classes C1, C2, . . . , Cr with
the property that any point belongs to a unique block of each class. A design is called
resolvable, if it has a parallelism.
Let (P,L, I) be a connected, finite point-line incidence geometry. The elements of P
and L are called points and lines, respectively, I ⊂ (P×L)∪(L×P) is a symmetric relation,
called incidence. A chain of length h is a sequence x0 Ix1 I . . . Ixh where xi ∈ P ∪ L.
The distance of the elements u and v, denoted by d(u, v), is the length of the shortest chain
joining them.
Definition 2.2. Let n > 1 be a positive integer. The incidence geometry G = (P,L, I) is
called a thick generalized n-gon if it satisfies the following axioms.
• d(x, y) ≤ n ∀ x, y ∈ P ∪ L.
• If d(x, y) = k < n, then there is a unique chain of length k joining x and y.
• ∀ x ∈ P ∪ L ∃ y ∈ P ∪ L such that d(x, y) = n.
• ∀ x ∈ P ∪ L there exist at least three elements yi ∈ P ∪ L such that d(x, yi) = 1.
For any finite thick generalized n-gon G there exist integers s, t ≥ 2 such that every
line is incident with exactly s+1 points and every point is incident with exactly t+1 lines.
The pair (s, t) is called the order of G.
In particular, for n = 3, the generalized 3-gons are the finite projective planes, for
n = 4, the generalized 4-gons are the finite generalized quadrangles (GQ-s for short). The
GQ-s have an alternative definition:
Definition 2.3. Let s > 1 and t > 1 be positive integers. A thick generalized quadrangle
of order (s, t) is a point-line incidence structure which satisfies the following axioms:
• every line is incident with exactly s+ 1 points;
• every point is incident with exactly t+ 1 lines;
512 Ars Math. Contemp. 23 (2023) #P4.01 / 509–530
• there exists a non-incident point-line pair;
• for every point P and every line ℓ not incident with P , there is exactly one line
through P which intersects ℓ.
2.2 Girth-biregular graphs
Let Γ denote a finite, connected, simple graph with vertex set V = V (Γ) and edge set
E = E(Γ). Let d denote the minimal path-length distance function of Γ and let D =
max{d(v, w) | v, w ∈ V } denote the diameter of Γ. For v ∈ V and an integer i we let
Γi(v) = {w ∈ V | d(v, w) = i}. We abbreviate Γ(v) = Γ1(v) and observe that Γi(v) = ∅
whenever i < 0 or i > D. For an edge uv of Γ, let Dij(u, v) = Γi(u) ∩ Γj(v).
We say that Γ is biregular with valencies k1, k2 (k ∈ Z), whenever Γ is bipartite with
bipartition sets A,B, and |Γ(v)| = k1 (|Γ(v)| = k2, respectively) for every v ∈ A (v ∈ B,
respectively). If Γ is not a tree, then the girth g of Γ is the length of a shortest cycle in Γ.
If C is a cycle of Γ of girth length g, then we refer to C as a girth cycle of Γ.
The incidence graph (also known as Levi graph) of a point-line incidence geometry is a
bipartite graph whose bipartition sets correspond to the set of points and lines, respectively,
and there is an edge between two vertices if and only if the corresponding point is incident
with the corresponding line.
The next ”folklore” statement gives an important correspondence between generalized
polygons and biregular graphs. The proof can be found for example in [11, Lemma 1.3.6],
or in [10, Chapter 12].
Theorem 2.4. A finite thick generalized n-gon G exists if and only if there exists a con-
nected bipartite biregular graph Γ of diameter n and girth 2n, such that each vertex has
degree at least three. In this case Γ is the incidence graph of G.
The following definition is a central definition of this paper.
Definition 2.5. Let Γ be a graph and let u, v be adjacent vertices of Γ. For the edge
e = uv of Γ let n(e) = n(uv) denote the number of girth cycles containing e. For a
vertex w of Γ let {e1, e2, . . . , ek(w)} be the set of edges incident to w ordered such that
n(e1) ≤ n(e2) ≤ · · · ≤ n(ek(w)). Then (n(e1), n(e2), . . . , n(ekw)) is called the signature
of w. The bipartite graph G is said to be girth-biregular if all of its vertices belonging to
the same bipartition have the same signature.
Observe that girth-biregular graphs are also biregular. The following straightforward
observation will be used through the rest of the paper frequently without explicitly referring
to it (see also [14, Subsection 2.2] and Figure 1).
Proposition 2.6. Let Γ be a biregular graph with valencies k1, k2 and girth 2d, d ≥ 2.
Let uv be an edge of Γ, such that the valency of u is k1 and valency of v is k2. Let
Dij = D
i
j(u, v). Then the following hold.
(i) If x, y are vertices of Γ with d(x, y) ≤ d − 1, then there is a unique path of length
d(x, y) between x and y.
(ii) Dii = ∅ for every integer i.
(iii) For 1 ≤ i ≤ d−1 and for z ∈ Dii+1 (resp. z ∈ D
i+1
i ), we have that |Γ(z)∩D
i−1
i | =
1 (resp. |Γ(z) ∩Dii−1| = 1) .
Gy. Kiss et al.: On girth-biregular graphs 513
(iv) For 0 ≤ i ≤ d − 2 and for z ∈ Dii+1, we have that |Γ(z) ∩D
i+1
i+2| = k1 − 1 if i is
even, and |Γ(z) ∩Di+1i+2| = k2 − 1 if i is odd.
(v) For 0 ≤ i ≤ d − 2 and for z ∈ Di+1i , we have that |Γ(z) ∩D
i+2
i+1| = k2 − 1 if i is
even, and |Γ(z) ∩Di+1i+2| = k1 − 1 if i is odd.
(vi) For 0 ≤ i ≤ d− 1 we have that
|Dii+1| =
{
(k1 − 1)i/2(k2 − 1)i/2 if i is even,
(k1 − 1)(i+1)/2(k2 − 1)(i−1)/2 if i is odd.
|Di+1i | =
{
(k1 − 1)i/2(k2 − 1)i/2 if i is even,
(k1 − 1)(i−1)/2(k2 − 1)(i+1)/2 if i is odd.
(vii) There are exactly n(uv) edges between Dd−1d and D
d
d−1.
1 1
11
edges
n(uv)
Ddd+1
Ddd-1 Dd+1dv
u 1 1
11
k -11 2 1 2
k -1
D12
D12
D23
D23
Dd-2d-1
Dd-1d-2
Dd-1dk-1 k -1
1
k -1
k -12 k -11 k -12
Figure 1: A biregular graph with valencies k1, k2 and girth 2d, d odd. The numbers near
the bubble representing the set Dij represent the number of neighbours that each vertex of
Dij has in the neighbouring bubble.
3 Some properties of girth-biregular graphs
In this section we continue to study girth-biregular graphs. We prove several results about
these graphs that are interesting on their own, and that will also be useful in the rest of
the paper. Keeping in mind Proposition 2.6, one can calculate the number of girth cycles
containing two fixed edges.
Lemma 3.1. Let Γ be a girth-biregular graph with valencies k1 ≥ k2 and girth g = 2d.
Let u1u2 and v1v2 be two edges of Γ. Without loss of generality we may assume that
d(u1, v1) = min{d(ui, vj) : 1 ≤ i, j ≤ 2}. Let m = d(u1, v1) + 1, and let c denote the
number of girth cycles containing both u1u2 and v1v2. Then c = 0 if m ≥ d+1 and c ≤ 1
if m = d. Moreover, if m ≤ d− 1, then
c ≤

(k1 − 1)(d−m)/2(k2 − 1)(d−m)/2, if m and d are of the same parity,
(k1 − 1)(d−1−m)/2(k2 − 1)(d+1−m)/2, if m is even and d is odd,
(k1 − 1)(d−1−m)/2(k2 − 1)(d+1−m)/2, if m is odd, d is even and valency of v2 is k2,
(k1 − 1)(d+1−m)/2(k2 − 1)(d−1−m)/2, if m is odd, d is even and valency of v2 is k1.
514 Ars Math. Contemp. 23 (2023) #P4.01 / 509–530
Proof. The statement is obvious if m ≥ d + 1. If m = d, then d − 1 = d(u1, v1) ≤
d(u2, v2), so there exists a girth cycle containing both u1u2 and v1v2 if and only if
d(u2, v2) = d− 1, hence c ≤ 1.
Suppose that m ≤ d − 1. Let Dij = Dij(u1, u2) and observe that v1 ∈ Dm−1m , v2 ∈
Dmm+1. Note that there is a unique path of length m − 1 between v1 and u1. Let F =
Dd−m−1d−m (v2, v1) and note that by Proposition 2.6(iii) we have that F ⊆ D
d−1
d . Let us
denote the valency of v2 by k and let k′ be the other valency of Γ. Then
|F | = (k − 1)⌈(d−m−1)/2⌉(k′ − 1)⌊(d−m−1)/2⌋,
and there is a unique path of length d −m − 1 between v2 and any element of F because
the girth of Γ is 2d. Now the number of girth cycles containing both u1u2 and v1v2 equals
to the number of edges between F and Ddd−1. Observe that this number is the same as the
number of (d −m)-arcs (v2, x1, . . . , f, r) where f ∈ F and r ∈ Dd−1d . Observe also that
the valency of f is k if d −m − 1 is even and it is k′ if d −m − 1 is odd. Therefore, we
have that c ≤ |F |(k − 1) if d −m − 1 is even, and c ≤ |F |(k′ − 1) if d −m − 1 is odd.
Now we distinguish four cases. If d and m are of the same parity, then d −m − 1 is odd,
and so
c ≤ |F |(k′ − 1) = (k − 1)(d−m)/2(k′ − 1)(d−m)/2 = (k1 − 1)(d−m)/2(k2 − 1)(d−m)/2.
If d is odd and m is even, then deg(u2) ̸= deg(v2), so we may assume deg(v2) = k2
(otherwise we interchange the roles of edges u1u2 and v1v2). Hence
c ≤ |F |(k − 1) = |F |(k2 − 1) = (k1 − 1)(d−m−1)/2(k2 − 1)(d−m+1)/2.
Finally, if d is even and m is odd, then
c ≤ |F |(k − 1) = (k − 1)(d−m+1)/2(k′ − 1)(d−m−1)/2,
and this gives the third and fourth estimates of the statement according as k = k1 or
k = k2.
Proposition 3.2. Let Γ be a girth-biregular graph with bipartition A,B and valencies k1 ≥
k2. Let us denote the signature of vertices from A by (a1, a2, . . . , ak1) and the signature of
vertices from B by (b1, b2, . . . , bk2). Then {a1, a2, . . . , ak1} = {b1, b2, . . . , bk2}.
Proof. As Γ is bipartite, each edge e of Γ is incident with one vertex from A and with
one vertex from B. It thus follows that n(e) ∈ {a1, a2, . . . , ak1} if and only if n(e) ∈
{b1, b2, . . . , bk2}. This shows that {a1, a2, . . . , ak1} = {b1, b2, . . . , bk2}.
Proposition 3.3. Let Γ be a girth-biregular graph with bipartition A,B and valencies k1 ≥
k2. Let us denote the signature of vertices from A by (a1, a2, . . . , ak1) and the signature of
vertices from B by (b1, b2, . . . , bk2). Pick a ∈ {a1, a2, . . . , ak1} = {b1, b2, . . . , bk2}. Let
aA (aB , respectively) denote the number of appearences of a in the signature (a1, a2, . . . ,
ak1) ((b1, b2, . . . , bk2), respectively). Then k2aA = k1aB .
Proof. Let us count the number of edges of Γ that are contained in exactly a girth cycles.
On the one hand, this number is equal to |A|aA, and on the other hand it is equal to |B|aB .
Recall also that |A|k1 = |B|k2. The claim follows.
Gy. Kiss et al.: On girth-biregular graphs 515
Let Γ be a girth-biregular graph with bipartition A,B and valencies k1 ≥ k2. Let us de-
note the signature of vertices from A by (a1, a2, . . . , ak1) and signature of vertices from B
by (b1, b2, . . . , bk2). Let us comment on the case k1 = k2. It follows from Proposition 3.3
that in this case we have aA = aB for every a ∈ {a1, a2, . . . , ak1} = {b1, b2, . . . , bk2}.
Therefore, Γ is in fact girth-regular graph. As girth regular graphs were studied in details
in [9] and [14], we will assume k1 > k2 for the rest of this paper.
Observe also that connected biregular graphs with valencies k1, k2 = 1 are just the
star graphs, which contain no cycles at all (and are therefore girth-biregular with signatures
(0, 0, . . . , 0) and (0)).
Let Γ be a girth-biregular graph with bipartition A,B and valencies k1 > k2 = 2.
Then for any vertex w ∈ B there are two edges, say u1w and u2w through w, hence a
cycle contains u1w if and only if it contains u2w. In particular, n(u1w) = n(u2w) which
implies b1 = b2. Now, define the graph Γ′ in the following way: V (Γ′) = A and there is
an edge between vertices u and v if and only if d(u, v) = 2 in Γ. Then Γ′ is an edge-girth-
regular graph with valency k1. These graphs were studied in [8]. Therefore, in the rest of
this paper we also assume k1 > k2 > 2.
The following theorem is a generalization of the result of Potočnik and Vidali [14,
Theorem 1.3].
Theorem 3.4. Let Γ be a girth-biregular graph with bipartition A,B, valencies k1 > k2 >
2 and girth 2d. Let us denote the signature of vertices from A by (a1, a2, . . . , ak1) and the
signature of vertices from B by (b1, b2, . . . , bk2). Let M = (k1 − 1)g/4(k2 − 1)g/4 if d is
even, and M = (k1 − 1)(g−2)/4(k2 − 1)(g+2)/4 if d is odd. Then ak1 = bk2 ≤ M .
When the upper bound is attained, ak1 = bk2 = M, the following (i)-(vii) hold.
(i) For every edge uv of Γ with u ∈ A and n(uv) = M we have Di+1i (u, v) = ∅ for
i ≥ d.
(ii) The signature of each vertex of Γ is (M,M, . . . ,M), hence n(e) = M for all e ∈
E(Γ).
(iii) Every path on d + 2 vertices of Γ, starting in a vertex that is contained in A, is
contained in a unique girth cycle;
(iv) If d is even and uv is an edge of Γ with u ∈ A, then Dii+1(u, v) = ∅ for i ≥ d.
(v) if d is odd and uv is an edge of Γ with u ∈ A, then Ddd+1(u, v) ̸= ∅ and Dii+1 = ∅
for i ≥ d+ 1.
(vi) if d is even, then Γ is the incidence graph of a generalized d-gon of order
(k1 − 1, k2 − 1);
(vii) if d = 3, then Γ is the incidence graph of a 2− (k1k2 − k1 + 1, k2, 1)-design.
Proof. Pick adjacent vertices u ∈ A, v ∈ B such that n(uv) = ak1 = bk2 .
We prove the upper bound on ak1 in the case when d is odd. The proof for the
case when d is even is similar. By Proposition 2.6(vi) we have that |Ddd−1(u, v)| =
(k1 − 1)(d−1)/2(k2 − 1)(d−1)/2. As Ddd−1(u, v) ⊆ B and as every vertex from Ddd−1(u, v)
has exactly one neighbour in Dd−1d−2(u, v), it follows that every vertex from
Ddd−1(u, v) has at most k2 − 1 neighbours in D
d−1
d (u, v). Therefore, there are at most
516 Ars Math. Contemp. 23 (2023) #P4.01 / 509–530
(k1 − 1)(d−1)/2(k2 − 1)(d+1)/2 edges between Ddd−1(u, v) and D
d−1
d (u, v). The result
now follows from Proposition 2.6(vii).
Now, suppose that ak1 = M.
(i): By Proposition 2.6(vii), there are M edges between Ddd−1(u, v) and D
d−1
d (u, v). Re-
call that by Proposition 2.6(iii), every vertex from Ddd−1(u, v) has exactly one neighbour
in Dd−1d−2(u, v). It follows that every vertex from D
d
d−1(u, v) has all other neighbours in
Dd−1d , and so D
d+1
d (u, v) = ∅. Consequently, D
i+1
i (u, v) = ∅ for every i ≥ d.
(ii): Let w ∈ D21(u, v) be any vertex. Then we have that
Dd−1d (v, w) = D
d−2
d−1(u, v) ∪
(
Ddd−1(u, v) \Dd−2d−1(w, v)
)
,
and, as Dd+1d (u, v) = ∅ by (i) above, also
Ddd−1(v, w) = D
d−1
d (u, v).
By Proposition 2.6(iv), the number of edges between Dd−2d−1(u, v) and D
d−1
d (u, v) is equal
to |Dd−2d−1(u, v)|(k2 − 1) if d is odd, and to |D
d−2
d−1(u, v)|(k1 − 1) if d is even. As every
vertex from Ddd−1(u, v) has exactly one neighbour in D
d−1
d−2(u, v) and as D
d+1
d (u, v) = ∅,
the number of edges between
(
Ddd−1(u, v) \D
d−2
d−1(w, v)
)
and Dd−1d (u, v) is equal to(
|Ddd−1(u, v)| − |Dd−2d−1(w, v)|
)
(k2 − 1) if d is odd,
and to (
|Ddd−1(u, v)| − |Dd−2d−1(w, v)|
)
(k1 − 1) if d is even.
Observe that by Proposition 2.6(vi) we have that |Dd−2d−1(u, v)| = |D
d−2
d−1(w, v)|, and so
Proposition 2.6(vii) and the above comments imply that n(vw) = (k2 − 1)|Ddd−1(u, v)| if
d is odd and n(vw) = (k1−1)|Ddd−1(u, v)| if d is even. Finally, Proposition 2.6(vi) implies
that n(vw) = M . Hence the signature of v is (M,M, . . . ,M), so the girth-biregularity of
Γ implies that n(e) = M for all e ∈ E(Γ).
(iii): Pick any path x0x1 . . . xd+1 with x0 ∈ A and consider the sets Dij(x0, x1). It follows
from Proposition 2.6 that xi ∈ Dii−1 for 1 ≤ i ≤ d. Recall that n(x0x1) = M by (ii)
above, and so Dd+1d (x0, x1) = ∅ by (i) above. It follows that xd+1 ∈ D
d−1
d . The result
now follows from Proposition 2.6(iii).
(iv): Recall that by (ii) above we have n(uv) = M , and so there are exactly M edges
between Dd−1d (u, v) and D
d
d−1(u, v). Recall also that by Proposition 2.6(iii), every vertex
from Dd−1d (u, v) has exactly one neighbour in D
d−2
d−1(u, v). It follows that every vertex
from Dd−1d (u, v) has all other neighbours in D
d
d−1, and so D
d
d+1(u, v) = ∅. Consequently,
Dii+1(u, v) = ∅ for every i ≥ d.
(v): By Proposition 2.6(vi) we have |Dd−1d (u, v)| = (k1−1)(d−1)/2(k2−1)(d−1)/2. As ver-
tices of Dd−1d (u, v) have valency k1, there are therefore k1(k1 − 1)(d−1)/2
(k2 − 1)(d−1)/2 edges going out of Dd−1d (u, v). As n(uv) = M by (ii) above, M =
(k1−1)(d−1)/2(k2−1)(d+1)/2 of these edges are between Dd−1d (u, v) and Ddd−1(u, v). By
Proposition 2.6(iii), (k1−1)(d−1)/2(k2−1)(d−1)/2 of these edges are between Dd−1d (u, v)
and Dd−2d−1(u, v). It follows that there are exactly (k1 − k2)(k1 − 1)(d−1)/2(k2 − 1)(d−1)/2
edges between Dd−1d (u, v) and D
d
d+1(u, v). As k1 > k2 ≥ 3, this number is nonzero,
implying that Ddd+1(u, v) ̸= ∅.
Gy. Kiss et al.: On girth-biregular graphs 517
Assume now that Dd+1d+2(u, v) ̸= ∅. Pick w ∈ D
d+1
d+2(u, v) and let ux1x2 . . . xdw be
arbitrary path between u and w such that xi ∈ Dii+1 for 1 ≤ i ≤ d. Note that this path
is not contained in a girth cycle of Γ, contradicting (iii) above. Therefore Dd+1d+2(u, v) = ∅
and consequently Dii+1(u, v) = ∅ for every i ≥ d+ 1.
(vi): Observe that (i), (ii) and (iv) above implies that the diameter of Γ is d. As k1 > k2 ≥
3, Theorem 2.4 implies that Γ is the incidence graph of a generalized d-gon.
(vii): Finally, suppose that d = 3. We call the vertices in A points and the the vertices in B
lines and we use the geometric terminology. We claim that there is a unique line through
any pair of distinct points. As the girth of Γ is 6, there is at most one line through any pair
of points. Pick now distinct points x, y ∈ A. Pick an arbitrary line z through x. It follows
from (i) and (v) above, that either y ∈ D21(x, z) or y ∈ D23(x, z). If y ∈ D21(x, z) , then z
is the unique line through x and y. If however y ∈ D23(x, z), then, by Proposition 2.6(iii),
there is a unique line w ∈ D12(x, z) which is adjacent to both x and y in Γ. Therefore, in
this case w is the unique line through x and y.
In the rest of this paper we use the following notation.
Notation 3.5. Let Γ be a girth-biregular graph with bipartition A,B, valencies k1 >
k2 ≥ 3, girth g = 2d, signatures (a1, a2, . . . , ak1) and (b1, b2, . . . , bk2). Let M =
(k1 − 1)g/4(k2 − 1)g/4 if d is even, and M = (k1 − 1)(g−2)/4(k2 − 1)(g+2)/4 if d is odd
and suppose that ak1 = M − ε for some ε < k2 − 1. Let uv be an edge with u ∈ A, v ∈ B
and n(uv) = ak1 , and let D
i
j = D
i
j(u, v). Note that D
i
i = ∅ for every i and that there are
no edges between Di−1i and D
i
i−1 for 1 ≤ i ≤ d− 1.
For every r ∈ Ddd−1 (s ∈ D
d−1
d , respectively) we let h(r) = |Γ(r) ∩ D
d+1
d | (h(s) =
|Γ(s)∩Ddd+1|, respectively). Let {r1, r2, . . . , rm} ⊆ Ddd−1 be the set of vertices of Ddd−1,
for which the value of the function h is positive, that is, the set of vertices of Ddd−1, that
have a neighbour in Dd+1d . Choose the indices in such a way that h(ri) ≤ h(rj) for i < j.
Similarly, let {s1, s2, . . . , sn} ⊆ Dd−1d be the set of vertices of D
d−1
d , for which the value
of the function h is positive. Again, choose the indices in such a way that h(si) ≤ h(sj)
for i < j. We also set γ = h(rm), σ = h(sn), µ = h(r1) and ν = h(s1).
Proposition 3.6. Suppose that g = 2d with d even. With reference to Notation 3.5, we have∑
r∈Ddd−1
h(r) =
m∑
i=1
h(ri) =
∑
s∈Dd−1d
h(s) =
n∑
i=1
h(si) = ε. (3.1)
Proof. The first and the third of the above equalities are clear. We now prove that∑n
i=1 h(si) = ε. The proof that
∑m
i=1 h(ri) = ε is similar. Let E denote the set
of edges, that have one endpoint in Dd−1d , and the other endpoint in D
d
d+1. Note that
E =
∑n
i=1 h(si), and so it is enough to prove |E| = ε. As d is even, it follows from Propo-
sition 2.6(vi) that |Dd−1d | = (k1 − 1)d/2(k2 − 1)(d−2)/2. As D
d−1
d ⊆ B, there are total
(k1−1)d/2(k2−1)(d−2)/2k2 edges, having one endpoint in Dd−1d . By Proposition 2.6(iii),
(k1 − 1)d/2(k2 − 1)(d−2)/2 of these edges have the other endpoint in Dd−2d−1 . Since ak =
M − ε, it follows from Proposition 2.6(vii) that there are (k1 − 1)d/2(k2 − 1)d/2 − ε edges
between Dd−1d and D
d
d−1. Combining these observations, we get the desired result.
Lemma 3.7. Suppose that g = 2d with d even. With reference to Notation 3.5, we have
m ≥ σ and n ≥ γ.
518 Ars Math. Contemp. 23 (2023) #P4.01 / 509–530
Proof. Set Γ(u) \ {v} = {u1, u2, . . . , uk1−1} and Γ(v) \ {u} = {v1, v2, . . . , vk2−1}.
Moreover, for 1 ≤ i ≤ k1 − 1 (1 ≤ i ≤ k2 − 1, respectively) set Ui = Γd−2(ui) ∩Dd−1d
(Vi = Γd−2(vi) ∩ Ddd−1, respectively). Note that as girth of Γ is 2d, the sets Ui (Vi,
respectively) are pairwise disjoint, and |Ui| = |Vi| = (k1 − 1)(d−2)/2(k2 − 1)(d−2)/2.
Moreover, each r ∈ Ddd−1 (s ∈ D
d−1
d , respectively) could have at most one neighbour in
Ui (Vi, respectively) for each i. It is now clear that if s ∈ Dd−1d has no neighbours in Vi
for some 1 ≤ i ≤ k2 − 1, then there is at least one vertex r ∈ Vi with h(r) ≥ 1. It follows
m ≥ σ. Similarly we show that n ≥ γ.
Equation (3.1) and Lemma 3.7 obviously imply the following inequalities:
µσ ≤ µm ≤ ε, νγ ≤ νn ≤ ε. (3.2)
If γ ≤ σ, then observe also that it follows from the above comments that
µ2 ≤ µγ ≤ µσ ≤ µm ≤ ε,
while if σ ≤ γ then
ν2 ≤ νσ ≤ νγ ≤ νn ≤ ε.
This shows that if γ ≤ σ then µ ≤
√
ε, while if σ ≤ γ then ν ≤
√
ε.
First, we give a lower bound on a1 using the vertex u.
Lemma 3.8. With reference to Notation 3.5 we have that
a1 ≥ (k1−1)(d−2)/2(k2−1)(d−2)/2 max{(k2−1−σ)(k1−1), (k1−1−γ)(k2−1)}−ε.
(3.3)
Proof. We prove that a1 ≥ (k1 − 1)d/2(k2 − 1)(d−2)/2(k2 − 1 − σ) − ε. The proof of
a1 ≥ (k1 − 1)(d−2)/2(k2 − 1)d/2(k1 − 1− γ)− ε is similar.
Recall that n(uv) = ak and that Dij = D
i
j(u, v). Let s ̸= v be a neighbour of u such
that n(us) = a1. Abbreviate K = Dd−1d ∩ Γd−2(s). For s′ ∈ K abbreviate L(s′) =
Ddd−1 ∩ Γ(s′). Note that as girth of Γ is 2d, we have that sets L(s′) are pairwise disjoint,
and so by (3.1) we have that ∑
s′∈K
∑
r′∈L(s′)
h(r′) ≤ ε.
Pick r′ ∈ L(s′) and observe that for each r̃ ∈ (Γ(r′) ∩ (Dd−1d ∪D
d−1
d−2)) \ {s′}, there is a
unique girth cycle containing the arc us and the 2-arc s′r′r̃. Note that
Gy. Kiss et al.: On girth-biregular graphs 519
|K| = (k1 − 1)(d−2)/2(k2 − 1)(d−2)/2, and so, by (3.1), we have
a1 = n(us) ≥
∑
s′∈K
∑
r′∈L(s′)
(k1 − 1− h(r′))
=
∑
s′∈K
∑
r′∈L(s′)
(k1 − 1)−
∑
s′∈K
∑
r′∈L(s′)
h(r′)
≥ (k1 − 1)
∑
s′∈K
(k2 − 1− h(s′))− ε
≥ (k1 − 1)
∑
s′∈K
(k2 − 1− σ)− ε
= (k1 − 1)(k1 − 1)(d−2)/2(k2 − 1)(d−2)/2(k2 − 1− σ)− ε
= (k1 − 1)d/2(k2 − 1)(d−2)/2(k2 − 1− σ)− ε.
4 The case g = 4
In this section we consider the case g = 4. Throughout this section we will use Nota-
tion 3.5. Recall that m (n, respectively) denotes the number of vertices of Ddd−1 (D
d−1
d ,
respectively), for which the value of the function h is positive.
Lemma 4.1. Assume that g = 4 and ε ≥ 1. Pick 1 ≤ i ≤ n, 1 ≤ j ≤ m, w ∈ Γ(si) ∩D23
and w̃ ∈ Γ(rj) ∩D32 . Then the following (i) – (iv) holds.
(i) There are at most (h(si) − 1)(k1 − 1) girth cycles of the form (w, si, x, y, w) such
that x ∈ Γ(si) ∩D23 .
(ii) There are at most ε girth cycles of the form (w, si, x, y, w) such that x ∈ D21 and
y ̸∈ D12 .
(iii) There are at most (h(rj) − 1)(k2 − 1) girth cycles of the form (w̃, rj , x, y, w̃) such
that x ∈ Γ(rj) ∩D32 .
(iv) There are at most ε girth cycles of the form (w̃, rj , x, y, w̃) such that x ∈ D12 and
y ̸∈ D21 .
Proof. (i): Note that there are h(si) − 1 choices for x, and for each such choice there are
at most k1 − 1 choices for y. The result follows.
(ii): As x ∈ D21 and y ̸∈ D12 , it follows that y ∈ D32 . It follows from Proposition 3.6 that
there are at most ε choices for the edge xy. For each such edge xy there is clearly at most
one girth cycle containing also the edge wsi. The result follows.
(iii), (iv): Similar to the proofs of (i) and (ii) above.
Lemma 4.2. Assume that g = 4 and ε ≥ 1. Then m ≥ 2 and n ≥ 2.
Proof. We prove that n ≥ 2. The proof that m ≥ 2 is similar. Suppose on the contrary
that n = 1. Note that in this case σ = ν = ε, γ = 1, m = ε and h(ri) = 1 for
1 ≤ i ≤ m. Let w be the unique neighbour of r1 in D32 . Let t = |Γ(w) ∩ D21| and note
that t ≤ m = ε. Note that the girth cycles containing the edge r1w are exactly the cycles
of form (w, r1, x, y, w), where x ∈ {v} ∪ (D12 \ {s1}) and y ∈ (Γ(w) ∩ D21) \ {r1}.
520 Ars Math. Contemp. 23 (2023) #P4.01 / 509–530
Therefore, n(r1w) ≤ (k1 − 1)(t− 1) ≤ (k − 1)(ε− 1). Since γ = 1 and σ = ε, we have
by Lemma 3.8 that
a1 ≥ max{(k2 − 1− ε)(k1 − 1), (k1 − 2)(k2 − 1)} − ε ≥ (k1 − 2)(k2 − 1)− ε,
and so
(k1 − 2)(k2 − 1)− ε ≤ a1 ≤ n(r1w) ≤ (k1 − 1)(ε− 1).
It follows that k1k2 − 2k2 + 1 ≤ k1ε, and so
k2 − 2 +
1
k1
≤ k2 −
2k2
k1
+
1
k1
≤ ε < k2 − 1,
contradicting the fact that ε is an integer.
We now give an upper bound for a1.
Lemma 4.3. Assume that g = 4 and ε ≥ 1. Let α = h(sn−1) and β = h(rm−1). Then
a1 ≤ (α− 1)(k1 − 1) + ε+ (k2 − α)(ε− α− σ + 1). (4.1)
and
a1 ≤ (β − 1)(k2 − 1) + ε+ (k1 − β)(ε− β − γ + 1). (4.2)
Proof. We prove inequality (4.1). The proof of inequality (4.2) is similar. Let {w1, . . . , wα}
= Γ(sn−1) ∩D23 . We estimate n(sn−1w1). To do this we split the girth cycles (w1, sn−1,
x, y, w1) into two types depending on the vertex x. We say that the girth cycle is of type 1
if x ∈ {w2, . . . , wα}, and of type 2 if x ∈ {u} ∪D21 . By Lemma 4.1(i) there are at most
(α− 1)(k1− 1) girth cycles of type 1. To estimate the number of girth cycles of type 2, we
further split these girth cycles into two subfamilies depending on the vertex y. Let us say
that the girth cycle (w1, sn−1, x, y, w1) with x ∈ {u} ∪D21 is of type 2a if y ∈ D12 , and of
type 2b if y ∈ D32 .
If the girth cycle is of type 2b, then x ∈ D21 , and so by Lemma 4.1(ii) there are at most
ε such girth cycles. To estimate the number of girth cycles of type 2a, observe that sn−1
has k2 − α neighbours in {u} ∪D21 , and that w1 has at most ε− α− σ + 1 neighbours in
D12 \{sn−1}. This shows that the number of girth cycles of type 2a is at most (k2−α)(ε−
α− σ + 1). As a1 ≤ n(sn−1w1), the result follows.
Lemma 4.4. Assume that g = 4 and ε ≥ 1. Then ε = k2 − 2 and k2 − 1 ≥ 2k1/3.
Proof. As in Lemma 4.3, let α = h(sn−1). Then, by Lemmas 3.8 and 4.3, we get that
(k1 − 1)(k2 − 1)− σ(k1 − 1)− ε ≤ (α− 1)(k1 − 1) + ε+ (k2 − α)(ε− α− σ + 1).
Rearranging the above inequality we find this is equivalent to
(k1 − 1)(k2 − 1) ≤ (k1 − k2 − 1 + α)(α+ σ − 1) + ε(k2 − α+ 2). (4.3)
Taking into account that α+ σ ≤ ε and that α ≥ 1, inequality (4.3) implies that
(k1 − 1)(k2 − 1) ≤ (k1 + 1)ε− k1 + 1 + k2 − α ≤ (k1 + 1)ε− k1 + k2,
Gy. Kiss et al.: On girth-biregular graphs 521
and so
ε ≥ (k1 − 1)(k2 − 1) + k1 − k2
k1 + 1
= k2 −
3k2 − 1
k1 + 1
. (4.4)
As k1 ≥ k2, the above inequality yields
ε ≥ k2 −
3k1 − 1
k1 + 1
= k2 − 3 +
4
k1 + 1
> k2 − 3.
Recall that ε < k2 − 1 by assumption, and so ε = k2 − 2 as claimed. Plugging ε = k2 − 2
into (4.4) we easily get that k2 − 1 ≥ 2k1/3.
Theorem 4.5. Assume that g = 4. Then ε = 0 and Γ is the complete bipartite graph
Kk1,k2 .
Proof. Suppose on the contrary that ε ≥ 1. Recall that ε = k2 − 2. As in Lemma 4.3, let
β = h(rm−1). Then, by Lemmas 3.8 and 4.3, we get that
(k1 − 1)(k2 − 1)− γ(k2 − 1)− ε ≤ (β − 1)(k2 − 1) + ε+ (k1 − β)(ε− β − γ + 1).
Rearranging the terms of the above inequality we get
(k1 − 1)(k2 − 1) ≤ ε(k1 − β + 2) + (β + γ − 1)(k2 − k1 + β − 1). (4.5)
If β = 1, then inequality (4.5) together with ε = k2 − 2 yields k1 − 1 ≤ γ(k2 − k1). But
this is a contradiction as k1 > k2 > 0.
If k2 − k1 + β − 1 ≤ 0, then inequality (4.5) together with ε = k2 − 2 and β ≥ 2 yields
(k1 − 1)(k2 − 1) ≤ (k2 − 2)(k1 − β + 2) ≤ (k2 − 2)k1,
implying k1 ≤ k2 − 1, a contradiction.
Therefore, we have that k2 − k1 + β − 1 > 0 and β ≥ 2. Recall that β + γ ≤ ε = k2 − 2,
and so inequality (4.5) gives us
(k1−1)(k2−1) ≤ ε(k1−β+2)+(ε−1)(k2−k1+β−1) = (k2−2)(k2+1)−k2+k1−β+1.
It follows that 2 ≤ β ≤ k22 − k2 − 2 + 2k1 − k1k2, or
k1(k2 − 2) ≤ k22 − k2 − 4.
As k1 ≥ k2 +1 this yields −2 ≤ −4, a contradiction. This shows that ε = 0 as claimed. It
is now easy to see that Γ is isomorphic to the complete bipartite graph Kk1,k2 .
5 The case g = 2d ≥ 8, where d is even
In this section we study girth-biregular graphs with girth g = 2d ≥ 8, d even. Throughout
this section we will use Notation 3.5. Assume that g = 2d ≥ 8. For every z ∈ D21 we
define
β(z) =
∑
r∈Ddd−1∩Γd−2(z)
h(r).
522 Ars Math. Contemp. 23 (2023) #P4.01 / 509–530
Note that for z ∈ D21 we have |Ddd−1 ∩ Γd−2(z)| = (k1 − 1)(d−2)/2(k2 − 1)(d−2)/2 and
that for z, z′ ∈ D21 (z ̸= z′), the sets Ddd−1 ∩ Γd−2(z) and Ddd−1 ∩ Γd−2(z′) are disjoint
as the girth of Γ is 2d. Therefore,∑
z∈D21
β(z) =
∑
r∈Ddd−1
h(r) = ε. (5.1)
In particular, β(z) ≤ ε. Recall also that for an edge e of Γ we denoted by n(e) the number
of girth cycles passing through e.
Lemma 5.1. Assume that g = 2d ≥ 8 and ε ≥ 1. Then
a1 ≥ (k1 − 1)d/2(k2 − 1)d/2 − k2ε.
Proof. Abbreviate ℓ = (k1 − 1)(d−2)/2(k2 − 1)(d−2)/2. Pick z ∈ D21 with n(vz) = a1
and let w1, w2, . . . , wℓ be the vertices of Ddd−1 ∩ Γd−2(z). For 1 ≤ j ≤ ℓ consider the
2d-cycles of the form (v, z, . . . , wj , b, r, r′, . . .) with b ∈ Dd−1d , where (v, z, . . . , wj) is the
unique path from v to wj of length d− 1. Observe that for fixed wj and r, there is only one
such cycle (recall that as g ≥ 8, wj and r have a unique common neighbour), and that for
fixed wj and b, we could choose r in k2 − 1− h(b) different ways. Therefore,
a1 = n(vz) ≥
ℓ∑
j=1
∑
b∈Γ(wj)∩Dd−1d
(k2 − 1− h(b))
=
ℓ∑
j=1
∑
b∈Γ(wj)∩Dd−1d
(k2 − 1)−
ℓ∑
j=1
∑
b∈Γ(wj)∩Dd−1d
h(b).
(5.2)
Furthermore, observe that for a fixed wj we could choose b in (k1 − 1 − h(wj)) different
ways, and so
ℓ∑
j=1
∑
b∈Γ(wj)∩Dd−1d
(k2−1) = (k2−1)
ℓ∑
j=1
(k1−1−h(wj)) = ℓ(k1−1)(k2−1)−(k2−1)β(z).
Finally, the sets Γ(wj)∩Dd−1d and Γ(wℓ)∩D
d−1
d are disjoint if j ̸= ℓ (otherwise we would
get a cycle of length 2d− 2), and so
ℓ∑
j=1
∑
b∈Γ(wj)∩Dd−1d
h(b) ≤
∑
b∈Dd−1d
h(b) = ε.
This, together with β(z) ≤ ε, shows that
a1 = n(vz) ≥ ℓ(k1− 1)(k2− 1)− (k2− 1)β(z)− ε ≥ (k1− 1)d/2(k2− 1)d/2−k2ε.
Lemma 5.2. Assume that g = 2d ≥ 8 and ε ≥ 1. Then
a1 < (k1 − 1)(d−2)/2(k2 − 1)(d−2)/2
(
k1ε− k1 + 2
)
.
Gy. Kiss et al.: On girth-biregular graphs 523
Proof. Let
D =
d−1⋃
i=0
(
Dii+1 ∪Di+1i
)
.
For vertices x, y ∈ D, let dD(x, y) denote the distance between x and y in the subgraph
Γ[D], that is, in the subgraph of Γ, that is induced by D. Observe that dD(x, y) ≤ 2d − 1
for all x, y ∈ D.
Pick a vertex r ∈ Ddd−1 with h(r) ≥ 1 and abbreviate α = h(r). Pick w ∈ Γ(r)∩D
d+1
d
and consider the set C of 2d-cycles (x0 = w, x1 = r, x2, . . . , x2d−1, w) through wr. Note
that, as w ̸∈ D at most 2d − 2 edges of such a cycle have both endpoints in D. For
1 ≤ i ≤ 2d − 1 let Ci denote the subset of C defined as follows. A cycle (x0 = w, x1 =
r, x2, . . . , x2d−1, w) is an element of Ci if and only if {x1, . . . , xi} ⊆ D and xi+1 ̸∈ A,
where the addition in subscripts is computed modulo 2d. For example, cycles in C1 are
those 2d-cycles (x0 = w, x1 = r, x2, . . . , x2d−1, w) , for which x2 ̸∈ D, while cycles in
C2d−1 are those for which {x1, x2, . . . , x2d−1} ⊆ D. Note that the sets Ci are pairwise
disjoint, and so
a1 ≤ n(wr) ≤ |C1|+ |C2|+ · · ·+ |C2d−1|.
Let us now estimate the above sum. To do this we introduce the following notation. For
i ∈ {1, 3, . . . , 2d− 1} we define
εi =
∑
b∈Dd−1d
dD(r,b)=i
h(b).
Note that as Γ[D] is bipartite with diameter at most 2d− 1, we have that
ε1 + ε3 + · · ·+ ε2d−1 = ε.
We also define
κ = |Γ(w) ∩ (Ddd−1 \ {r})| = |Γ(w) ∩Ddd−1| − 1.
Note that α+ κ ≤ ε.
Consider a 2d-cycle (x0 = w, x1 = r, x2, . . . , x2d−1, w) ∈ C1. Observe that there
are α − 1 choices for x2. For each such choice of x2, there are, by Lemma 3.1, at most
(k1 − 1)(d−2)/2(k2 − 1)d/2 girth cycles containing both edges wr and rx2. Therefore,
|C1| ≤ (α− 1)(k1 − 1)(d−2)/2(k2 − 1)d/2.
Consider a 2d-cycle (x0 = w, x1 = r, x2, . . . , x2d−1, w) ∈ C2 ∪ C4 ∪ · · · ∪ C2d−2.
Assume that this cycle is an element of C2j (1 ≤ j ≤ d − 1). Observe that in this
case we have that x2j ∈ Dd−1d and that dD(r, x2j) = 2j − 1 (otherwise there would be
a cycle of length less than 2d). Therefore, we could choose an edge x2jx2j+1 in ε2j−1
different ways. For each such choice of an edge x2jx2j+1, there are, by Lemma 3.1, at
most (k1 − 1)(d−2)/2(k2 − 1)(d−2)/2 girth cycles containing edges wr and x2jx2j+1, and
so
|C2|+ |C4|+ · · ·+ |C2d−2| ≤ (ε1 + ε3 + · · ·+ ε2d−3)(k1 − 1)(d−2)/2(k2 − 1)(d−2)/2
= ε(k1 − 1)(d−2)/2(k2 − 1)(d−2)/2.
524 Ars Math. Contemp. 23 (2023) #P4.01 / 509–530
Consider a 2d-cycle (x0 = w, x1 = r, x2, . . . , x2d−1, w) ∈ C3 ∪ C5 ∪ · · · ∪ C2d−3. If
this cycle is an element of C2j+1 (1 ≤ j ≤ d−2), then it is easy to see that x2j+1 ∈ Ddd−1,
and so x2j+2 ∈ Dd+1d \ {w}. Therefore, there are at most ε − κ − α choices for an edge
x2j+1x2j+2. For each such choice there are, by Lemma 3.1, at most (k1 − 1)(d−4)/2(k2 −
1)(d−2)/2 girth cycles containing edges wr and x2j+1x2j+2, and so
|C3|+ |C5|+ · · ·+ |C2d−3| ≤ (ε− κ− α)(k1 − 1)(d−4)/2(k2 − 1)(d−2)/2.
Finally, consider a 2d-cycle (x0 = w, x1 = r, x2, . . . , x2d−1, w) ∈ C2d−1. Note that
we have at most k1−α choices for a vertex x2. For each choice of vertices x2, x3, . . . , xi−1,
where i ≤ d, we have at most k1 − 1 choices for vertex xi if i is even, and k2 − 1 choices
for xi if i is odd. Therefore, there are at most (k1 − α)(k1 − 1)(d−2)/2(k2 − 1)(d−2)/2
choices for vertices x2, x3, . . . , xd. On the other hand, there are at most κ choices for a
vertex x2d−1. For each such choice of vertices x2, x3, . . . , xd and x2d−1, there is at most
one girth cycle containing the edges wr, rx2, x2x3, . . . xd−1xd and x2d−1w. Therefore,
|C2d−1| ≤ κ(k1 − α)(k1 − 1)(d−2)/2(k2 − 1)(d−2)/2.
To further estimate the sum |C1|+ |C2|+ · · ·+ |C2d−1|, we first note that
|C1|+ |C2d−1| ≤ (k1 − 1)(d−4)/2(k2 − 1)(d−2)/2(
(α− 1)(k1 − 1)(k2 − 1) + κ(k1 − α)(k1 − 1)
)
< (k1 − 1)(d−4)/2(k2 − 1)(d−2)/2(
(α− 1)(k1 − 1)2 + κ(k1 − α)(k1 − 1)
)
= (k1 − 1)(d−4)/2(k2 − 1)(d−2)/2(
(α− 1 + κ)(k1 − 1)2 − κ(α− 1)(k1 − 1)
)
≤ (k1 − 1)(d−4)/2(k2 − 1)(d−2)/2(α− 1 + κ)(k1 − 1)2
≤ (k1 − 1)d/2(k2 − 1)(d−2)/2(ε− 1),
while
|C2|+ |C3|+ · · ·+ |C2d−2| ≤ (k1 − 1)(d−4)/2(k2 − 1)(d−2)/2(ε(k1 − 1) + (ε− κ− α))
≤ (k1 − 1)(d−4)/2(k2 − 1)(d−2)/2
(
ε(k1 − 1) + ε− 1
)
= (k1 − 1)(d−4)/2(k2 − 1)(d−2)/2(k1ε− 1).
Therefore,
a1 ≤ n(wr) ≤ |C1|+ |C2|+ · · ·+ |C2d−1|
≤ (k1 − 1)(d−4)/2(k2 − 1)(d−2)/2
(
(ε− 1)(k1 − 1)2 + k1ε− 1
)
= (k1 − 1)(d−4)/2(k2 − 1)(d−2)/2
(
ε(k21 − k1) + ε− (k1 − 1)2 − 1
)
< (k1 − 1)(d−4)/2(k2 − 1)(d−2)/2
(
ε(k21 − k1) + (k2 − 1)− (k1 − 1)2
)
< (k1 − 1)(d−4)/2(k2 − 1)(d−2)/2
(
ε(k21 − k1) + (k1 − 1)− (k1 − 1)2
)
= (k1 − 1)(d−2)/2(k2 − 1)(d−2)/2
(
k1ε− k1 + 2
)
.
The result follows.
Gy. Kiss et al.: On girth-biregular graphs 525
Theorem 5.3. Assume that g = 2d ≥ 8 and d is even. Then ε = 0 and Γ is the incidence
graph of a finite thick generalized d-gon, hence either d = 4 or d = 8.
Proof. Suppose first that ε is positive. By Lemma 5.1 and 5.2 we have
(k1 − 1)d/2(k2 − 1)d/2 − k2ε ≤ a1 < (k1 − 1)(d−2)/2(k2 − 1)(d−2)/2
(
k1ε− k1 + 2
)
.
This implies
k2 − 1 > ε >
(k1 − 1)(d−2)/2(k2 − 1)(d−2)/2(k1k2 − k2 − 1)
k2 + k1(k1 − 1)(d−2)/2(k2 − 1)(d−2)/2
>
(k1 − 1)(d−2)/2(k2 − 1)(d−2)/2(k1k2 − k2 − 1)
k1
(
1 + (k1 − 1)(d−2)/2(k2 − 1)(d−2)/2
)
= k2 − 2 +
(k1 − 1)(d−2)/2(k2 − 1)(d−2)/2(2k1 − k2 − 1)− k1(k2 − 2)
k1
(
1 + (k1 − 1)(d−2)/2(k2 − 1)(d−2)/2
)
As k1(k2 − 2) < (k1 − 1)(k2 − 1) < (k1 − 1)(d−2)/2(k2 − 1)(d−2)/2(2k1 − k2 − 1), the
above inequality implies
k2 − 1 > ε > k2 − 2,
contradicting the fact that ε is an integer. Therefore, ε = 0.
6 The case g = 2d, where d is odd
In this section we consider the case g = 2d with d odd, in particular the case g = 6 when
we provide a characterization of affine planes. Unfortunately, the method we applied in the
proof of Lemma 5.2 for giving an upper estimate on b1 does not work for odd d, but we can
calculate the exact value of b1 if ε = 1. Throughout this section we will use Notation 3.5.
Theorem 6.1. Assume that d is odd and suppose that ak1 = bk2 = M − 1. Then b1 =
M − k2 + 1 and b2 = · · · = bk2 = M − 1.
Proof. Pick adjacent vertices u ∈ A, v ∈ B such that n(uv) = ak1 = bk2 = M − 1. Let
Dij denote D
i
j(u, v) and
D =
d−1⋃
i=0
(
Dii+1 ∪Di+1i
)
.
For vertices x, y ∈ D, let dD(x, y) denote the distance between x and y in the subgraph
Γ[D], that is, in the subgraph of Γ, that is induced by D. Observe that dD(x, y) ≤ 2d − 1
for all x, y ∈ D.
By Proposition 2.6(vi) and (vi) we have that
|Dd−1d | = |D
d
d−1| = (k1 − 1)(g−2)/4(k2 − 1)(g−2)/4 =
M
k2 − 1
,
and there are M − 1 edges between Dd−1d and Ddd−1. Hence all but one vertices in Ddd−1
have k2 − 1 neighbours in Dd−1d . Let p ∈ Ddd−1 denote the unique vertex which has only
k2 − 2 neighbours in Dd−1d .
526 Ars Math. Contemp. 23 (2023) #P4.01 / 509–530
We claim that all but one vertices in Dd−1d have k2 − 1 neighbours in Ddd−1, too.
Let x be any vertex in Dd−1d . Then for each vertex y ∈ D21 there is at most one vertex
z ∈ Ddd−1 ∩ Γ(x) so that d(y, z) = d − 2, because otherwise a cycle of length 2(d − 1)
would appear. Thus
|Γ(x) ∩Ddd−1| ≤ |D21| = k2 − 1.
This implies, by the pigeonhole principle, that there is a unique vertex r ∈ Dd−1d which has
only k2− 2 neighbours in Ddd−1. Then r has one neighbour in D
d−2
d−1 and it has k1− k2+1
neighbours outside D.
Now, let w be an arbitrary vertex in D21 and let S = D
d
d−1 \D
d−2
d−1(w, v). Then
Ddd−1(w, v) = D
d−2
d−1 ∪ S.
We now describe the set Dd−1d (w, v). Observe that
Dd−1d (w, v) ⊆ D
d−1
d ∪ {p1}, (6.1)
where p1 is the unique neighbour of p outside D. There are two possibilities we have
to consider, namely either w is the unique vertex of D21 for which dD(p, w) = d − 2,
or dD(p, w) = d. Let us first consider the case dD(p, w) = d − 2. Note that in this
case p1 ∈ Dd−1d (w, v), so there is a unique vertex w1 ∈ D
d−1
d which is not contained in
Dd−1d (w, v). Observe that every vertex from D
d−1
d , which has k2 − 1 neighbours in Ddd−1,
is at distance d− 1 from w, and so w1 = r. Therefore (6.1) implies
Dd−1d (w, v) =
(
Dd−1d \ {r}
)
∪ {p1}.
We now count the number of neighbours between Dd−1d (w, v) and D
d
d−1(w, v). Recall
that each vertex from Dd−1d has a unique neighbour in D
d−2
d−1 and that each vertex from
Dd−1d \ {r} has k2 − 1 neighbours in Ddd−1. Pick x ∈ D
d−1
d \ {r}. As x ∈ D
d−1
d (w, v), x
has at least one neighbour in Ddd−1\S. On the other hand, if x has more than one neighbour
in Ddd−1 \ S, then this would imply a cycle of length 2(d − 1), a contradiction. Using the
above observations we now have
n(vw) =
(
|Dd−1d | − 1
)
(k2 − 1)
=
(
(k1 − 1)(d−1)/2(k2 − 1)(d−1)/2 − 1
)
(k2 − 1)
= M − k2 + 1.
In the case when dD(p, w) = d − 2 we have that d(w, p1) = d + 1 (note that p is the
only neighbour of p1 in D), and so by (6.1) we have Dd−1d (w, v) = D
d−1
d . Observe also
that |S| = |Ddd−1| − |D
d−2
d−1(w, v)| = (k1 − 1)(d−1)/2(k2 − 1)(d−3)/2(k2 − 2). Similar
arguments as in the previous case now show that
n(vw) = |Dd−1d |+ |S|(k2 − 1)− 1
= (k1 − 1)(d−1)/2(k2 − 1)(d−1)/2 + (k1 − 1)(d−1)/2(k2 − 1)(d−1)/2(k2 − 2)− 1
= M − 1.
This proves the statement.
Gy. Kiss et al.: On girth-biregular graphs 527
Theorem 6.2. Assume that d is odd and k2 does not divide k1. If ak+1 = bk = M − ε for
a non-negative integer ε ≤ 1, then ε = 0 and a1 = · · · = ak+1 = b1 = · · · = bk = M.
Proof. We first assume ε = 1 and derive a contradiction. If ε = 1, then it follows from
Theorem 6.1 that the signature of any vertex from B is (M − k2 + 1,M − 1, . . . ,M − 1).
Now Proposition 3.2 yields that the signature of any vertex from A is (M−k2+1, . . . ,M−
k2+1,M−1, . . . ,M−1). Let a = M−k2+1 and let aA and aB be as in Proposition 3.3.
Observe that aB = 1 and so we have k2aA = k1 by Proposition 3.3. Hence k1 is divisible
by k2, a contradiction. Therefore ε = 0 and the result now follows from Theorem 3.4.
In particular, we consider the case k1−1 = k2 = k and d = 3. Then k1k2−k1+1 = k2
and it is well-known that a 2−(k2, k, 1) design is a finite affine plane of order k. Combining
Theorems 3.4(vii) and 6.2 we get the following characterization.
Corollary 6.3. Assume that k1 − 1 = k2 = k and that d = 3. If ak+1 = bk = M − ε for a
non-negative integer ε ≤ 1, then ε = 0 and Γ is the incidence graph of a finite affine plane
of order k.
7 Examples
In this section we provide some examples where ak1 is close to the upper bound given
in Theorem 3.4. In all cases, the signatures of the points are constants, hence each edge
is contained in the same number of girth cycles. So our examples are edge-girth-regular
graphs, too. Let us start with the g = 4 case.
Example 7.1. Let f1 > f2 ≥ 1 and h > 2 be integers and consider the complete bipartite
graph Γ′ = Kf1h,f2h with bipartition A and B. Label the vertices so that
A =
f1⋃
i=1
{u1,i, u2,i, . . . , uh,i} , B =
f2⋃
j=1
{v1,j , v2,j , . . . , vh,j} .
Let Γ denote a graph that is obtained from Γ′ by deleting all edges of the form uℓ,ivℓ,j ,
where ℓ ∈ {1, 2, . . . , h}, i ∈ {1, 2, . . . , f1} and j ∈ {1, 2, . . . , f2}. Then Γ is a bipartite
biregular graph with g = 4, k1 = f2(h− 1) and k2 = f1(h− 1).
Take any edge e = uℓ1,ivℓ2,j in Γ. Then ℓ1 ̸= ℓ2, and there are ((f2(h − 1) − 1)
(f1(h − 1) − 1) 3-arcs of Γ which contain e. Let us now count how many of these 3-arcs
are not contained in a 4-cycle. Let A = vℓ′,j′uℓ1,ivℓ2,juℓ′′,i′′ be any 3-arc containing edge
e. Note that ℓ′ ̸= ℓ1 and ℓ′′ ̸= ℓ2. Then A is not contained in a 4-cycle if and only if
vertices vℓ′,j′ and uℓ′′,i′′ are not adjacent in Γ, which happens if and only if ℓ′ = ℓ′′. As
ℓ′ ̸= ℓ1 and ℓ′′ ̸= ℓ2, there are h−2 choices for ℓ′ = ℓ′′, hence there are f1f2(h−2) 3-arcs
containing e, that are not contained in a 4-cycle. So the number of girth cycles through e
in Γ is ((f2(h− 1)− 1) (f1(h− 1)− 1)− f1f2(h− 2). It follows that Γ is girth-biregular
with
a1 = · · · = ak1 = b1 = · · · = bk2 = (k1− 1)(k2− 1)− f1f2(h− 2) = M − f1f2(h− 2).
For g = 6 we follow the examples of the paper [1].
528 Ars Math. Contemp. 23 (2023) #P4.01 / 509–530
Example 7.2. Take an affine plane of order q and remove i parallel classes. Consider
the incidence graph of this structure. The lines still have size q and the points have degree
q+1−i, so it is a bipartite biregular graph with valencies q and q+1−i. To count the girth
cycles containing the edge corresponding to an incident point-line pair (e0, P0), we have to
choose a point P0 ̸= P1 ∈ e0, and a line e0 ̸= e1 through P0 and complete it to a girth cycle
(of length 6) by choosing a point P0 ̸= P2 ∈ e1 and a line e2 joining P1 and P2. There are
q − 1 ways to choose P1 and q − i ways of choosing e1. For e2 we have to choose a line
different from e1, not parallel to e0, so we have (q− 1− i) possibilities, since the point P2
will just be the unique point of e0∩e2. So, in total there are M ′ = (q−1)(q− i)(q−1− i)
girth cycles through the edge (e0, P0).
In particular, when we have an affine plane of order q, its incidence graph is a bipartite
biregular graph with valencies q+1 and q, and we have M = (q−1)2q girth cycles through
an edge. If there is an affine plane of order q + 1 as well, then removing i = 2 parallel
classes will also give us a bipartite biregular graph with valencies q+1 and q and this graph
will have M ′ = q(q − 1)(q − 2) = M − q(q − 1) girth cycles through every edge.
Another construction from the paper [1] is the following.
Example 7.3. Let us consider a Steiner system on v points and line size k. Delete a point
P ∗ and all the lines through the deleted point. The incidence graph of the resulting structure
will be a bipartite biregular graph with valencies k and r−1, again with r = (v−1)/(k−1).
One can more or less copy the argument in the previous example: using the same notation,
the point P1 can be chosen in (k−1) ways. Now consider the line e∗ in the original Steiner
system that joins P1 and P ∗. If the line e1 intersects e∗, then we have (k − 2) choices for
P2 and e2, and there are (k−2) such lines in the original Steiner system. So, this case gives
(k− 1)(k− 2)2 girth cycles. There remain (r− 2)− (k− 2) = r− k lines through P0, not
intersecting e∗. If e1 is one of them, then there are (k−1) ways to extend it to a girth cycle.
This is (k− 1)2(r− k) possibility, so in total we have (k− 1)((k− 2)2 + (r− k)(k− 1))
girth cycles containing the edge (e0, P0).
It is easy to extend Example 7.2 to resolvable Steiner systems.
Example 7.4. Consider a resolvable Steiner system and denote by v the number of points,
by r the degrees of points, where r = (v − 1)/(k − 1). In this case k divides v, and the
original design will have (k − 1)2(r − 1) girth cycles through any edge. If we remove
i parallel classes of lines, then the incidence graph of the resulting structure will have
degrees k and r − i). For determining the number of girth cycles containing an edge start
from an incident point-line pair (P0, e0) as before. Take a point P1 on e0 and let U be the
set of points which are on the lines through P1 that belong to the deleted parallel classes.
This implies that |U | = i(k − 1). Let rj , j = 0, . . . , k − 1, be the number of lines
through P0 which intersect U in exactly j points. Clearly, we have
∑
j rj = r − 1, and∑
j jrj = |U | = i(k − 1). On a line ℓ through having j points in U , we can choose the
point P2 of the girth cycle in (k − 1− j) ways. This way we get in total
k−1∑
j=0
(k − 1− j)rj = (k − 1)(r − 1)− i(k − 1)
girth cycles for a given choice of P1, so the total number of girth cycles will be
(k − 1)2((r − 1)− i). For small i this is close to our upper bound.
Gy. Kiss et al.: On girth-biregular graphs 529
In particular, we mention two examples arising from higher dimensional finite spaces.
1. Let n = 2m + 1. Remove the qm + 1 elements of a line spread from PG(n, q) and
denote the correponding point-line incidence graph by Γ. Then Γ is a girth-biregular
bipartite graph with g = 6, k1 = q2m + · · ·+ q and k2 = q + 1 and its signature is
a1 = · · · = ak1 = b1 = · · · = bk2 = q2(q2m + · · ·+ q − 2) = M − q2.
2. Let us remove the qn−1 elements of a class of parallel lines from AG(n, q) and
denote the corresponding point-line incident graph by Γ. Then Γ is a girth-biregular
bipartite graph with g = 6, k1 = qn−1 + · · ·+ q and k2 = q and its signature is
a1 = · · · = ak1 = b1 = · · · = bk2 = (q− 1)2(qn−1 + · · ·+ q− 2) = M − (q− 1)2.
In both cases the magnitude of ε is only k2/(n−1)1 .
In the case g = 8 our examples come from incidence graphs of generalized quadrangles.
For a detailed descriptions of generalized quadrangles, their ovoids and spreads, we refer
the reader to the book of Payne and Thas [13].
Example 7.5. Let G′ = (P,L, I) be a generalized quadrangle of order (s, t) and Γ′ be the
Levi graph of G′.
Suppose that G′ admits a spread S (a set of st + 1 lines, no two of which intersect).
Delete the lines of S. Then the Levi graph Γ of G = (P,L \ S, I) is a bipartite graph with
bipartition |A| = (s + 1)(st + 1) and |B| = t(st + 1), valencies s + 1 and t and g = 8.
We claim that it is also girth-biregular with
a1 = · · · = as+1 = b1 = · · · = bt = s2(t2 − 3t+ 2) = M − s2(t− 1).
Dually, if G′ admits an ovoid O (a set of st+ 1 points, no two of which are collinear),
then the Levi graph Γ of G = (P \ O,L, I) is a girth-biregular graph with valencies s and
t+ 1, and
a1 = · · · = as+1 = b1 = · · · = bt = t2(s2 − 3s+ 2) = M − t2(s− 1).
In G for any incident point-line pair (P, ℓ) there are (t − 1)s points in P which are
collinear with P but are not incident with ℓ, and there are s(t−1) lines in which meet ℓ but
are not incident with P . Let R be one of these points and e be one of these lines. Then there
is a unique point-line pair (T, f) in G′ so that R I f IT I e. Thus in Γ′ there are s2(t− 1)2
girth cycles through the edge which corresponds to the pair (P, ℓ). For a fixed R there is a
unique element f ∈ S through R. All the s other points on f determines a unique 8-cycle
which contains (P, ℓ). No two elements of S intersect, hence there are (t− 1)s · s deleted
8-cycles. Thus in Γ′ the total number of girth cycles through the edge corresponding to
(P, ℓ) is
s2(t− 1)2 − s(t− 1)s = s2(t2 − 3t+ 2) = s2(t− 1)(t− 2).
Among the known generalized quadrangles only a few admit a spread or an ovoid. In
particular, the classical generalized quadrangle Q(5, q) admits a spread. In this case Γ has
valencies q+1 and q2, and the number of girth cycles through every edge is q2(q2−1)(q2−
2) = M − q2(q2 − 1). So the magnitude of ε is M2/3.
530 Ars Math. Contemp. 23 (2023) #P4.01 / 509–530
ORCID iDs
György Kiss https://orcid.org/0000-0003-3312-9575
Štefko Miklavič https://orcid.org/0000-0002-2878-0745
Tamás Szőnyi https://orcid.org/0000-0001-7184-8496
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doi:10.1002/jcd.21836, https://doi.org/10.1002/jcd.21836.
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1684.b0d.
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P4.02 / 531–551
https://doi.org/10.26493/1855-3974.2698.6fe
(Also available at http://amc-journal.eu)
An extension of the Erdős-Ko-Rado theorem to
uniform set partitions
Karen Meagher * , Mahsa N. Shirazi
Department of Mathematics and Statistics, University of Regina,
Regina, SK, S4S 0A2, Canada
Brett Stevens †
School of Mathematics and Statistics, Carleton University,
Ottawa, ON, K1S 5B6, Canada
Received 14 September 2019, accepted 25 December 22, published online 27 February 2023
Abstract
A (k, ℓ)-partition is a set partition which has ℓ blocks each of size k. Two (k, ℓ)-
partitions P and Q are said to be partially t-intersecting if there exist blocks Pi in P and
Qj in Q such that |Pi ∩Qj | ≥ t. In this paper we prove a version of the Erdős-Ko-
Rado theorem for partially 2-intersecting (k, ℓ)-partitions. In particular, we show for ℓ
sufficiently large, the set of all (k, ℓ)-partitions in which a block contains a fixed pair is the
largest set of 2-partially intersecting (k, ℓ)-partitions. For k = 3, we show this result holds
for all ℓ.
Keywords: Erdős-Ko-Rado Theorem, uniform set partitions, ratio bound, clique, coclique, quotient
graphs.
Math. Subj. Class. (2020): 05E30, 05C50, 05C25
1 Introduction
In 1961, Erdős, Ko, and Rado proved that if F is a t-intersecting family of k-subsets of
{1, 2, . . . , n}, then
(
n−t
k−t
)
is a tight upper bound on the size of F , provided that n is suffi-
ciently large [7]. This result has motivated consideration of “intersecting” families of many
*Corresponding author. Research supported by NSERC Discovery Research Grant, Application No.: RGPIN-
2018-03952.
†Research supported by NSERC Discovery Research Grant, Application No.: RGPIN-2017-06392.
E-mail addresses: karen.meagher@uregina.ca (Karen Meagher), mahsa.nasrollahi@gmail.com (Mahsa N.
Shirazi), brett@math.carleton.ca (Brett Stevens)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
532 Ars Math. Contemp. 23 (2023) #P4.02 / 531–551
other combinatorial objects using diverse proof techniques and has developed into an ac-
tive and broad area of research. There are many recent results giving analogs of the EKR
theorem; see, for example, [9, 13, 16, 20, 26] or [12] and the references within. In this
work, we prove an extension of the EKR theorem to systems of uniform set partitions.
A (k, ℓ)-partition is a set partition of {1, 2, . . . , kℓ} with exactly ℓ blocks each of size
k. These are also called uniform set partitions. We use Uk,ℓ to denote the set of all (k, ℓ)-
partitions, and uk,ℓ = |Uk,ℓ|. It is easy to see that
uk,ℓ =
1
ℓ!
(
kℓ
k
)(
kℓ− k
k
)(
kℓ− 2k
k
)
· · ·
(
k
k
)
=
(kℓ)!
(k!)ℓℓ!
. (1.1)
In [8], Erdős and Székely considered different types of intersection for partitions. In
one of these types, and the one we consider here, two partitions P and Q are intersecting in
a pair if there exist blocks Pi in P , and Qj in Q such that |Pi ∩Qj | ≥ 2. In [20], Meagher
and Moura generalized this definition: two partitions P and Q are partially t-intersecting if
there exist Pi in P , and Qj in Q such that |Pi ∩Qj | ≥ t. The work of Meagher and Moura
is different than that of Erdős and Székely since only uniform partitions are considered
in [20].
A set of partitions is partially t-intersecting if the partitions in the set are pairwise par-
tially t-intersecting. Meagher and Moura [20] conjectured if P ⊂ Uk,ℓ is a set of partially
t-intersecting partitions, with t ≤ k, then |P| ≤
(
kℓ−t
k−t
)
uk,ℓ−1. A set of this size can be
formed by fixing a t-subset T and taking all (k, ℓ)-partitions with a block that contains
T ; such a set is called canonically t-intersecting. Meagher and Moura further conjec-
tured that only the canonically t-intersecting (k, ℓ)-partitions attain this maximum size. As
pointed out by Brunk in [2], this conjecture additionally requires that k ≤ ℓ(t− 1), since if
k > ℓ(t− 1), then any two (k, ℓ)-partitions are t-partially intersecting.
If k = t = 2, then the (2, ℓ)-partitions are perfect matchings in the complete graph
on 2ℓ vertices, and partially 2-intersecting is equivalent to intersecting (as sets of edges).
The Meagher-Moura conjecture has been proven in this case in [13], so in this paper we
only consider k ≥ 3. In particular, we prove the Meagher-Moura conjecture for t = 2
with k = 3 and all values of ℓ, and for all k ≥ 4, provided that ℓ is sufficiently large. Our
approach is to define a graph in which the cocliques (also known as independent sets) are
equivalent to partially 2-intersecting (k, ℓ)-partitions from Uk,ℓ. Then we use a version of
the algebraic method from [13] to find the size of a maximum coclique in the graph. This
is an approach that has been very effective in proving many EKR-type results, indeed it
is the main topic of the book [12]. This method is particularly effective when considering
intersecting permutations in groups and it has been applied to many families of groups, see
for example [5, 6, 17, 21, 22, 23, 25].
2 Overview of method
In a graph X a clique is a set of vertices which induce a complete subgraph; and a coclique
is a set of vertices which induce an empty subgraph. The size of a largest clique and
a largest coclique are denoted by ω(X) and α(X), respectively. The adjacency matrix
A(X) of X is a matrix in which rows and columns are indexed by the vertices in X and
the (i, j)-entry is 1 if i and j are adjacent, and 0 otherwise. The eigenvalues of X refer
to the eigenvalues of its adjacency matrix. We use 1 to denote the all-ones vector; for any
d-regular graph, the all-ones vector is an eigenvector with eigenvalue d.
K. Meagher et al.: An extension of the Erdős-Ko-Rado theorem to uniform set partitions 533
In general, finding the largest coclique of a graph X is known to be NP-hard, but the
Delsarte-Hoffman (ratio) bound gives an upper bound on α(X). This bound is based on
the ratio between the largest and the smallest eigenvalue of the adjacency matrix of the
graph. A proof of this result can be found in [4] or in [12, Section 2.4], we also recommend
Haemers’ paper [15] on the history of this bound.
Theorem 2.1 (Delsarte-Hoffman bound [4]). Let A be the adjacency matrix for a d-regular
graph X on vertex set V (X). If the least eigenvalue of A is τ , then
α(X) ≤ |V (X)|
1− dτ
.
If equality holds for some coclique S with characteristic vector νS , then
νS −
|S|
|V (X)|
1
is an eigenvector with eigenvalue τ .
Define Xk,ℓ to be the graph with Uk,ℓ as its vertex set, in which two partitions P and Q
are adjacent if every pair of blocks, one from P and one from Q, have at most 1 element
in common. The group Sym(kℓ) acts transitively on the vertices of Xk,ℓ and preserves the
edges. This means the Xk,ℓ is vertex transitive and regular. We will denote the degree by
dk,ℓ, or simply d when the context is clear.
A resolvable packing design on kℓ points with block size k and index λ = 1 is equiva-
lent to a clique in this graph. Further, a resolvable balanced incomplete block design on kℓ
points with block size k and index λ = 1, if it exists, gives a maximum clique.
For any distinct i, j ∈ {1, . . . , kℓ}, let Si,j be the subset of partitions in Uk,ℓ for which
the elements i and j are in the same block. Then Si,j is a coclique in the graph Xk,ℓ and
the size of Si,j is
1
(ℓ− 1)!
(
kℓ− 2
k − 2
)(
kℓ− k
k
)
· · ·
(
k
k
)
.
The main goal in this paper is to prove, using the ratio bound, that Si,j is a maximum
coclique in Xk,ℓ. For the ratio bound to hold with equality, we need to prove if τ is the
least eigenvalue of Xk,ℓ, then
1− dk,ℓ
τ
=
uk,ℓ
|Si,j |
=
kℓ− 1
k − 1
.
Thus we need to prove two facts: first that τ = −dk,ℓ(k−1)k(ℓ−1) is an eigenvalue of Xk,ℓ; and
second that τ is the least eigenvalue of Xk,ℓ.
In Section 3, we show how the eigenvalues of Xk,ℓ are related to the irreducible charac-
ters of Sym(kℓ), and we prove some bounds on the degrees of these irreducible characters.
Next, in Section 4, we calculate three of the eigenvalues of Xk,ℓ; one of these eigenval-
ues is the τ above. Next we prove if there is an eigenvalue of Xk,ℓ that is strictly smaller
than τ , there is a function that is an upper bound on the eigenvalue’s multiplicity. In Sec-
tion 6 we prove that this function is bounded by
(
kℓ
3
)
−
(
kℓ
2
)
for ℓ sufficiently large. This
uses the result from Section 5, that the limit of ratio uk,ℓ/dk,ℓ is finite as ℓ goes to ∞.
The bounds from Section 3 then prove that no such eigenvalues exist and this proves the
534 Ars Math. Contemp. 23 (2023) #P4.02 / 531–551
Meagher-Moura Conjecture with t = 2, for all values of k, provided that ℓ is sufficiently
large. Finally, in Section 7, we prove a weaker bound on uk,ℓ/dk,ℓ when k = 3 that holds
for all ℓ. Thus we prove the Meagher-Moura Conjecture for t = 2, k = 3 for all values
of ℓ.
3 Representations of the symmetric group
In this section we will explain the connection between the eigenvalues of the graph Xk,ℓ
and the irreducible characters of the symmetric group. We also recall some results on the
degree of the irreducible characters that are involved in the eigenvalues.
For any character χ of Sym(n), we can consider its restriction to H ≤ Sym(n) which
is denoted by res (χ)H . Similarly if χ is a character of H ≤ Sym(n), then its induced
character on Sym(n) is denoted by ind (χ)Sym(n). The trivial character on a group H is
denoted by 1H .
The stabilizer of a partition in Uk,ℓ is the group Sym(k) ≀ Sym(ℓ) (this is called the
wreath product of Sym(k) and Sym(ℓ)). The cosets Sym(kℓ)/(Sym(k) ≀ Sym(ℓ)) are
in one-to-one correspondence with the partitions of Uk,ℓ. The action of Sym(kℓ) on the
partitions is equivalent to the action of Sym(kℓ) on the cosets Sym(kℓ)/(Sym(k)≀Sym(ℓ))
and this action is clearly transitive. The permutation representation of this action is
ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ)
.
The module for this representation can be thought of as the vector space of length-uk,ℓ
vectors with the characteristic vectors of P ∈ Uk,ℓ, denoted by vP , as its basis. The group
Sym(kℓ) acts on this vector space by the action on the partitions, for any σ ∈ Sym(kℓ) the
action is σ(vP ) = vPσ .
This representation can be decomposed as the sum of irreducible representations of
Sym(kℓ). If the multiplicity of each irreducible representation in the decomposition is
equal to 1, then the representation is called multiplicity-free. In general, the group Sym(k) ≀
Sym(ℓ) is not multiplicity free in Sym(kℓ). In fact it is not multiplicity free unless k = 2,
ℓ = 2, or (k, ℓ) is one of (3, 3), (4, 3), (5, 3) or (3, 4) [11].
3.1 Orbital association scheme
The set of orbitals of the action of a group G on a set Ω is the set of orbits of the action of
G on Ω×Ω. Each orbital of Sym(kℓ) on Sym(kℓ)/(Sym(k) ≀Sym(ℓ)) can be represented
by an object called a meet table. The meet table for two (k, ℓ)-partitions is a ℓ× ℓ array in
which the (i, j)-entry is |Pi ∩Qj |. Two meet tables are isomorphic if one can be obtained
from the other by permuting the rows and the columns. In [12, Section 15.4] it is shown that
the set of non-isomorphic meet tables corresponds to the set of orbitals. For each orbital
O there is a corresponding meet table M ; this means for P,Q ∈ Uk,ℓ the meet table of
P and Q is M if and only if (P,Q) ∈ O. Further, each orbital can be represented as a
uk,ℓ × uk,ℓ matrix, with the (P,Q)-entry equal to 1 if and only if the meet table of P and
Q is isomorphic to the table representing the orbital. The set of these uk,ℓ × uk,ℓ-matrices
of the orbitals forms an association scheme if and only if ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ) is
multiplicity-free. In general, these matrices form a homogeneous coherent configuration.
K. Meagher et al.: An extension of the Erdős-Ko-Rado theorem to uniform set partitions 535
The graph Xk,ℓ is the union of the orbitals from the action of Sym(kℓ) on the cosets
Sym(kℓ)/(Sym(k) ≀ Sym(ℓ)) that are represented by a meet table that has no entry greater
than 1. This means for every permutation σ ∈ Sym(kℓ) its action on the partitions is an
automorphism of Xk,ℓ. In particular, if Mσ is the permutation representation of σ, then
Mσ−1 A(Xk,ℓ)Mσ = A(Xk,ℓ).
Further, if v is any θ-eigenvector of Xk,ℓ, then Mσv is also a θ-eigenvector. This implies
the eigenspaces of Xk,ℓ are invariant under the action of Sym(kℓ) and thus a union of
irreducible modules in the decomposition of
ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ)
.
We say that an eigenvalue θ belongs to a module if the module is a subspace of the θ-
eigenspace.
3.2 Degree of the irreducible characters of Sym(kℓ)
In this section we will give some results on the irreducible representations of Sym(n). We
refer the reader to [24], or any similar reference on this topic, for details and background.
It is well-known that the irreducible representations of Sym(n) correspond to integer parti-
tions on n. We will use λ ⊢ n to indicate that λ is an integer partition of n, this means that
λ = [λ1, λ2, . . . , λj ], each λi is an integer with
∑j
i=1 λi = n. We will use χλ to represent
the irreducible character of Sym(n) corresponding to the partition λ.
From [13] we have a list of irreducible representations of the symmetric group with
small degree.
Lemma 3.1. For n ≥ 9, let χ be a character of Sym(n) with degree less than (n2−n)/2.
If χλ is a constituent of χ, then λ is one of the following partitions of n:
[n], [1n], [n− 1, 1], [2, 1n−2], [n− 2, 2], [2, 2, 1n−4], [n− 2, 1, 1], [3, 1n−3].
This proof uses the branching rule, which we state here. For a proof of this rule
see [3, Corollary 3.3.11].
Lemma 3.2. Let λ ⊢ n, then
res (χλ)Sym(n−1) =
∑
χλ− ,
where the sum is taken over all partitions λ− of n − 1 that have a Young diagram which
can be obtained by the deletion of a single box from the Young diagram of λ. Further,
ind (χλ)
Sym(n+1)
=
∑
χλ+ ,
where the sum is taken over partitions λ+ of n + 1 that have a Young diagram which can
be obtained by the addition of a single box to Young diagram of λ.
Using the same approach as the proof for Lemma 3.1 we can get a second family of
irreducible characters with slightly larger, but still small degree.
536 Ars Math. Contemp. 23 (2023) #P4.02 / 531–551
Lemma 3.3. For n ≥ 13, let χ be an irreducible character of Sym(n) with degree less
than
(
n
3
)
−
(
n
2
)
. If χλ is a constituent of χ, then λ is one of the following partitions of n:
[n], [1n], [n− 1, 1], [2, 1n−2], [n− 2, 2], [2, 2, 1n−4],
[n− 2, 1, 1], [3, 1n−3], [n− 3, 3], [2, 2, 2, 1n−6].
Proof. The hook length formula confirms that each of the 10 characters above have degree
less than or equal to
(
n
3
)
−
(
n
2
)
.
We prove this result by induction. For n = 13 and 14 this can be calculated directly
using the GAP character table library [10]. We assume for n ≥ 14 that the lemma holds
for n and n− 1, and we will prove that the lemma holds for n+ 1.
Assume that χ is an irreducible character of Sym(n+ 1) that has dimension less than(
n+ 1
3
)
−
(
n+ 1
2
)
=
(n+ 1)n(n− 4)
6
,
but is not one of the ten irreducible characters listed in the statement of the lemma. We will
show that such a χ cannot exist.
If one of the ten irreducible characters of Sym(n) with degree less than
(
n
3
)
−
(
n
2
)
is a
constituent of res (χ)Sym(n), then we can determine the possible constituents of χ with the
branching rule.
Constituent of res (χ)Sym(n) Constituents of χ
[n] [n+ 1], [n, 1]
[n− 1, 1] [n, 1], [n− 1, 2], [n− 1, 1, 1]
[n− 2, 2] [n− 1, 2], [n− 2, 3], [n− 2, 2, 1]
[n− 2, 1, 1] [n− 1, 1, 1], [n− 2, 2, 1], [n− 2, 1, 1, 1]
[n− 3, 3] [n− 2, 3], [n− 3, 4], [n− 3, 3, 1]
[1n] [2, 1n−1], [1n+1]
[2, 1n−2] [3, 1n−2], [2, 2, 1n−3], [2, 1n−1]
[2, 2, 1n−4] [3, 2, 1n−4], [2, 2, 2, 1n−5], [2, 2, 1n−3]
[3, 1n−3] [4, 1n−3], [3, 2, 1n−4], [3, 1n−2]
[2, 2, 2, 1n−6] [3, 2, 2, 1n−6], [2, 2, 2, 2, 1n−7], [2, 2, 2, 1n−5]
Table 1: Constituents of χ, if res (χ)Sym(n) has a constituent with degree less than(
n
3
)
−
(
n
2
)
.
By Frobenius reciprocity, for any character ϕ of Sym(n)
⟨res (χ)Sym(n), ϕ⟩Sym(n) = ⟨χ, ind (ϕ)
Sym(n+1)⟩Sym(n+1).
K. Meagher et al.: An extension of the Erdős-Ko-Rado theorem to uniform set partitions 537
Character Degree
[n− 3, 4] (n+ 1)n(n− 1)(n− 7)/24
[n− 3, 3, 1] (n+ 1)n(n− 2)(n− 5)/8
[n− 2, 2, 1] (n+ 1)(n− 1)(n− 3)/3
[n− 2, 1, 1, 1] n(n− 1)(n− 2)/6
[2, 2, 2, 2, 1n−8] (n+ 1)n(n− 1)(n− 7)/24
[3, 2, 2, 1n−6] (n+ 1)n(n− 2)(n− 5)/8
[3, 2, 1n−4] (n+ 1)(n− 1)(n− 3)/3
[4, 1n−3] n(n− 1)(n− 2)/6
Table 2: Degrees of the characters from Table 1 that are larger than (n+1)n(n−4)6 for n ≥ 13.
This means if ϕ is a constituent of res (χ)Sym(n), then χ is one of the constituents of
ind (ϕ)
Sym(n+1). The possible constituents of ind (ϕ)Sym(n+1) are recorded in Table 1;
the second column lists the irreducible characters that, according to the branching rule, are
constituents of representation of Sym(n+ 1) induced by the character in the first column.
From these lists, and the degrees of the characters given in Table 2, we see that either
χ is one of the ten listed in the theorem, or the degree of χ is larger than
(
n
3
)
−
(
n
2
)
(again,
the degrees are calculated using the hook length formula). Thus res (χ)Sym(n) does not
contain any of the ten irreducible characters of Sym(n) in the statement of the theorem.
Next consider the case where the decomposition of res (χ)Sym(n) contains at least two
irreducible characters of Sym(n) which are not in the list of the ten irreducible characters
with dimension less
(
n
3
)
−
(
n
2
)
= n(n− 1)(n− 5)/6. In this case, the degree of χ must be
at least n(n− 1)(n− 5)/3. But since n > 7, this is strictly larger than (n+1)n(n− 4)/6.
Finally we need to consider the case where res (χ)Sym(n) contains exactly one irre-
ducible character of Sym(n), which is not one of the ten listed in the theorem. By the
branching rule the only irreducible characters of Sym(n+ 1) for which res (χ)Sym(n) con-
tains only one irreducible character have a rectangular Young diagram, so χ = χ[st] for
some s and t.
Next consider res (χ)Sym(n−1), this is the restriction of χ = χ[st] to Sym(n− 1). By
the branching rule, this can contain only the irreducible characters of n−1 that correspond
to the partitions λ′ = [st−1, s− 2] and λ′′ = [st−2, s− 1, s− 1].
If λ′ is one of the ten partitions that correspond to irreducible characters of Sym(n− 1)
with degree less than
(
n−1
3
)
−
(
n−1
2
)
, then one of the following cases must hold:
• t = 1 and λ′ = [n− 1] and s = n+ 1,
• t = 2 and λ′ = [n− 1, 1], [n− 2, 2] or [n− 3, 3], and s ≤ 5, or
• 2 < t < 4 and and s ≤ 2.
The first of these cases implies χ = [n + 1], which contradicts the degree of χ, and none
of the other cases can happen, since n = st and n is assumed to be at least 13.
Similarly, assume λ′′ = [st−2, s−1, s−1] is one of the partitions corresponding to the
ten characters of Sym(n− 1) that have degree less than
(
n−1
3
)
−
(
n−1
2
)
. Then one of the
following cases must hold:
538 Ars Math. Contemp. 23 (2023) #P4.02 / 531–551
• t = 2 and λ′′ = [s− 1, s− 1] and s ≤ 4,
• 2 < t ≤ 5 and λ′′ = [st−1, 1, 1] and s ≤ 2, or
• s = 1.
The first two cases imply that n ≤ 10 and the final case implies that χ = [1(n+1)] which
has degree 1.
Thus res (χ)Sym(n−1) has two characters with degree at least
(
n−1
3
)
−
(
n−1
2
)
, so the
degree of χ is at least (n− 1)(n− 2)(n− 6)/3, which is strictly greater than (n+1)n(n−
4)/6 for n ≥ 13. This is a contradiction, so no such χ exists.
Next we will show that there are only three irreducible characters in the decomposition
of ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ) that have degree no more than
(
kℓ
3
)
−
(
kℓ
2
)
. To do this we
will consider the action of different Young subgroups on Uk,ℓ. For any integer partition
λ ⊢ n we will denote the Young subgroup by
Sym(λ) = Sym(λ1)× Sym(λ2)× · · · × Sym(λk).
Theorem 3.4. Assume kℓ ≥ 13 and k ≥ 3. Then the only partitions in the decomposition
of ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ) with dimension less than or equal to
(
kℓ
3
)
−
(
kℓ
2
)
are
χ[kℓ], χ[kℓ−2,2], χ[kℓ−3,3].
Proof. Lemma 3.3 lists the 10 irreducible representations of Sym(kℓ) with dimension no
more than
(
kℓ
3
)
−
(
kℓ
2
)
. We only need to show which of these representations are in the
decomposition. The tool we use is Frobenius reciprocity along with the action of different
Young subgroups on Uk,ℓ.
By Frobenius reciprocity〈
ind (1Sym(λ))
Sym(kℓ)
, ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ)
〉
Sym(kℓ)
=〈
1, res (ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ)
)
Sym(λ)
〉
Sym(λ)
.
The second inner product above gives the number of orbits of the action of Sym(λ) on the
cosets Sym(kℓ)/(Sym(k) ≀ Sym(ℓ)); or, equivalently, the number of orbits of Sym(λ) on
the partitions in Uk,ℓ. Using this fact with different Young subgroups will allow us to deter-
mine that many of the representations with small degree do not occur in the decomposition
of ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ).
To start, it is clear that Sym(kℓ) has one orbit on the (k, ℓ)-partitions, so χ[kℓ] has
multiplicity 1 in the decomposition. Next consider the group Sym([kℓ− 1, 1]), it is also
straight-forward that this group only has one orbit on the partitions. Using the definition of
the Specht modules and the labelling of the irreducible characters of the symmetric group
it is straight forward to see that
ind (1Sym([kℓ−1,1]))
Sym(kℓ)
= χ[kℓ] + χ[kℓ−1,1],
so we have that 〈
χ[kℓ] + χ[kℓ−1,1], ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ)
〉
= 1.
Since we know that χ[kℓ] occurs in this decomposition with multiplicity 1, this implies that
χ[kℓ−1,1] does not occur in the decomposition of ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ).
K. Meagher et al.: An extension of the Erdős-Ko-Rado theorem to uniform set partitions 539
Next we consider the group Sym([kℓ− 2, 2]). This group has two orbits on the par-
titions of Uk,ℓ. Again, from the definition of the Specht modules and the labelling of the
irreducible characters, we have that〈
ind (1Sym([kℓ−2,2]))
Sym(kℓ)
, ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ)
〉
=〈
χ[kℓ] + χ[kℓ−1,1] + χ[kℓ−2,2], ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ)
〉
= 2.
This implies that χ[kℓ−2,2] occurs in the decomposition of ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ) with
multiplicity 1.
We continue this process with the group Sym([kℓ− 2, 1, 1]). It has two orbits on the
partitions of Uk,ℓ. Since
ind (1Sym([kℓ−2,1,1]))
Sym(kℓ)
= χ[kℓ] + χ[kℓ−1,1] + χ[kℓ−2,2] + χ[kℓ−2,1,1],
we conclude that χ[kℓ−2,1,1] does not occur in the decomposition.
Next, we consider the group Alt(kℓ) ∩ Sym([kℓ− 2, 2]). This group has two orbits
on the partitions of Uk,ℓ. Again the decomposition of ind (1Alt(kℓ)∩Sym([kℓ−2,2]))
Sym(kℓ) is
well-known (a proof can be found in [11, Proposition 1.4]) and we have
ind (1Alt(kℓ)∩Sym([kℓ−2,2]))
Sym(kℓ)
= χ[kℓ] + χ[kℓ−1,1] + χ[kℓ−2,2] + χ[kℓ−2,1,1]
+ χ[1kℓ] + χ[2,1kℓ−2] + χ[2,2,1kℓ−4] + χ[3,1kℓ−3].
This implies that none of χ[1kℓ], χ[2,1kℓ−2], χ[2,2,1kℓ−4] and χ[3,1kℓ−3] occur in the decom-
position of ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ).
Next we consider the group Sym([kℓ− 3, 3]). This group has three orbits on the parti-
tions of Uk,ℓ and from the decomposition of ind (1Sym([kℓ−3,3]))
Sym(kℓ) we have that
⟨ind (1Sym([kℓ−3,3]))
Sym(kℓ)
, ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ)⟩ =
⟨χ[kℓ] + χ[kℓ−1,1] + χ[kℓ−2,2] + χ[kℓ−3,3], ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ)⟩ = 3.
This implies that χ[kℓ−3,3] occurs in the decomposition of ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ) with
multiplicity 1.
Next we consider the group Alt(kℓ) ∩ Sym([kℓ− 2, 1, 1]). This group has 2 orbits on
the partitions of Uk,ℓ and
ind (1Alt(kℓ)∩Sym([kℓ−2,1,1]))
Sym(kℓ)
= χ[kℓ] + χ[kℓ−1,1] + χ[kℓ−2,2] + χ[kℓ−2,1,1]
χ[1kℓ] + χ[2,1kℓ−2] + χ[2,2,1kℓ−4] + χ[3,1kℓ−3].
Since χ[kℓ], and χ[kℓ−2,2] are in the decomposition, none of the irreducible representations
χ[1kℓ], χ[2,1kℓ−2], χ[2,2,1kℓ−4], or χ[3,1kℓ−3] occur in the decomposition of
ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ).
Finally, we consider the group Alt(kℓ)∩ Sym([kℓ− 3, 3]). This group has three orbits
on the partitions of Uk,ℓ and
ind (1Alt(kℓ)∩Sym([kℓ−3,3]))
Sym(kℓ)
= χ[kℓ] + χ[kℓ−1,1] + χ[kℓ−2,2] + χ[kℓ−3,3]
+ χ[1kℓ] + χ[2,1kℓ−2] + χ[22,1kℓ−4] + χ[23,1kℓ−6].
Which shows χ[2,2,2,1kℓ−6] is not in the decomposition of ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ).
540 Ars Math. Contemp. 23 (2023) #P4.02 / 531–551
4 Eigenvalues of Xk,ℓ with k ≥ 3
In this section we will find three of the eigenvalues of Xk,ℓ. For ease of notation, we will
denote the irreducible representation of χλ by the λ-module. Also, the number of vertices
in Xk,ℓ, which is equal to uk,ℓ, will be denoted simply by v and the degree of the graph
Xk,ℓ will be simply written as d, rather than dk,ℓ.
Any subgroup H ≤ Sym(kℓ) acts on the vertices of Xk,ℓ and the orbits of this action
form an equitable partition. From any equitable partition, we can form a quotient graph and
the eigenvalues of this quotient graph will be eigenvalues of the Xk,ℓ (details can be found
in [13, Section 2.2]). The trivial case is H = Sym(kℓ), since this group is transitive, the
equitable partition has all the vertices of Xk,ℓ in a single part. The quotient graph for this
is simply the 1 × 1 matrix with the single entry d. The eigenvalue of this matrix is simply
d, and the eigenvector is the all ones vector and the eigenspace is isomorphic to the trivial
representation of Sym(kℓ). So d belongs to the [kℓ]-module.
Since the subgroup Sym([kℓ− 1, 1]) has only one orbit on the vertices of Xk,ℓ, the next
subgroup we consider is the Young subgroup Sym([kℓ− 2, 2]), considered as the stabilizer
of the set {1, 2}. The action of Sym([kℓ− 2, 2]) on the partitions will give us another
eigenvalue of the graph.
Lemma 4.1. For integers k and ℓ, with k, ℓ ≥ 2, τ = − (k−1)dk(ℓ−1) is an eigenvalue of Xk,ℓ
with multiplicity at least
(
kℓ
2
)
−
(
kℓ
1
)
.
Proof. The action of Sym([kℓ− 2, 2]) on the (k, ℓ)-partitions has exactly 2 orbits: S1, the
set of all partitions that have 1 and 2 in the same block, and S2, the set of all partitions in
which 1 and 2 are in different blocks. The orbit S1 is a coclique in Xk,ℓ so the quotient
matrix for this partition has the form(
0 d
−τ d+ τ
)
. (4.1)
The eigenvalues of the quotient matrix (4.1) are d and τ . We can calculate the value of
τ by counting edges between S1 and S2. Since S1 is a coclique, each vertex in it is adjacent
to exactly d vertices in S2, and each vertex in S2 is adjacent to −τ vertices in S1. Using
the sizes of S1 and S2, we have that the number of edges between S1 and S2 is equal to
|S1| d =
(
kℓ− 2
k − 2
)
uk,ℓ−1 d
and also to
|S2|(−τ) =
(
kℓ− 2
k − 1
)(
kℓ− k − 1
k − 1
)
uk,ℓ−2(−τ).
Thus
τ = − (k − 1)d
k(ℓ− 1)
(4.2)
is a second eigenvalue for Xk,ℓ. Since this eigenvalue arises from the action of
Sym([kℓ− 2, 2]), it belongs to a module that is common between the two representations
ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ) and ind (1Sym([kℓ−2,2]))
Sym(kℓ). Thus it belongs to the mod-
ule [kℓ − 2, 2], as this is the only common module, and must have dimension at least(
kℓ
2
)
−
(
kℓ
1
)
.
K. Meagher et al.: An extension of the Erdős-Ko-Rado theorem to uniform set partitions 541
We denote this eigenvalue by τ since in Section 6 it will be shown that τ is the least
eigenvalue of Xk,ℓ, provided that ℓ is sufficiently large. We also note that a second irre-
ducible module may also have τ as the eigenvalue belonging to it, so the multiplicity of τ
could be higher than the degree of the [kℓ− 2, 2]-module.
Next we will consider the Young subgroup Sym([kℓ− 3, 3]), thought of as the group
that stabilizes the set {1, 2, 3}. The action of this subgroup on Uk,ℓ has 3 orbits: T1, the set
of all partitions with 1, 2, 3 in the same block; T2 the set of all partitions in which 1, 2, 3 are
in exactly two different blocks; and T3 the set of all partitions in which 1, 2, 3 are in three
different blocks. Any vertex in T1 is adjacent only to vertices in T3. Similarly, a vertex in
T2 can be adjacent to vertices in T2 and T3. The quotient graph for this equitable partition
is
M =
0 0 d0 a d− a
b c d− b− c
 ,
where a, b, c are all non-negative.
The eigenvalues for this quotient graph will be the eigenvalues that belong to modules
that are both in the decomposition of ind (1Sym([kℓ−3,3]))
Sym(kℓ) and the decomposition
of ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ). Thus the eigenvalues will belong to the [kℓ], [kℓ − 2, 2]
and [kℓ − 3, 3] modules. We have already seen that the eigenvalue for [kℓ] is d, and the
eigenvalue for [kℓ− 2, 2] is τ . We will denote the eigenvalue belonging to [kℓ− 3, 3] by θ.
Since the trace of the matrix is the sum of the eigenvalues we have that
d+ a− b− c = d+ τ + θ. (4.3)
The number of edges between T1 and T3 is equal to
d|T1| = d
(
kℓ− 3
k − 3
)
uk,ℓ−1,
and also to
b|T3| = b
(
kℓ− 3
k − 1
)(
kℓ− k − 2
k − 1
)(
kℓ− 2k − 1
k − 1
)
uk,ℓ−3.
Setting these equations equal to each other, then expanding the binomial coefficients and
rearranging yields
(k − 1)(k − 2)
k2(ℓ− 1)(ℓ− 2)
d = b.
Replacing d = −k(ℓ−1)k−1 τ shows that
b = − (k − 1)(k − 2)
k2(ℓ− 1)(ℓ− 2)
k(ℓ− 1)
(k − 1)
τ = − k − 2
k(ℓ− 2)
τ. (4.4)
Putting this into Equation 4.3 produces the following formula
θ = a+
k − 2
k(ℓ− 2)
τ − c− τ = a− c+ (k − 2)− k(ℓ− 2)
k(ℓ− 2)
τ. (4.5)
542 Ars Math. Contemp. 23 (2023) #P4.02 / 531–551
Similarly, counting the number of edges between T2 and T3 yields
3
(
kℓ− 3
k − 2
)(
kℓ− k − 1
k − 1
)
uk,ℓ−2(d− a) =(
kℓ− 3
k − 1
)(
kℓ− k − 2
k − 1
)(
kℓ− 2k − 1
k − 1
)
uk,ℓ−3(c).
Again, expanding the binomial coefficients and rearranging shows that
a = d− (ℓ− 2)k
3(k − 1)
c.
The characteristic polynomial of M is
x3 + (−a+ b+ c− d)x2 + (−ab+ ad− bd− cd)x+ abd.
Substituting in the values we have computed for b and c, and using the fact that τ is a root
of the characteristic polynomial we get
a =
2(k − 1)
k(ℓ− 1)
d. (4.6)
From this we can compute that
c =
3(kℓ− 3k + 2)(k − 1)
k2(ℓ− 1)(ℓ− 2)
d. (4.7)
Lemma 4.2. For integers k and ℓ, with k, ℓ ≥ 3,
θ =
2(k − 1)(k − 2)d
k2(ℓ− 1)(ℓ− 2)
is an eigenvalue of Xk,ℓ with multiplicity at least
(
kℓ
3
)
−
(
kℓ
2
)
.
Proof. By Equations (4.5), (4.6) and (4.7), we can calculate that
θ =
2(k − 1)(k − 2)d
k2(ℓ− 1)(ℓ− 2)
. (4.8)
From the comments above, θ = 2(k−1)(k−2)dk2(ℓ−1)(ℓ−2) is the eigenvalue belonging to the unique
[kℓ − 3, 3]-module in ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ). Since the dimension of the irreducible
representation of [kℓ− 3, 3] is
(
kℓ
3
)
−
(
kℓ
2
)
, the multiplicity of θ is at least
(
kℓ
3
)
−
(
kℓ
2
)
.
5 Bound on degree of Xk,ℓ
In this section we will find a lower bound on the degree of Xk,ℓ for all sufficiently large ℓ.
If P and Q are two partitions that are adjacent in Xk,ℓ, then the meet table of P and Q is
an ℓ × ℓ matrix with entries either 0 or 1, and further, the entries in each row and column
in the meet table sum to k. We define Mk,ℓ to be the set of all such meet tables, so all
ℓ× ℓ matrices with entries either 0 or 1, and row and columns sums equal to k. To find the
K. Meagher et al.: An extension of the Erdős-Ko-Rado theorem to uniform set partitions 543
degree of Xk,ℓ, we first state a result on the number of such meet tables. Next, for a fixed
partition P and a meet table M ∈ Mk,ℓ, we count the number of partitions Q for which
the meet table of P and Q is M .
Bender [1] determined the asymptotic cardinality of Mk,ℓ. (In fact, Bender found a
much more general result, but we only state the result that we need here.)
Theorem 5.1 (Bender [1]). For positive integers k, ℓ
lim
ℓ→∞
(k!)2ℓ
(kℓ)!
|Mk,ℓ| = e−
(k−1)2
2 .
To get a lower bound on dk,ℓ, we fix a partition P in Uk,ℓ, then for each M ∈ Mk,ℓ,
we will count the number of Q so that the meet table of P and Q is M , then we use
Theorem 5.1 to bound the size of Mk,ℓ.
Lemma 5.2. For positive integers k, ℓ with k ≤ ℓ,
dk,ℓ =
k!ℓ
ℓ!
|Mk,ℓ|.
Proof. Fix a partition P ∈ Uk,ℓ. Define a bipartite multigraph with the vertices in one part
the meet tables in Mk,ℓ, and the vertices in the other part the partitions in the neighbour-
hood of P in Xk,ℓ. Two vertices M and Q are adjacent if the meet table of P and Q is M .
By counting the number of edges in this graph in two ways, we will determine the size of
the neighbourhood of P in terms of |Mk,ℓ|.
For any M ∈ Mk,ℓ, with M = [mi,j ] assume that row i corresponds to the block
Pi ∈ P . Construct a partition Q = {Q1, Q2, . . . , Qℓ} so that the block Qj corresponds to
column j of M and |Pi∩Qj | = mi,j . Since the entries of a row in M are either 0 or 1, and
sum to k, there are k! ways to select how the elements from Pi will be distributed to the
blocks of Q. So for each meet table M , there are k!ℓ partitions Q that can be constructed
this way. It is possible that some of these partitions are equal, once the blocks are reordered,
so this is a multigraph.
For every Q in the neighbourhood of P , there are ℓ! ways to order the blocks of Q,
once the blocks are ordered the meet table for P and Q is uniquely defined. In the bipartite
graph, Q is adjacent to each of these tables in the graph (again, these tables may not be
distinct, so the graph is a multigraph). The degree of every vertex Q is ℓ!.
Thus we have that the number of edges in the multigraph is
ℓ!dk,ℓ =
∑
M∈Mk,ℓ
k!ℓ,
and the result follows.
Using Theorem 5.1 we have the asymptotic size of dk,ℓ.
Corollary 5.3. For a fixed integer k with k ≥ 2,
lim
ℓ→∞
uk,ℓ
dk,ℓ
= e
(k−1)2
2 .
544 Ars Math. Contemp. 23 (2023) #P4.02 / 531–551
Proof. From Equation (1.1) and Lemma 5.2,
uk,ℓ
dk,ℓ
=
(kℓ)!
(k!)ℓℓ!
ℓ!
k!ℓ|Mk,ℓ|
=
(kℓ)!
(k!)2ℓ|Mk,ℓ|
.
The result then follows from Theorem 5.1.
Thus for every ϵ > 0, there exists an ℓ′ such that for all ℓ ≥ ℓ′,
uk,ℓ
dk,ℓ
≤ e
(k−1)2
2 + ϵ.
6 A bound on the multiplicity of eigenvalues with large absolute value
In Section 4 we found three eigenvalues, d, τ , and θ of Xk,ℓ, and the ratio between d and
τ is dτ =
k(1−ℓ)
k−1 . If τ is the least eigenvalue of the graph, then by the ratio bound any
coclique will have size no more than
|V (Xk,ℓ)|
1− dτ
=
|V (Xk,ℓ)|
1− k(1−ℓ)k−1
= uk,ℓ−1.
This is exactly the size of a set of canonically 2-intersecting (k, ℓ)-partitions. Thus our goal
in this section is to show that τ is the least eigenvalue of Xk,ℓ. To this end, we first show if
Xk,ℓ has an eigenvalue λ with λ2 > τ2, then there is a bound on the multiplicity of λ.
Let
{d(1), τ (mτ ), θ(mθ), λ(m2)2 , . . . , λ
(mj)
j }
be the spectrum of the matrix Xk,ℓ, where the values mi represent the multiplicities of the
eigenvalues. By squaring A and taking the trace, we have
vd = d2 +mττ
2 +mθθ
2 +
j∑
i=2
miλ
2
i .
Hence for every 2 ≤ i ≤ j we have
vd− d2 −mττ2 −mθθ2 ≥ miλ2i .
Assume λi is an eigenvalue of Xk,ℓ with λ2i > τ
2, and also that λi is not the eigenvalue
belonging to the [kℓ], [kℓ− 2, 2] or [kℓ− 3, 3] modules, then
vd− d2 −mττ2 −mθθ2
τ2
≥ mi.
Expanding θ using Equation (4.8) in the above equation produces the following equation(v
d
− 1
) k2(ℓ− 1)2
(k − 1)2
−mθ
4(k − 2)2
k2(ℓ− 2)2
−mτ ≥ mi.
Further, by Lemmas 4.1 and 4.2, it is known that mτ ≥
(
kℓ
2
)
−
(
kℓ
1
)
and mθ ≥
(
kℓ
3
)
−
(
kℓ
2
)
,
so this bound becomes(v
d
− 1
) k2(ℓ− 1)2
(k − 1)2
− (kℓ)(kℓ− 1)(kℓ− 5)
6
4(k − 2)2
k2(ℓ− 2)2
− (kℓ)(kℓ− 3)
2
≥ mi.
K. Meagher et al.: An extension of the Erdős-Ko-Rado theorem to uniform set partitions 545
Our next step is to show that this upper bound on mi is smaller than
(
kℓ
3
)
−
(
kℓ
2
)
. This
will be a contradiction with Theorem 3.4 since we have assumed that λ does not belong to
any of the [kℓ], [kℓ− 2, 2], and [kℓ− 3, 3] modules. In other words, we need to prove that
v
d
− 1 <
ℓ(k − 1)2
(
k2(ℓ− 2)2(kℓ− 4)(kℓ+ 1) + 4(k − 2)2(kℓ− 1)(kℓ− 5)
)
6k3(ℓ− 1)2(ℓ− 2)2
. (6.1)
In the next result, we will see that this will follow from Corollary 5.3.
Theorem 6.1. Fix an integer k ≥ 3. For ℓ sufficiently large, the largest set of partially
2-intersecting uniform (k, ℓ)-partitions has size(
kℓ− 2
k − 2
)
uk,ℓ−1.
Proof. For any distinct i, j ∈ {1, . . . , kℓ}, the set Si,j of all (k, ℓ)-partitions with i and j
in the same block form a set of partially 2-intersecting (k, ℓ)-partitions of the size given in
the theorem.
Corollary 5.3 shows that vd approaches a fixed constant, namely e
(k−1)2
2 , as ℓ goes
to infinity. Since the right hand side of Equation (6.1) grows linearly in ℓ, we have that
Equation (6.1) holds for ℓ sufficiently large. This implies if there is an eigenvalue λ of Xk,ℓ
with λ ≤ τ , then the multiplicity of λ is less than or equal to
(
kℓ
3
)
−
(
kℓ
2
)
.
By Theorem 3.4, eigenspaces with dimension less than or equal to
(
kℓ
3
)
−
(
kℓ
2
)
can only
include the [kℓ], [kℓ− 2, 2] or the [kℓ− 3, 3]-modules. The degree, d, is the eigenvalue be-
longing to the [kℓ]-module, and Lemma 4.1 and Lemma 4.2 shows that τ is the eigenvalue
belonging to the [kℓ − 2, 2]-module and θ belongs to the [kℓ − 3, 3]-module. So we can
conclude that τ = − (k−1)dk(ℓ−1) is the least eigenvalue of Xk,ℓ and that τ belongs only to the
[kℓ− 2, 2]-module.
By the ratio bound, Theorem 2.1, the maximum size of coclique in Xk,ℓ is
|V (Xk,ℓ)|
1− dτ
=
v
1− d
− (k−1)d
k(ℓ−1)
=
v
1 + k(ℓ−1)k−1
=
v(k − 1)
kℓ− 1
=
(
kℓ− 2
k − 2
)
uk,ℓ−1.
The previous result shows that the sets Si,j are the largest intersecting sets. We further
conjecture that these sets are the only maximum intersecting sets.
Conjecture 6.2. For k ≥ 3 and ℓ sufficiently large, the only sets of partially 2-intersecting
(k, ℓ)-partitions with size
(
kℓ−2
k−2
)
uk,ℓ−1 are the sets Si,j .
We can make a step towards this conjecture with the following weaker characterization
of the maximum intersecting sets. Denote the characteristic vectors of the sets Si,j by vi,j .
Corollary 6.3. For a fixed integer k ≥ 3 and ℓ sufficiently large, let S be any maximum
partially 2-intersecting set of (k, ℓ)-partitions. Then the characteristic vector of S is a
linear combination of the vectors vi,j .
Proof. For k ≥ 3 and ℓ sufficiently large, Si,j is a maximum coclique in Xk,ℓ and equality
holds in the ratio bound. Let vi,j be the characteristic vector of Si,j . Since we have equality
in the ratio bound, this implies that
vi,j −
k − 1
kℓ− 1
1
546 Ars Math. Contemp. 23 (2023) #P4.02 / 531–551
is a τ -eigenvector (where 1 is the all ones vector). Since no other modules have have
eigenvalue τ , these vectors are in the [kℓ− 2, 2]-module. Further, the set of vectors{
vi,j −
k − 1
kℓ− 1
1 | i, j ∈ {1, . . . , kℓ}
}
is invariant under the action of Sym(kℓ), so they form a module. Since the [kℓ − 2, 2]-
module is irreducible, these vectors span the entire [kℓ − 2, 2]-module; this also implies
that the vectors {vi,j | i, j ∈ {1, . . . , kℓ}} span the [kℓ] and [kℓ− 2, 2]-modules.
Let S be a partially 2-intersecting set of (k, ℓ)-partition of maximum size, and let vS
denote the characteristic vector of S. Then vS − k−1kℓ−1 1 is in the [kℓ− 2, 2]-module. Thus
vS is in the span of the [kℓ] and [kℓ − 2, 2]-module, so vS is a linear combination of the
vectors vi,j .
7 Exact result for k = 3
In this section we will prove Theorem 6.1 holds for all ℓ ≥ 3, when k = 3. It is already
known, see Corollary 7.5.6 in [19], that Theorem 6.1 holds in the case where k = 3 and ℓ
is odd; this follows from the existence of resolvable packing designs of an appropriate size.
For k = 3, we observed experimentally that the ratio u3,ℓ/d3,ℓ converges to e
(k−1)2
2 =
e2 surprisingly quickly. If the sequence of u3,ℓ/d3,ℓ was non-increasing this would be
sufficient, but we have no proof of this. Rather, in this section we show an upper bound
on the ratio u3,ℓ/d3,ℓ for all ℓ, or, equivalently, a lower bound on d3,ℓ. This bound holds
for ℓ > 10, and we simply directly check the theorem for the specific graphs with smaller
values of ℓ.
Lemma 7.1. For ℓ > 10, the degree, d3,ℓ is greater than u3,ℓ/24.
Proof. We will use a truncated inclusion-exclusion argument to bound the degree. Since
Xk,ℓ is vertex transitive, we obtain a bound on the degree by counting the neighbours of an
arbitrary partition P ∈ U3,ℓ.
Fix a partition P ∈ U3,ℓ and let J be the set of pairs {x, y} of elements in {1, 2, . . . , 3ℓ}
that are contained in the same block of P . Note that |J | = 3ℓ. For a pair {x, y} ∈ J ,
we define A{x,y} to be the set of all partitions which contain x and y in the same block.
Further, for a subset J ⊆ J , define
N(J) = |∩{x,y}∈JA{x,y}|
and for 0 ≤ j ≤ 3ℓ let Nj =
∑
J,|J|=j N(J). By inclusion-exclusion,
d3,ℓ =
3ℓ∑
j=0
−1jNj . (7.1)
Next we calculate Nj . First, we note that N0 = N(∅) = u3,ℓ.
For any subset J ⊆ J , each block in P contains either 0, 1, 2 or 3 of the pairs from
J . For i = 0, 1, 2, 3, let ni be the number of blocks in P that have exactly i of their
pairs in J . We call the 4-tuple (n0, n1, n2, n3) the pair distribution of J and note that
n0 + n1 + n2 + n3 = ℓ. For each block of P its block type relative to J is the number of
pairs in J that are in the block. With this terminology, we can find Nj .
K. Meagher et al.: An extension of the Erdős-Ko-Rado theorem to uniform set partitions 547
First, fix a subset J ⊆ J with pair distribution (n0, n1, n2, n3) and count the number
of partitions Q ∈ ∩{x,y}∈JA{x,y}. Each block of P with block type n3 relative to J
determines exactly which three elements are in a block of Q, as do the blocks of type n2.
Each of the blocks of type n1 determines two of the three points in the block of Q. One
more point must be chosen to complete each of these blocks, and this choice is ordered
since each pair of type n1 from J uniquely labels its corresponding block. Each of the
blocks of type n0 does not determine any points in Q. Thus the number of partitions Q
which contain the pairs from J is given by the multinomial coefficient
1
n0!
(
3ℓ− 3(n3 + n2)− 2n1
1(n1), 3(n0)
)
(where the exponent in braces indicates the number is repeated that many times).
We now count the number of possible J which have pair distribution (n0, n1, n2, n3).
The number of ways to select the type of each block in P is equal to the multinomial
coefficient (
ℓ
n0, n1, n2, n3
)
,
since we are choosing the blocks from P that have either 0, 1, 2 or 3 pairs in J . Each of
the blocks of P with type n3 has all of its three pairs in J , while for each block of type n2
there are three ways to choose which two of the three possible pairs are in J . Similarly,
for each block of type n1 there is one pair in J , and there are three ways to chose this pair.
Finally, each of the n0 blocks does not contribute any pairs to J . Thus there are
3n1+n2
different sets J in with pair distribution (n0, n1, n2, n3).
Finally, we sum the number of partitions Q ∈ ∩{x,y}∈JA{x,y} over all possible pair
distributions that J can have. Each pair distribution (n0, n1, n2, n3) is an ordered partition
of ℓ into exactly four non-negative parts. The pair distribution (n0, n1, n2, n3) corresponds
to a set J of size n1+2n2+3n3. Define C(ℓ, j) to be the set of compositions of ℓ into four
parts with n1 + 2n2 + 3n3 = j.
Then from our previous counting we have that
Nj =
∑
(n0,n1,n2,n3)∈C(ℓ,j)
3n1+n2
(
ℓ
n0, n1, n2, n3
)
1
n0!
(
3ℓ− 3(n3 + n2)− 2n1
1(n1), 3(n0)
)
.
When we put this value in Equation 7.1 and truncate this sum after an odd j we will get
a lower bound on d3,ℓ. Taking j up to 5 we sum over the following list of pair distributions:
C(ℓ, 0) = {(ℓ, 0, 0, 0)}
C(ℓ, 1) = {(ℓ− 1, 1, 0, 0)}
C(ℓ, 2) = {(ℓ− 1, 0, 1, 0), (ℓ− 2, 2, 0, 0)}
C(ℓ, 3) = {(ℓ− 1, 0, 0, 1), (ℓ− 2, 1, 1, 0), (ℓ− 3, 3, 0, 0)}
C(ℓ, 4) = {(ℓ− 2, 1, 0, 1), (ℓ− 2, 0, 2, 0), (ℓ− 3, 2, 1, 0), (ℓ− 4, 4, 0, 0)}
C(ℓ, 5) = {(ℓ− 2, 0, 1, 1), (ℓ− 3, 2, 0, 1), (ℓ− 3, 1, 2, 0), (ℓ− 4, 3, 1, 0), (ℓ− 5, 5, 0, 0)}
548 Ars Math. Contemp. 23 (2023) #P4.02 / 531–551
Expanding this becomes
d3,ℓ ≥
5∑
j=0
−1j
∑
(n0,n1,n2,n3)∈C(ℓ,j)
3n1+n2
(
ℓ
n0, n1, n2, n3
)(3ℓ−3(n3+n2)−2n1
1(n1),3(n0)
)
n0!
=
(
ℓ
ℓ
)(
3ℓ
3(ℓ)
)
(ℓ)!
+
−3
(
ℓ
ℓ−1,1
)(
3ℓ−2
1,3(ℓ−1)
)
(ℓ− 1)!
+
3
(
ℓ
ℓ−1,1
)(
3ℓ−3
3(ℓ−1)
)
(ℓ− 1)!
+
32
(
ℓ
ℓ−2,2
)(
3ℓ−4
1(2),3(ℓ−2)
)
(ℓ− 2)!
+
−
(
ℓ
ℓ−1,1
)(
3ℓ−3
3(ℓ−1)
)
(ℓ− 1)!
+
−32
(
ℓ
ℓ−2,1(2)
)(
3ℓ−5
1,3(ℓ−2)
)
(ℓ− 2)!
+
−33
(
ℓ
ℓ−3,3
)(
3ℓ−6
1(3),3(ℓ−3)
)
(ℓ− 3)!
+
3
(
ℓ
ℓ−2,1(2)
)(
3ℓ−5
1,3(ℓ−2)
)
(ℓ− 2)!
+
32
(
ℓ
ℓ−2,2
)(
3ℓ−6
3(ℓ−2)
)
(ℓ− 2)!
+
33
(
ℓ
ℓ−3,2,1
)(
3ℓ−7
1(2),3(ℓ−3)
)
(ℓ− 3)!
+
34
(
ℓ
ℓ−4,4
)(
3ℓ−8
1(4),3(ℓ−4)
)
(ℓ− 4)!
+
−3
(
ℓ
ℓ−2,1(2)
)(
3ℓ−6
3(ℓ−2)
)
(ℓ− 2)!
+
−32
(
ℓ
ℓ−3,2,1
)(
3ℓ−7
1(2),3(ℓ−3)
)
(ℓ− 3)!
+
−33
(
ℓ
ℓ−3,1,2
)(
3ℓ−8
1,3(ℓ−3)
)
(ℓ− 3)!
+
−34
(
ℓ
ℓ−4,3,1
)(
3ℓ−9
1(3),3(ℓ−4)
)
(ℓ− 4)!
+
−35
(
ℓ
ℓ−5,5
)(
3ℓ−10
1(5),3(ℓ−5)
)
(ℓ− 5)!
=
(243ℓ6 − 2997ℓ5 + 13905ℓ4 − 32355ℓ3 + 42732ℓ2 − 32728ℓ+ 11200)(3ℓ− 10)!
80(6ℓ−4)(ℓ− 10)!(ℓ6 − 39ℓ5 + 625ℓ4 − 5265ℓ3 + 24574ℓ2 − 60216ℓ+ 60480)
.
Thus
u3,ℓ
d3,ℓ
≤ 5(729ℓ
6 − 6561ℓ5 + 23085ℓ4 − 40095ℓ3 + 35586ℓ2 − 14904ℓ+ 2240)
243ℓ6 − 2997ℓ5 + 13905ℓ4 − 32355ℓ3 + 4273ℓ2 − 32728ℓ+ 11200
.
For ℓ > 10 this gives that u3,ℓ/d3,ℓ < 24.
Theorem 7.2. For k = 3 and all ℓ ≥ 3 the largest set of partially 2-intersecting uniform
partitions has size
(3ℓ− 2)u3,ℓ−1.
Proof. For ℓ = 3 all the eigenvalues of X3,3 have long been known to be {36, 8, 2, −4,
−12} [18]. The ratio bound holds with equality, and the only irreducible representation
that belongs to the least eigenvalue is χ[7,2].
For ℓ = 4, all the eigenvalues of X3,4 are {1296, 96, 72, 48, 32, 0,−24,−48,−288}.
These can be calculated by making a quotient graph of X3,4 from the action of Sym(3) ≀
Sym(4) on the partitions. This equitable partition has a cell of size 1, so the eigenvalues
of the quotient graph are exactly the eigenvalues of X3,4. Further, the multiplicities of the
eigenvalues can be calculated using the formulas in [14, Section 5.3] and the [10, 2]-module
is the only module to which the eigenvalue −288 belongs.
For ℓ ∈ {5, . . . , 12} the only irreducible representations with dimension less then(
3ℓ
3
)
−
(
3ℓ
2
)
in the decomposition of ind (1Sym(3)≀Sym(ℓ))
Sym(3ℓ) are the three listed in Theo-
rem 3.4—this can be checked using GAP [10]. Thus Theorem 3.4 holds for all 5 ≤ ℓ ≤ 12
when k = 3.
For all ℓ > 10, Lemma 7.1 shows that uk,ℓ/dk,ℓ − 1 < 23. In this same range, the right
hand side of Equation (6.1) is at least 26. Thus the inequality from Equation (6.1) holds for
all ℓ > 10.
K. Meagher et al.: An extension of the Erdős-Ko-Rado theorem to uniform set partitions 549
For 5 ≤ ℓ ≤ 10 the degrees d3,ℓ can be directly computed
d3,5 = 132192, d3,7 = 3829057920, d3,9 = 333973115062272,
d3,6 = 19258560, d3,8 = 1001695548672, d3,10 = 138348645213579264,
and the inequality from Equation (6.1) directly checked.
8 Further work
In this paper we only consider partially 2-intersecting partitions, but the conjecture in [20]
is for partial t-intersection sets of partitions with k ≤ ℓ(t − 1). It is possible that the
approach in this paper could be applied for larger values of t, but there are some steps that
we predict will be complicated.
It is straight-forward to generalize the definition of Xk,ℓ to partially t-intersecting par-
titions by defining the graph Xt,k,ℓ. This graph will also have Uk,ℓ as its vertex set, and two
partitions P and Q are adjacent if and only if for all pairs of blocks Pi ∈ P and Qj ∈ Q
we have |Pi ∩Qj | < t. A partially t-intersecting set of partitions is a coclique in Xt,k,ℓ.
The conjecture is if k < ℓ(t − 1), then the maximum cocliques in Xt,k,ℓ are exactly
the canonical partially t-intersecting sets. The Young subgroup Sym([kℓ− t, t]) is the
stabilizer of a canonically partially t-intersecting set. The most significant complication is
that for t > 2, there are more than two irreducible representations in both
ind (1Sym([kℓ−t,t]))
Sym(kℓ) and ind (1Sym(k)≀Sym(ℓ))
Sym(kℓ)
. (8.1)
For the approach given in this paper to work, we believe the eigenvalues belonging to all
the irreducible representations common to these two induced representations, except the
trivial representation, should be the least eigenvalue of Xt,k,ℓ. To make this happen we
suspect that a weighted adjacency matrix of Xt,k,ℓ would be needed in the ratio bound,
rather than just the adjacency matrix; the weighting would have to be chosen so that the
common modules (except the trivial) in the representations in (8.1) all belong to the same
eigenvalue. Another complication is that potentially more of the eigenvalues of Xt,k,ℓ
would have to be calculated, at the very least all the eigenvalues belonging to the common
representations would need to be known.
Bender’s theorem is much more general than the version we stated here. We only state
Bender’s theorem for matrices with 01-entries, but the full theorem applies to matrices with
entries less than t. Using the full theorem we would be able to approximate the degree of
Xt,k,ℓ for t ≥ 2.
ORCID iDs
Karen Meagher https://orcid.org/0000-0002-7948-9149
Mahsa N. Shirazi https://orcid.org/0000-0003-3924-0848
Brett Stevens https://orcid.org/0000-0003-4336-1773
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P4.03 / 553–562
https://doi.org/10.26493/1855-3974.2683.5f3
(Also available at http://amc-journal.eu)
Almost simple groups as flag-transitive
automorphism groups of symmetric designs with
λ prime*
Seyed Hassan Alavi † , Ashraf Daneshkhah , Fatemeh Mouseli
Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran
Received 19 August 2021, accepted 10 January 2023, published online 28 February 2023
Abstract
In this article, we study symmetric designs with λ prime admitting a flag-transitive and
point-primitive automorphism group G of almost simple type with socle X . We prove that
either D is one of the six well-known examples of biplanes and triplanes, or D is the point-
hyperplane design of PG(n−1, q) with λ = (qn−2−1)/(q−1) prime and X = PSLn(q).
Keywords: Almost simple group, automorphism group, flag-transitive, point-primitive, symmetric de-
sign.
Math. Subj. Class. (2020): 05B05, 05B25, 20B25, 20D05
1 Introduction
Symmetric designs admitting flag-transitive automorphism groups are of most interest.
Kantor [14] classified flag-transitive symmetric (v, k, 1) designs known as projective planes
of order n, and showed that either D is a Desarguesian projective plane and PSL3(n)⊴G,
or G is a sharply flag-transitive Frobenius group of odd order (n2 + n+ 1)(n+ 1), where
n is even and n2 + n + 1 is prime. Regueiro gave a classification of nontrivial symmetric
designs with λ = 2 (biplanes) admitting flag-transitive automorphism groups apart from
those groups contained in a 1-dimensional affine group [17, 18, 19, 20, 21]. Dong, Fang and
Zhou studied flag-transitive automorphism groups G of nontrivial symmetric (v, k, 3) de-
signs (triplanes), and in conclusion, excluding the case where G ⩽ AΓL1(q) where q = pm
with p ⩾ 5 prime, they determined all such possible symmetric designs [13, 25, 26, 27, 28].
*The authors would like to thank anonymous referees for providing us helpful and constructive comments and
suggestions.
†Corresponding author.
E-mail addresses: alavi.s.hassan@basu.ac.ir (Seyed Hassan Alavi), adanesh@basu.ac.ir (Ashraf
Daneshkhah), f.mouseli@sci.basu.ac.ir (Fatemeh Mouseli)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
554 Ars Math. Contemp. 23 (2023) #P4.03 / 553–562
Table 1: Some symmetric designs with λ prime.
Line v k λ X H G Designs References∗
1 7 4 2 PSL2(7) S4 PSL2(7) Complement of the
Fano plane
[2, 8]
2 11 5 2 PSL2(11) A5 PSL2(11) Hadamard [2, 8]
3 11 6 3 PSL2(11) A5 PSL2(11) Complement of line 2 [2, 8]
4 15 7 3 A7 PSL3(2) A7 PG2(3, 2) [8, 26, 29]
5 45 12 3 PSU4(2) 2.(A4×A4).2 PSU4(2) - [7, 11, 22]
6 45 12 3 PSU4(2) 2.(A4×A4).2:2 PSU4(2):2 - [7, 11, 22]
Note: The last column addresses to references in which a design with the parameters in the line
has been constructed.
Recently, Z. Zhang, Y. Zhang and Zhou in [24] proved that if D is a nontrivial symmet-
ric (v, k, λ) design with λ prime and G is a flag-transitive and point-primitive automor-
phism group of D, then G must be of affine or almost simple type. In this paper, we study
symmetric designs with λ prime admitting a flag-transitive and point-primitive automor-
phism group of almost simple type. Indeed, we have already shown in [4] that almost
simple exceptional groups of Lie type give rise to no possible symmetric designs with λ
prime. In addition, we investigated the case where G is an almost simple group with so-
cle X being a finite simple classical group of Lie type, and proved that D is either the
point-hyperplane design of a projective space PG(n− 1, q), or it is of parameters (7, 4, 2),
(11, 5, 2), (11, 6, 3) or (45, 12, 3). Here, we focus on the case where G is an almost simple
group with socle X an alternating group:
Theorem 1.1. Let D be a nontrivial symmetric (v, k, λ) design with λ prime and α a point
of D, and let G be a flag-transitive and point-primitive automorphism group of D of almost
simple type. If the socle of G is an alternating group Ac with c ⩾ 5, then D = PG2(3, 2)
with parameters (15, 7, 3) and G = A7 with the point-stabiliser Gα = PSL3(2).
By Theorem 1.1 and the main results of [3, 23], we consequently obtain all symmetric
designs with λ prime admitting flag-transitive and point-primitive automorphism groups of
almost simple type:
Corollary 1.2. Let D be a nontrivial symmetric (v, k, λ) design with λ prime. Suppose
that G is a flag-transitive and point-primitive automorphism group of D of almost simple
type with socle X . Then λ = 2, 3 and (D, G) is as in one of the lines of Table 1, or D
is the point-hyperplane design of PG(n − 1, q) with λ = (qn−2 − 1)/(q − 1) prime and
X = PSLn(q).
In order to prove Theorem 1.1, for the case where λ ⩽ 100 or gcd(k, λ) = 1, by
[29, 30], we obtain the designs in the statement. Then we assume that λ > 100 and λ
divides k (as λ is prime). In this case, we show that there is no symmetric deign with
λ prime and flag-transitive and point-primitive automorphism group G. Here, we first
observe that the point-stabiliser H of G has to be large, that is to say, |G| ⩽ |H|3, see
Corollary 2.2. The possibilities for H can be read off from [5]. In Section 3, we examine
these possibilities and achieve our desired result. In Section 4, we give a detailed proof of
Corollary 1.2 which follows immediately from Theorem 1.1 and the main results of [3, 23].
S. H. Alavi et al.: Almost simple groups as flag-transitive automorphism groups . . . 555
1.1 Definitions and notation
All groups and incidence structures in this paper are finite. A group G is said to be almost
simple with socle X if X ⊴ G ⩽ Aut(X), where X is a nonabelian simple group. Sym-
metric and alternating groups on c letters are denoted by Sc and Ac, respectively. We write
“n” for group of order n. A symmetric (v, k, λ) design D is a pair (P,B) with a set P of v
points and a set B of v blocks such that each block is a k-subset of P and each pair of dis-
tinct points is contained in exactly λ blocks. We say that D is nontrivial if 2 < k < v − 1.
A flag of D is a point-block pair (α,B) such that α ∈ B. An automorphism of D is a
permutation on P which maps blocks to blocks and preserving the incidence. The full au-
tomorphism group Aut(D) of D is the group consisting of all automorphisms of D. For
G ⩽ Aut(D), G is called flag-transitive if G acts transitively on the set of flags. The group
G is said to be point-primitive if G acts primitively on P . For a given positive integer n
and a prime divisor p of n, we denote the p-part of n by np, that is to say, np = pt with
pt | n but pt+1 ∤ n. Further notation and definitions in both design theory and group theory
are standard and can be found, for example in [6, 9, 12, 15, 16].
2 Preliminaries
In this section, we state some useful facts in both design theory and group theory. If a group
G acts on a set P and α ∈ P , the subdegrees of G are the length of orbits of the action of
the point-stabiliser Gα on P .
Lemma 2.1 ([2, Lemma 2.1]). Let D be a symmetric (v, k, λ) design, and let G be a
flag-transitive automorphism group of D. If α is a point of D, then
(i) k(k − 1) = λ(v − 1);
(ii) k divides |Gα|, and λv < k2;
(iii) k | λd, for all nontrivial subdegrees d of G.
For a point-stabiliser H of a flag-transitive automorphism group G of a design D, by
Lemma 2.1(ii), we conclude that λ|G| ⩽ |H|3, and so we have that
Corollary 2.2. Let D be a symmetric (v, k, λ) design with a flag-transitive automorphism
group G and α a point of D. Then |G| ⩽ |Gα|3.
Lemma 2.3. Suppose that s and t are positive integers. Then
(i) if t > s ⩾ 9, then
(
s+t
s
)
> s2t3;
(ii) if s ⩾ 4 and there exists t0 ⩾ 7 such that
(
s+t0
s
)
> s2t30, then
(
s+t
s
)
> s2t3 for all
t ⩾ t0.
Proof. (i) If t > s = 9, then we observe that the inequality
(
s+t
s
)
=
(
t+9
9
)
> 81t3 = s2t3
holds. If t > s ⩾ 10, then 10 ⩽ s ⩽ s+t2 , and so that
(
s+t
s
)
⩾
(
10+t
10
)
> t5. Since t > s,
we have that t5 > s2t3, and hence
(
s+t
s
)
> t5 > s2t3.
556 Ars Math. Contemp. 23 (2023) #P4.03 / 553–562
(ii) It suffices to show that
(
s+t0+1
s
)
> s2(t0 + 1)
3. Note that(
s+ t0 + 1
s
)
=
(
s+ t0
s
)
(s+ t0 + 1)
(t0 + 1)
> s2t30
(s+ t0 + 1)
(t0 + 1)
= s2(t0 + 1)
3 (s+ t0 + 1)t
3
0
(t0 + 1)4
.
Since t0 ⩾ 7 and s ⩾ 4, it follows that (s+ t0+1)t30 ⩾ (t0+5)t
3
0 > (t0+1)
4. Therefore,(
s+t0+1
s
)
> s2(t0 + 1)
3.
3 Proof of Theorem 1.1
Suppose that G is a flag-transitive and point-primitive automorphism group of a symmetric
(v, k, λ) design D with λ prime. Suppose that X is the alternating group Ac of degree
c ⩾ 5 on Ω = {1, . . . , c} and that H := Gα with α a point of D. Then H is maximal in G
by [12, Corollary 1.5A], and since G = HX , we conclude that
v =
|X|
|H ∩X|
. (3.1)
If λ ⩽ 100 or gcd(k, λ) = 1, then by [29, 30], we conclude that D is PG2(3, 2) with
parameters (15, 7, 3), and G = A7 with the point-stabiliser H = PSL3(2). Therefore, we
can assume that λ > 100 and gcd(k, λ) ̸= 1. Since k(k−1) = λ(v−1) and gcd(k, λ) ̸= 1,
we conclude that λ | k, and so by Lemma 2.1(ii), the parameter λ divides |H|. In what
follows, assuming that λ > 100 divides k and gcd(k, λ) ̸= 1, we show that there is no
flag-transitive and point-primitive automorphism group of a symmetric (v, k, λ) design D
with λ prime.
Let H0 := H ∩X . Then by [5, Theorem 2 and Proposition 6.1], one of the following
holds:
(i) H0 is intransitive on Ω = {1, . . . , c};
(ii) H0 is transitive and imprimitive on Ω = {1, . . . , c};
(iii) G = Sc and (c,H) is one of the following:
(5,AGL1(5)) , (6,PGL2(5)) , (7,AGL1(7)) , (8,PGL2(7)) ,
(9,AGL2(3)) ,
(
10,A6·22
)
, (12,PGL2(11)) ;
(iv) G = A6·2 = PGL2(9) and H is D20 or a Sylow 2-subgroup P of G of order 16;
(v) G = A6·2 = M10 and H is AGL1(5) or a Sylow 2-subgroup P of G of order 16;
(vi) G = A6·22 = PΓL2(9) and H is AGL1(5)×2 or a Sylow 2-subgroup P of G of
order 32;
(vii) G = Ac and (c,H) is one of the following:
(5,D10), (6,PSL2(5)), (7,PSL2(7)), (8,AGL3(2)),
(9, 32·SL2(3)), (9,PΓL2(8)), (10,M10), (11,M11),
(12,M12), (13,PSL3(3)), (15,A8), (16,AGL4(2)),
(24,M24).
S. H. Alavi et al.: Almost simple groups as flag-transitive automorphism groups . . . 557
Since λ is a prime divisor of k, it follows from Lemma 2.1(ii) that λ is a prime divisor of
|H|. For the possibilities recorded in (iii) – (vii), we then have λ ∈ {2, 3, 5, 7, 11, 13, 23},
and this violates our assumption that λ > 100, see Table 2. Therefore, H0 is either intran-
sitive, or imprimitive.
Table 2: The possibilities for λ in cases (iii) – (vii) in Section 3.
Line H |H| λ
1 AGL1(5) 2
2 · 5 2, 5
2 PGL2(5) 2
3 · 3 · 5 2, 3, 5
3 AGL1(7) 2 · 3 · 7 2, 3, 7
4 PGL2(7) 2
4 · 3 · 7 2, 3, 7
5 AGL2(3) 2
4 · 33 2, 3
6 A6 · 22 25 · 32 · 5 2, 3, 5
7 PGL2(11) 2
3 · 3 · 5 · 11 2, 3, 5, 11
8 D10 2 · 5 2, 5
9 PSL2(5) 2
2 · 3 · 5 2, 3, 5
10 PSL2(7) 2
3 · 3 · 7 2, 3, 7
11 AGL3(2) 2
6 · 3 · 7 2, 3, 7
12 32 · SL2(3) 23 · 33 2, 3
13 PΓL2(8) 2
3 · 33 · 7 2, 3, 7
14 M10 2
4 · 32 · 5 2, 3, 5
15 M11 2
4 · 32 · 5 · 11 2, 3, 5, 11
16 M12 2
6 · 33 · 5 · 11 2, 3, 5, 11
17 PSL3(3) 2
4 · 33 · 13 2, 3, 13
18 A8 2
6 · 32 · 5 · 7 2, 3, 5, 7
19 AGL4(2) 2
10 · 32 · 5 · 7 2, 3, 5, 7
20 M24 2
10 · 33 · 5 · 7 · 11 · 23 2, 3, 5, 7, 11, 23
21 D20 2
2 · 5 2, 5
22 AGL1(5) 2
2 · 5 2, 5
23 P 24 2
24 AGL1(5)× 2 23 · 5 2, 5
25 P 25 2
Note: In line 23, P is a Sylow 2-subgroup of G = A6 · 2 of order 16.
In line 25, P is a Sylow 2-subgroup of G = A6 · 22 of order 32.
(I) Suppose that H0 = (Ss×Sc−s)∩Ac is intransitive on Ω = {1, . . . , c} with 1 ⩽ s < c/2.
In this case, H = (Ss × Sc−s) ∩ G. Note that H is maximal in G as long as s ̸= c − s.
Since λ is an odd prime divisor of |H|, it follows that λ divides s! or (c − s)!, and since
s < c− s, we conclude that
λ ⩽ max{s, c− s} = c− s. (3.2)
Note that H0 contains all the even permutations of H , and hence H0 = H if G = Ac, or
the index of H0 in H is 2 if G = Sc. Since G is flag-transitive, H is transitive on the set of
blocks passing through α. Hence H fixes exactly one point in P , and so stabilizes exactly
558 Ars Math. Contemp. 23 (2023) #P4.03 / 553–562
one s-subset, say S, in Ω. Therefore, we can identify the point α of P with the unique s-
subset S of Ω stabilized by H . Thus v =
(
c
s
)
. Since H0 acting on Ω is intransitive, it has at
least two orbits. According to [10, page 82], two points of P are in the same orbit under H0
if and only if the corresponding s-subsets S1 and S2 of Ω intersect S in the same number
of points. Thus G acting on P has rank s + 1, and each H0-orbit Oi on P corresponds to
a possible size i ∈ {0, 1, . . . , s} and these are precisely the families of s-subsets of Ω that
intersect S, see also [1, Proposition 2.5]. Then if di is the length of a G-orbit on P , then
d0 = 1, and dj =
(
s
j−1
)(
c−s
s−j+1
)
when G = Ac or dj =
(
s
j−1
)(
c−s
s−j+1
)
/2 when G = Sc for
j = 1, . . . , s.
By Lemma 2.1(iii), we have that k divides λdj for all nontrivial subdegrees dj of G.
By taking j = s, we have that k divides λs(c− s), and so k ⩽ λs(c− s). As λv < k2 by
Lemma 2.1(ii), it follows from (3.2) that
v =
(
c
s
)
< λs2(c− s)2 ⩽ s2(c− s)3.
Set t := c− s. Thus (
s+ t
s
)
< s2t3. (3.3)
Applying Lemma 2.3(i), we conclude that (3.3) holds only for s ⩽ 8.
If s = 1, then v = c ⩾ 5. Note that G is (v− 2)-transitive on P . Since 2 < k ⩽ v− 2,
G acts k-transitively on P . Then
(
c
k
)
= |BG| = |B| = v = c for every block B ∈ B. This
implies that k = 1 or k = c − 1. Since k ⩾ λ > 100, we conclude that k = c − 1, that is
to say, D is a trivial design, which is a contradiction.
If s = 2, then the subdegrees are 1,
(
c−2
2
)
, 2(c − 2) if G = Ac, or 1,
(
c−2
2
)
/2, (c − 2)
if G = Sc. Thus G is a primitive permutation group of rank 3. Therefore, the possibilities
for D can be read off from [11], which gives no example with λ > 100 prime.
If s = 3, then by Lemma 2.1(iii), k divides λd3 = 3λ(c− 3), and so k = 3λ(c− 3)/u
for some positive integer u. We apply Lemma 2.1(i) and since v − 1 =
(
c
3
)
− 1 = (c −
3)(c2 + 2)/6, we deduce that
3λ(c− 3)
u
·
(
3λ(c− 3)
u
− 1
)
=
λ(c− 3)(c2 + 2)
6
,
and so
(c2 + 2)u2 + 18u− 54(c− 3)λ = 0, (3.4)
for some positive integer u. Define f(x) := (c2 + 2)x2 + 18x − 54(c − 3)λ with x ⩾ 1.
Note here that c > 2s = 6 and λ ⩽ c − 3 by (3.2). Then f ′(x) = 2(c2 + 2)x + 18 > 0
for x ⩾ 1. If x ⩾ 8, then since λ ⩽ c − 3, we have that f(x) = (c2 + 2)x2 + 18x −
54(c − 3)λ ⩾ 10c2 + 324c − 214 > 0 for c > 6. Therefore, we cannot find any positive
integer u ⩾ 8 satisfying (3.4), and hence 1 ⩽ u ⩽ 7. Thus by (3.4), we have that 54λ =
u2(c + 3) + (11u2 + 18u)/(c − 3), and so c − 3 divides 11u2 + 18u, where 1 ⩽ u ⩽ 7.
Moreover, λ > 100 is a prime number less than 665 as λ ⩽ c − 3 ⩽ 11u2 + 18u ⩽ 665.
For these values of u, c and λ, it is easy to check that (3.4) does not hold.
S. H. Alavi et al.: Almost simple groups as flag-transitive automorphism groups . . . 559
Table 3: An upper bound and a lower bound for t and c when 4 ⩽ s ⩽ 8.
s Bounds for t Bounds for c
4 5 ⩽ t ⩽ 373 9 ⩽ c ⩽ 377
5 6 ⩽ t ⩽ 46 11 ⩽ c ⩽ 51
6 7 ⩽ t ⩽ 22 13 ⩽ c ⩽ 28
7 8 ⩽ t ⩽ 14 15 ⩽ c ⩽ 21
8 9 ⩽ t ⩽ 11 17 ⩽ c ⩽ 19
If 4 ⩽ s ⩽ 8, then we apply (3.3) and Lemma 2.3(ii), and so we can find a lower bound
and an upper bound for t as in the second column of Table 3. Since also c = t − s, we
can find a lower bound and an upper bound for c as in the third column of Table 3. For
example, if s = 4, then we take t0 = 374 and observe that
(
s+t0
s
)
=
(
378
4
)
= 837222750 >
837017984 = 42 ·3743 = s2t30, then Lemma 2.3(ii) implies that if t ⩾ 374, then (3.3) does
not hold, which is a contradiction. Thus t ⩽ 373. Moreover, it is easy to check that (3.3)
holds for t ⩾ 5. Thus 5 ⩽ t ⩽ 373. Note that t = c − 4, and hence 9 ⩽ c ⩽ 377. This
follows the first row of Table 3. For each t, s and c as in Table 3, we note by (3.2) that
λ ⩽ c− s = t. Then we obtain v =
(
c
s
)
and all the possibilities for prime λ > 100. But it
is easy to check that for such parameters v and λ, we cannot find any possible parameters
set (v, k, λ) satisfying Lemma 2.1.
(II) Suppose now that H0 is transitive and imprimitive on Ω = {1, . . . , c}. In this case,
H = (Ss ≀ Sc/s) ∩ G is imprimitive, where s divides c and 2 ⩽ s ⩽ c/2. Indeed, H0 is
transitive and imprimitive on Ω = {1, . . . , c}, H0 acting on Ω preserves a partition Σ of Ω
into t classes of size s with t ⩾ 2, s ⩾ 2 and c = st. Thus H0 ⩽ GΣ < G. Since G is
isomorphic to Sc or Ac and since both natural actions of G and X on Ω are primitive, we
conclude that H0 contains all the even permutations of Ω preserving the partition Σ. By the
same argument as in [10, Case 2], [17, (3.2)] and [29, pages 1489-1490], the imprimitive
partition Σ is the only nontrivial partition of Ω preserved by H0. Since X acts transitively
on all the partitions of Ω into t classes of size s, we can identify the points of the D with
the partitions of Ω into t classes of size s, and so v =
(
ts
s
)(
(t−1)s
s
)
· · ·
(
3s
s
)(
2s
s
)
/(t!), that is
to say,
v =
(
ts− 1
s− 1
)(
(t− 1)s− 1
s− 1
)
· · ·
(
3s− 1
s− 1
)(
2s− 1
s− 1
)
. (3.5)
We note that the suborbits of G on Ω can be described by the notion of j-cyclics intro-
duced in [10, page 84]. Indeed, if a partition Σ1 of Ω is a point of P , then for j = 2, . . . , t,
the set Γj of j-cyclic partitions with respect to Σ1 is a union of H-orbits on P , see [10,
Case 2] and [29, pages 1490-1491]. Therefore, by Lemma 2.1(iii), k divides λds, where
ds =
{
s2
(
t
2
)
, if s ⩾ 3;
t(t− 1), if s = 2.
(3.6)
Therefore, by Lemma 2.1(iii), we have that k divides λds, where ds is as in (3.6). Note that
λ is a prime divisor of k, and so by Lemmas 2.1(ii), we conclude that λ ⩽ c = st.
560 Ars Math. Contemp. 23 (2023) #P4.03 / 553–562
If s = 2, then t ⩾ 3 as c = st ⩾ 5. By (3.5), we have that v =
∏t−2
i=0[2t − (2i + 1)]
and since k divides λd2 = λt(t− 1) and λ ⩽ c = 2t, it follows from Lemma 2.1(ii) that
t−2∏
i=0
[2t− (2i+ 1)] < λd22 ⩽ 2t3(t− 1)2 < 2t5.
This forces t ⩽ 6, and hence λ ⩽ 2t = 12, which is a contradiction as λ > 100.
If s ⩾ 3, then since,(
is− 1
s− 1
)
=
is− 1
s− 1
· is− 2
s− 2
· · · is− (s− 1)
1
> is−1
with 2 ⩽ i ⩽ t, by (3.5), we conclude that v > t(s−1)(t−1). Since also k divides λds =
λs2
(
t
2
)
and λ ⩽ st, we deduce by Lemma 2.1(ii) that
t(s−1)(t−1) < s5t
(
t
2
)2
.
Thus t(s−1)(t−1)−5 < s5. Note that s ⩾ 3. Then this inequality holds only for
t = 2 and 3 ⩽ s ⩽ 30;
t = 3 and 3 ⩽ s ⩽ 8;
t = 4 and s = 3, 4;
t = 5 and s = 3.
However, for such pairs (t, s), we easily observe that λ ⩽ st < 100, which is a contradic-
tion. This completes the proof.
4 Proof of Corollary 1.2
Let D be a nontrivial symmetric (v, k, λ) design with λ prime. Suppose that G is a flag-
transitive and point-primitive automorphism group of D of almost simple type with socle
X . Since λ is prime, by [23], the socle X cannot be a sporadic simple group. If the socle
X is a simple group of Lie type, then by [3, Theorem 1], D is the point-hyperplane design
of PG(n − 1, q) with λ = (qn−2 − 1)/(q − 1) prime and X = PSLn(q), or (D, G) is
as in one of the lines 1-3 and lines 5-6 of Table 1. If the socle X is an alternating group
Ac of degree c ⩾ 5, then Theorem 1.1 implies that D is the unique design PG2(3, 2) with
parameters (15, 7, 3), and G = A7 with the point-stabiliser PSL3(2), and this follows line
4 of Table 1. This finishes the proof.
ORCID iDs
Seyed Hassan Alavi https://orcid.org/0000-0002-6442-2507
Ashraf Daneshkhah https://orcid.org/0000-0001-8355-6853
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P4.04 / 563–575
https://doi.org/10.26493/1855-3974.2905.c94
(Also available at http://amc-journal.eu)
A classification of connected cubic
vertex-transitive bi-Cayley graphs over
semidihedral group*
Jianji Cao †
School of Applied Mathematics, Shanxi University of Finance and Economics,
Taiyuan Shanxi 030006, P. R. China
Young Soo Kwon ‡
Department of Mathematics, Yeungnam University, Gyeongsan 38541, R. Korea
Mimi Zhang §
School of Mathematical Science, Hebei Normal University,
Shijiazhuang 050024, P. R. China
Received 14 June 2022, accepted 15 January 2023, published online 10 March 2023
Abstract
A graph Γ is said to be a bi-Cayley graph over a group H if there exists a subgroup
of Aut(Γ) isomorphic to H acting semiregularly on its vertex set with two orbits. In
this paper, we give a complete classification of connected cubic vertex-transitive bi-Cayley
graphs over semidihedral group. As a byproduct, we construct a family of vertex-transitive
non-Cayley graphs.
Keywords: Semidihedral group, bi-Cayley graph, vertex-transitive.
Math. Subj. Class. (2020): 05C25, 20B25.
*The authors would like to thank the referee for his/her valuable suggestions and useful comments contributed
to the final version of this paper.
†The first author was supported by the China Scholarship Council Foundation (CSC No:201908140049), the
National Natural Science Foundation of China (12171302) and the Natural Science Foundation of Shanxi Province
(202103021224287).
‡Corresponding author. The second author was supported by the Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B05048450)
and (2021K2A9A2A11101586).
§The third author was supported by the National Natural Science Foundation of China (12101181) and the
Natural Science Foundation of Hebei Province (A2019205180).
E-mail addresses: 13994371056@163.com (Jianji Cao), ysookwon@ynu.ac.kr (Young Soo Kwon),
14118412@bjtu.edu.cn (Mimi Zhang)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
564 Ars Math. Contemp. 23 (2023) #P4.04 / 563–575
1 Introduction
Throughout this paper, all groups are assumed to be finite, and all graphs are assumed to be
finite, connected, simple and undirected. For a graph Γ, let V (Γ), E(Γ) and A(Γ) denote
vertex set, edge set and arc set of Γ, respectively. A graph Γ is said to be vertex-transitive,
edge-transitive and arc-transitive if the full automorphism group Aut (Γ) acts transitively
on V (Γ), E(Γ) and A(Γ), respectively. For other terminology related to group theory and
graph theory not defined here, we refer the reader to [1, 11].
Let G be a group and S be a subset of G such that S−1 = S and 1 /∈ S. Then the
Cayley graph Γ = Cay(G,S) over G with respect to S is defined as the graph with vertex
set V (Γ) = G and edge set E(Γ) = {{g, sg} | g ∈ G, s ∈ S}. Similarly, for a given group
H , letR, L and S be subsets ofH such thatR−1 = R, L−1 = L andR∪L does not contain
the identity element of H . The bi-Cayley graph over H denoted by BiCay(H , R, L, S) is
the graph having vertex set the union of the right part H0 = {h0 | h ∈ H} and the left part
H1 = {h1 | h ∈ H}, and edge set the union of the right edges {{h0, g0} | gh−1 ∈ R},
the left edges {{h1, g1} | gh−1 ∈ L} and the spokes {{h0, g1} | gh−1 ∈ S}. When
|R| = |L| = s, BiCay(H , R, L, S) is said to be an s-type bi-Cayley graph.
The triple (R,L, S) of three subsets R,L, S of a group H is called bi-Cayley triple
if R = R−1, L = L−1 and 1 ∈ S. Two bi-Cayley triples (R,L, S) and (R′, L′, S′)
of a group H are said to be equivalent, denoted by (R,L, S) ≡ (R′, L′, S′), if either
(R′, L′, S′) = (R,L, S)α or (R′, L′, S′) = (L,R, S−1)α for some automorphism α of H .
By Proposition 2.1(3)–(4), the bi-Cayley graphs corresponding to two equivalent bi-Cayley
triples of the same group are isomorphic.
In the study of bi-Cayley graphs, a considerable attention was given to the follow-
ing problem: for a given finite group H , classify bi-Cayley graphs over H with specific
symmetry properties. For example, vertex-transitive (edge-transitive) generalized Petersen
graphs had been classified in [4, 9]. Marušič and Pisanski in [6] classified all cubic arc-
transitive bi-Cayley graphs over dihedral group. All tetravalent edge-transitive bicirculants
(bi-Cayley group over cyclic group) were characterized in [5], a classification of cubic
edge-transitive bi-Cayley graphs over inner abelian p-groups were presented in [10], and
all cubic vertex-transitive bi-Cayley graphs over abelian groups were classified in [13].
Recently, Zhang and Zhou in [12] gave a classification of cubic edge-transitive bi-Cayley
graphs over dihedral groups.
Motivated by the works listed above, in this paper, we shall investigate cubic bi-Cayley
graphs over semidihedral groups. Recall that the semidihedral group of order 4n with n an
even is defined as follows:
SD4n = ⟨a, b | a2n = b2 = 1, b−1ab = an−1⟩.
Note that all cubic bi-Cayley graphs over abelian groups have been classified in [13]. So
we assume that n ≥ 4.
The Petersen graph is a bi-Cayley graph over a cyclic group of order 5, and the Petersen
graph is also a vertex-transitive non-Cayley graph. There are many research focusing on
the classification of vertex-transitive non-Cayley graphs, see [3, 4, 7, 8]. By Magma, we
found some examples of vertex-transitive non-Cayley graph Γ, where Γ is a cubic vertex-
trasntive bi-Cayley graph over SD4n. So another motivation for us to consider cubic
vertex-transitive bi-Cayley graphs over SD4n is to construct some kind of vertex-transitive
non-Cayley graphs.
J. Cao et al.: A classification of connected cubic vertex-transitive bi-Cayley graphs . . . 565
In [2] a classification of cubic edge-transitive bi-Cayley graphs over semidihedral group
is given. For the completeness of the results, we list the main theorem in [2] in the follow-
ing. ( For the definition of CQ(t, n), we refer the reader to [12])
Theorem 1.1 ([2, Theorem 1]). Let Γ be a cubic connected bi-Cayley graph over semidi-
hedral group SD4n. Then Γ is edge-transitive if and only if (R,L, S) is equivalent to one
of the following triples. Furthermore, all of the corresponding graphs are arc-transitive.
(1) (R,L, S) ≡ ({b}, {ba4}, {1, a}) with n = 4 and Γ is isomorphic to F032A.
(2) (R,L, S) ≡ ({b}, {ba2}, {1, a}) with n = 6 and Γ is isomorphic to F048A.
(3) (R,L, S) ≡ ({b}, {ba2t}, {1, a}), with t an odd, 3 ≤ t ≤ n − 3, n | 2(t2 + t + 1)
and Γ is isomorphic to CQ(t, n).
(4) (R,L, S) ≡ ({b, ba2}, {a, a−1}, {1}) with n = 4 and Γ is isomorphic to F032A.
(5) (R,L, S) ≡ ({b, ba6}, {a, a−1}, {1}) with n = 10 and Γ is isomorphic to F080A.
(6) (R,L, S) ≡ ({b, ba2}, {a, a−1}, {1}) with n = 12 and Γ is isomorphic to F096A.
In this paper, we determine all cubic vertex-transitive bi-Cayley graphs over semidihe-
dral group SD4n. The main results are in the following.
Theorem 1.2. Let Γ be a 0-type cubic connected bi-Cayley graph over SD4n. Then Γ is a
Cayley graph and Γ is isomorphic to BiCay(SD4n, ∅, ∅, S), where S = {1, a, b}, {1, ba, b}
or {1, ba, a}.
Theorem 1.3. Let Γ be a 1-type cubic connected bi-Cayley graph over SD4n. Then Γ is
a vertex-transitive graph if and only if one of the followings holds. Furthermore all of the
corresponding graphs are Cayley graphs.
(1) (R,L, S) ≡ ({b}, {bai}, {1, aj}) with i an even, j an odd and the greatest common
divisor of i, j and n is equal to 1.
(2) (R,L, S) ≡ ({b}, {bal}, {1, ba}) with (l − 1)2 ≡ 1, n− 1 (mod 2n).
Theorem 1.4. Let Γ be a 2-type cubic connected bi-Cayley graph over SD4n. Then Γ is a
vertex-transitive graph if and only if one of the followings holds:
(1) (R,L, S) ≡ ({b, ban}, {ba, ban+1}, {1});
(2) (R,L, S) ≡ ({a, a−1}, {b, ba2l}, {1}) with 2l2 ≡ 2 (mod 2n);
(3) (R,L, S) ≡ ({a, a−1}, {b, ba2l}, {1}) with 2l2 ≡ −2 (mod 2n).
Furthermore, the graphs corresponding to (1) and (2) are Cayley graphs and the graph
corresponding to (3) is a non-Cayley graph.
Theorem 1.1 gives a classification of cubic edge-transitive and arc-transitive bi-Cayley
graphs over SD4n. Theorems 1.2, 1.3 and 1.4 give a classification of cubic vertex-transitive
bi-Cayley graphs over SD4n. As a byproduct, we construct a family of vertex-transitive
non-Cayley graphs which correspond to (3) in Theorem 1.4.
566 Ars Math. Contemp. 23 (2023) #P4.04 / 563–575
2 Preliminary
In this section, we give two properties of bi-Cayley graph.
Proposition 2.1 ([14, Lemma 3.1]). For a bi-Cayley graph BiCay(H,R,L, S) over H , the
following hold.
(1) H is generated by R ∪ L ∪ S.
(2) Up to graph isomorphism, S can be chosen to contain the identity of H .
(3) For any automorphism α of H , BiCay(H, R, L, S) ∼= BiCay(H, Rα, Lα, Sα).
(4) BiCay(H, R, L, S) ∼= BiCay(H, L, R, S−1).
Let Γ = BiCay(H, R, L, S). For an automorphism α of H and x, y, g ∈ H , define
two permutations of V (Γ) = H0 ∪H1 as follows:
δα,x,y : h0 7→ (xhα)1, h1 7→ (yhα)0, ∀h ∈ H,
σα,g : h0 7→ (hα)0, h1 7→ (ghα)1, ∀h ∈ H.
Set
I = {δα,x,y | α ∈ Aut (H) s.t. Rα = x−1Lx, Lα = y−1Ry, Sα = y−1S−1x},
F = {σα,g | α ∈ Aut (H) s.t. Rα = R, Lα = g−1Lg, Sα = g−1S}.
Proposition 2.2 ([14, Theorem 1.1]). Let Γ = BiCay(H, R, L, S) be a bi-Cayley graph
over the group H . Then NAut (Γ)(R(H)) = R(H)⋊ F if I = ∅ and NAut (Γ)(R(H)) =
R(H)⟨F, δα,x,y⟩ if I ̸= ∅ and δα,x,y ∈ I . Furthermore, for any δα,x,y ∈ I , we have the
following:
(1) ⟨R(H), δα,x,y⟩ acts transitively on V (Γ);
(2) if α has order 2 and x = y = 1, then Γ is isomorphic to the Cayley graph
Cay(H̄, R ∪ αS), where H̄ = H ⋊ ⟨α⟩.
3 Proof of main theorems
In the beginning of this section, firstly we give some basic properties of SD4n in the fol-
lowing lemma without proof which are needed in the proof of our main Theorems.
Lemma 3.1. The following hold.
(1) SD4n = ⟨a⟩ ∪ b⟨a⟩. Where b⟨a⟩ = {ba2i} ∪ {ba2i+1} with 0 ⩽ i ⩽ n − 1,
and furthermore every element of set {ba2i} has order 2 and every element of set
{ba2i+1} has order 4.
(2) Aut (SD4n) is transitive on sets {ba2i} and {ba2i+1} with 0 ⩽ i ⩽ n− 1.
(3) If SD4n = ⟨x, y⟩, then there exists α ∈ Aut (SD4n) mapping {x, y} to one of the
following subsets:{a, b}, {ba, b} {ba, a}.
J. Cao et al.: A classification of connected cubic vertex-transitive bi-Cayley graphs . . . 567
For any integers i, j satisfying (i, 2n) = 1 and j is even, we have ⟨ai, baj⟩ = ⟨a, b⟩ =
SD4n and the map
ψi,j : a 7→ ai, b 7→ baj
can induce an automorphism of SD4n. In the following of this section we shall use ψi,j to
denote the automorphism of SD4n induced by the above map.
Proof of Theorem 1.2. Since Γ is a 0-type bi-Cayley graph, R = L = ∅. By Propo-
sition 2.1(1) and (2), let S = {1, g, h} with SD4n = ⟨g, h⟩. By Lemma 3.1(3), S is
equivalent to one of the following three subsets: {1, a, b}, {1, ba, b}, {1, ba, a}. It is easy
to check that ψ−1,0, ψn+1,0 and ψ−1,n+2 are three automorphisms of SD4n of order 2 such
that {1, a, b}ψ−1,0 = {1, a, b}−1, {1, ba, b}ψn+1,0 = {1, ba, b}−1 and {1, ba, a}ψ−1,n+2 =
{1, ba, a}−1. By Proposition 2.2, Γ is a Cayley graph. □
In order to get the classification of 1-type graph Γ, we give the following Lemma.
Lemma 3.2 ([12, Proposition 5.1]). Let K = ⟨a, b | an = b2 = (ab)2 = 1⟩ be a dihedral
group of order 2n, and Γ1 = BiCay(K, {b}, {bai}, {1, baj}) be a cubic bi-Cayley graph
over K. If Γ1 is vertex-transitive, then (j, n) = 1 and j2 ≡ ±(j − i)2 (mod n).
Proof of Theorem 1.3. Assume that Γ is a 1-type cubic vertex-transitive bi-Cayley graph
over SD4n. By the definition of 1-type bi-Cayley graph, we can let R = {x}, L = {y}
and S = {1, z}. As R = R−1 and L = L−1, both x and y are involutions. Since Γ is
connected, by Proposition 2.1(1), SD4n = ⟨x, y, z⟩. We confirm that there is at least one
of x and y is not contained in ⟨a⟩. If not, x = y = an implies that SD4n = ⟨an, z⟩ for
some z ∈ SD4n, a contradiction. Without loss of generality, assume that x ∈ b⟨a2⟩. By
Lemma 3.1(2), Aut (SD4n) acts transitively on the set {ba2k | 0 ≤ k ≤ n − 1}. So we
assume that x = b. Now y = an or y = bai for some even i.
Case 1: y = an ∈ ⟨a⟩.
In this case, z = am or bam. Since SD4n = ⟨x, y, z⟩, one has (m, 2n) = 1. The
map am 7→ a, b 7→ b can induce an automorphism α of SD4n, such that (R,L, S)α =
({b}, {an}, {1, a}) or ({b}, {an}, {1, ba}).
If (R,L, S) ≡ ({b}, {an}, {1, a}), then (10, 11, (an)1, (an)0, (an+1)1, a1) is the unique
6-cycle passing through 10. On the other hand, there exists a 6-cycle (11, (an)1, (an−1)0,
(an−1)1, (a
−1)1, (a
−1)0) passing through 11 but not passing through 10, contrary to the
vertex-transitivity of Γ.
Suppose that (R,L, S) ≡ ({b}, {an}, {1, ba}). Then there is a 5-cycle (11, (an)1,
(an)0, (ba
n+1)1, (ba
n+1)0) passing through 11 but not passing through 10. On the other
hand, (10, 11, (an)1, (ba)0, (ba)1) and (10, 11, (ban+1)0, (ban+1)1, (ba)1) are all 5-cycles
passing through 10, and these also passing through 11, contrary to the vertex-transitivity
of Γ.
Case 2: y = bai ∈ b⟨a2⟩ for some even i.
In this case, z = aj or baj for some odd j.
Subcase 2.1: z = aj ∈ ⟨a⟩ for some odd j.
In this case, it is easy to check that ψ−1,i is an automorphism of SD4n of order 2 such
that Rψ−1,i = L, Lψ−1,i = R and Sψ−1,i = S−1. By Proposition 2.2(2), Γ is a Cayley
graph. If (j, 2n) ̸= 1, then ⟨a⟩ = ⟨ai, aj⟩. So the greatest common divisor of i, j and n is
equal to 1. This is the graph corresponding to (1) in the theorem.
568 Ars Math. Contemp. 23 (2023) #P4.04 / 563–575
Subcase 2.2: z = baj ∈ b⟨a⟩ for some odd j.
Suppose that j = n2 with odd
n
2 . By the connectivity of Γ, ⟨a⟩ = ⟨a
n
2 , ai⟩, and hence
⟨ai⟩ = ⟨a2⟩ or ⟨a4⟩. By Proposition 2.1(3), we may assume that ai = a2 or a4. Now Γ is
isomorphic to a bi-Cayley graph with (R,L, S) ≡ ({b}, {ba2}, {1, ban2 }) or (R,L, S) ≡
({b}, {ba4}, {1, ban2 }). In the above two cases, we can find that there are two 6-cycles pass-
ing through 10, namely (10, b0, b1, (a
3n
2 )0, (ba
3n
2 )0, 11) and (10, b0, (a
n
2 )1, (a
n
2 )0, (ba
n
2 )0,
(ba
n
2 )1). On the other hand, (11, 10, b0, b1, (a
3n
2 )0, (ba
3n
2 )0) is the unique 6-cycle passing
through 11, contrary to the vertex-transitivity of Γ. So, we may assume that z ̸= ba
n
2 .
Suppose that (j, 2n) ̸= 1. Since ⟨a⟩ = ⟨ai, aj⟩ with i an even and j an odd, the greatest
common divisor of i, j and n is equal to 1. We consider the subgraphs of Γ induced by the
vertices at distance from 10 and 11 at most 4 respectively. Since j ̸≡ n2 ( mod 2n), one
can see that (10, 11, (ban+j)0, (ban+j)1, (an)0, (an)1, (baj)0, (baj)1) is the unique 8-cycle
passing through 10 and it is also the unique 8-cycle passing through 11 too.
Since (an)0 and (an)1 are the unique vertices that have the longest distance from 10
and 11 in the 8-cycle, respectively, {10, (an)0} and {11, (an)1} are blocks of Aut (Γ) on
V (Γ). Let
C0 = {{10, (an)0}R(h) | h ∈ SD4n}, C1 = {{11, (an)1}R(h) | h ∈ SD4n}
and C = C0 ∪ C1. Then C is a complete block system of Aut (Γ).
Let ΓC be the quotient graph. Let N = ⟨R(an)⟩ and let K be the kernel of Aut (Γ)
acting on C. Now one can show that K = N and ΓC has valence 3 and K = N is
semiregular. Since R(SD4n) acts on V (Γ) semiregularly with two orbits, R(SD4n)/N
acts on C semiregularly with two orbits C0 and C1. So the quotient graph ΓC is a bi-
Cayley graph over R(SD4n)/N . Let SD4n = R(SD4n)/N and let h̄ = hN for any
h ∈ R(SD4n). Now SD4n = ⟨ā, b̄ | ān = b̄2 = (āb̄)2 = 1̄⟩ ∼= D2n. Also we can assume
that
V (ΓC) = {h̄0 | h̄ ∈ SD4n} ∪ {h̄1 | h̄ ∈ SD4n}.
Note that for any h̄ ∈ SD4n
NΓC (h̄0) = {bh0, h̄1, bajh1}, NΓC (h̄1) = {baih1, h̄0, ban+jh0}.
So we may view ΓC as the bi-Cayley graph BiCay(SD4n, {b̄}, {bai}, {1̄, baj}). Since Γ
is vertex-transive, the quotient graph ΓC is also vertex-transitive. Since ΓC is a 1-type
cubic vertex-transitive bidihedrant, ΓC is a Cayley graph by [12, Proposition 5.1]. Also by
Lemma 3.2, (j, n) = 1 and j2 ≡ ±(j−i)2( mod n). This implies that (j, 2n) = 1, which
is contrary to (j, 2n) ̸= 1.
So we can assume that (j, 2n) = 1. The map aj 7→ a, b 7→ b can induce an automor-
phism β of SD4n, such that (R,L, S)β ≡ ({b}, {bal}, {1, ba}). We consider the subgraphs
of Γ induced by the vertices at distance from 10 and 11 at most 4, respectively. Considering
possible 8-cycles containing 10 and 11, we see that if l ̸≡ 2, n, n + 2, n2 + 1 (mod 2n),
then (10, 11, (ban+1)0, (ban+1)1, (an)0, (an)1, (ba)0, (ba)1) is the unique 8-cycle passing
through 10 and it is the unique 8-cycle passing through 11 too. In this case, any automor-
phism of Γ sends spokes to spokes. Furthermore, {10, (an)0} and {11, (an)1} are blocks
of Aut (Γ) on V (Γ) and by a similar way, one can see that (l − 1)2 ≡ ±1 (mod n). So
(l− 1)2 ≡ ±1 (mod 2n), (l− 1)2 ≡ n+ 1 (mod 2n) or (l− 1)2 ≡ n− 1 (mod 2n). It
is easy to find that l ≡ 2, n, n+2 (mod 2n) also satisfy (l− 1)2 ≡ ±1 (mod 2n). In the
following, we divide the proof into four subcases.
J. Cao et al.: A classification of connected cubic vertex-transitive bi-Cayley graphs . . . 569
Subsubcase 2.2.1: (R,L, S) ≡ ({b}, {ban2 +1}, {1, ba}) with n2 an odd.
In this case, we can find that there are two 6-cycles passing through 11, namely (11,
(ba
n
2 +1)1, (ba
n
2 +1)0, (a
3n
2 )1, (ba)1, 10) and (11, (ba
n
2 +1)1, (a
n
2 )0, (a
n
2 )1, (ba
n+1)1,
(ban+1)0). On the other hand, (11, (ba
n
2 +1)1, (ba
n
2 +1)0, (a
3n
2 )1, (ba)1, 10) is the unique
6-cycle passing through 10, contrary to the vertex-transitivity of Γ.
Subsubcase 2.2.2: (R,L, S) ≡ ({b}, {bal}, {1, ba}) with (l − 1)2 ≡ ±1 (mod 2n).
If (l−1)2 ≡ −1 (mod 2n), then l2−2l+2 ≡ 0 (mod 2n) implies that n|( l
2
2 − l+1).
Since l is even, l
2
2 − l + 1 is odd. This implies that n is also odd, a contradiction.
Assume that (l − 1)2 ≡ 1 (mod 2n). Then ((l − 1)2, 2n) = 1, and furthermore
(n+1− l, 2n) = 1. It is easy to check that ψn+1−l,l is an automorphism of SD4n of order
2 such that Rψn+1−l,l = L, Lψn+1−l,l = R and Sψn+1−l,l = S−1. By Proposition 2.2(2), Γ
is a Cayley graph. This is the graph of type (2) in the theorem.
Subsubcase 2.2.3: (R,L, S) ≡ ({b}, {bal}, {1, ba}) with (l − 1)2 ≡ n+ 1( mod 2n).
Suppose that Γ is vertex-transitive. Then there is an automorphism ω2 of Γ such that
10
ω2 = 11. Note that 11ω2 = 10 or 11ω2 = (ban+1)0.
Suppose that 11ω2 = 10. Then b0ω2 = (bal)1 and (ba
l
1)
ω2
= b0. We consider
the subgraphs of Γ induced by the vertices at distance from 10 and 11 at most 5, respec-
tively. It is easy to find that both (bal)0 and (al−1)0 are adjacent with (bal)1, furthermore
{b0, (an−1)1}ω2 = {(bal)1, (bal)0} or {b0, (an−1)1}ω2 = {(bal)1, (al−1)0}.
Assume that {b0, (an−1)1}ω2 = {(bal)1, (bal)0}. Then (an−1)1
ω2 = (bal)0 and
(an−1)0
ω2 = (an+l−1)1. There is a unique 10-cycle passing through vertexes 11, 10, b0,
(an−1)1 and (an−1)0, that is (11, 10, b0, (an−1)1, (an−1)0, (ban)1, (ban)0, (an)0,
(ban+1)1, (ba
n+1)0). On the other hand, there are two 10-cycles passing through vertexes
10, 11, (ba
l)1, (ba
l)0 and (an+l−1)1, that are (10, 11, (bal)1, (bal)0, (an+l−1)1, (an+l−1)0,
(ban+l−1)0, (ba
n+l−1)1, (a
n−1)1, b0) and (10, 11, (bal)1, (bal)0, (an+l−1)1, (an+l−1)0,
(ban+l)1, (a
n)1, (ba)0, (ba)1), a contradiction. So {b0, (an−1)1}ω2 = {(bal)1, (al−1)0},
which implies (an−1)1
ω2 = (al−1)0. By a similar reason, one can show that
{(bal)1, (al−1)0}ω2 = {b0, (an−1)1}, and hence (al−1)0
ω2
= (an−1)1. Therefore
10
ω2
11
ω2
10, b0
ω2
(bal)1
ω2
b0, (a
l−1)0
ω2
(an−1)1
ω2
(al−1)0;
Now let us consider the subgraphs of Γ induced by the vertices at distance from (al−1)0
and (an−1)1 at most 5, respectively. We can get
(an−1)0
ω2
(al−1)1
ω2
(an−1)0,
(ban−1)0
ω2
(ba2l−1)1
ω2
(ban−1)0,
(a2(l−1))0
ω2
(a2(n−1))1
ω2
(a2(l−1))0;
By a similar way, we can get (ak(n−1))1
ω2
= (ak(l−1))0 for any k. By inserting k =
l − 1, we have (a(l−1)(n−1))1
ω2
= (an+1−l)1
ω2
= (an+1)0. On the other hand, (10, 11,
(ban+1)0, (ba
n+1)1, (a
n)0, (a
n)1, (ba)0, (ba)1) is the unique 8-cycle passing through 10
and 11 implies that (10, 11, (ban+1)0, (ban+1)1, (an)0, (an)1, (ba)0, (ba)1) is mapped to
(11, 10, (ba)1, (ba)0, (a
n)1, (a
n)0, (ba
n+1)1, (ba
n+1)0) by ω2. So (ban+1)1
ω2 = (ba)0,
furthermore (an+1−l)1
ω2
= a0, a contradiction.
570 Ars Math. Contemp. 23 (2023) #P4.04 / 563–575
Suppose that 11ω2 = (ban+1)0. Then b0ω2 = (bal)1 and (ba
l
1)
ω2
= (an+1)0. We
consider the subgraphs of Γ induced by the vertices at distance from 10 and 11 at most
5, respectively. Since both (an+1)1 and (ban+2)1 are adjacent with (an+1)0, we have
{bal1, (al−1)0}ω2 = {(an+1)0, (an+1)1} or {(bal)1, (al−1)0}ω2 = {(an+1)0, (ban+2)1}.
Assume that {(bal)1, (al−1)0}ω2 = {(an+1)0, (ban+2)1}. Then (al−1)0
ω2
=
(ban+2)1 and (al−1)1
ω2
= (ban+2)0. There is a unique 10-cycle passing through ver-
texes 10, 11, (bal)1, (al−1)0 and (al−1)1, that is (10, 11, (bal)1, (al−1)0, (al−1)1, (ban+l)0,
(ban+l)1, (a
n)1, (ba)0, (ba)1). On the other hand, there are two 10-cycles passing through
vertexes 11, (ban+1)0, (an+1)0, (ban+2)1 and (ban+2)0, that are (11, (ban+1)0, (an+1)0,
(ban+2)1, (ba
n+2)0, a1, a0, (ba)0, (ba)1, 10) and (11, (ban+1)0, (an+1)0, (ban+2)1,
(ban+2)0, a1, (ba
l+1)1, (a
l)0, (ba
l)0, (ba
l)1), a contradiction. So {(bal)1, (al−1)0}ω2 =
{(an+1)0, (an+1)1}, and hence (al−1)0
ω2
= (an+1)1. Therefore ω2 maps the path
P1 : 10, 11, (ba
l)1, (a
l−1)0 to the path Q1 : 11, (ban+1)0, (an+1)0, (an+1)1 in this order,
namely 10ω2 = 11, 11ω2 = (ban+1)0, (bal)1
ω2
= (an+1)0 and (al−1)0
ω2
= (an+1)1.
Similarly, we consider the subgraphs of Γ induced by the vertices at distance from (al−1)0
and (an+1)1 at most 5, respectively. One can show that ω2 maps the path P2 : (al−1)0,
(al−1)1, (ba
2l−1)1, (a
2(l−1))0 to the path Q2 : (an+1)1, (ba2)0, (a2)0, (a2)1 in this order.
By a similar way, we can get (ak(l−1))0
ω2
= (ak(n+1))1 for any k. By inserting k = l− 1,
we have (a(l−1)(l−1))0
ω2
= (an+1)0
ω2 = (a(l−1)(n+1))1 = (a
n+l−1)1. On the other
hand, (10, 11, (ban+1)0, (ban+1)1, (an)0, (an)1, (ba)0, (ba)1) is the unique 8-cycle pass-
ing through 10 and 11 implies that (10, 11, (ban+1)0, (ban+1)1, (an)0, (an)1, (ba)0, (ba)1)
is mapped to (11, (ban+1)0, (ban+1)1, (an)0, (an)1, (ba)0, (ba)1, 10) by ω2. So
(ban+1)0
ω2 = (ban+1)1, which implies (an+1)0
ω2 = (an+1−l)1, a contradiction. There-
fore Γ is not vertex-transitive, a contradiction.
Subsubcase 2.2.4: (R,L, S) ≡ ({b}, {bal}, {1, ba}) with (l − 1)2 ≡ n− 1 (mod 2n).
Note that n ≡ 2 (mod 4) in this case. One can check that the map
ω3 : (a
k)0 7→ (ak(1−l))1, (ak)1 7→ (ban+1+k(1−l))0,
(bak)0 7→ (bak(1−l)+l)1, (bak)1 7→ (a(k−1)(1−l))0,
with 0 ≤ k < 2n is a permutation on V (Γ) with order 8. Furthermore, for any 0 ≤ k < 2n,
we have
NΓ((a
k)0)
ω3 = {(ban+1+k(1−l))0, (ak(1−l))0, (bak(1−l)+l)1} = NΓ((ak(1−l))1),
NΓ((a
k)1)
ω3 = {(ak(1−l))1, (ban+1+k(1−l))1, (an+1+k(1−l))0} = NΓ((ban+1+k(1−l))0),
NΓ((ba
k)0)
ω3 = {(bak(1−l)+l)0, (a(k−1)(1−l))0, (ak(1−l))1} = NΓ((bak(1−l)+l)1),
NΓ((ba
k)1)
ω3 = {(a(k−1)(1−l))1, (bak(1−l)+l)1, (ba(k−1)(1−l))0} = NΓ((a(k−1)(1−l))0).
So ω3 induces an automorphism of Γ of order 8. Denote H01 = {(ak)0}, H02 =
{(bak)0}, and H11 = {(ak)1}, H12 = {(bak)1} with 0 ≤ k < 2n. We have the following:
H01
ω3
H11
ω3
H02
ω3
H12
ω3
H01
Note that ⟨R(a)⟩ acts transitively on the sets H01, H02, H11, H12, respectively. So M2 =
⟨R(a), ω3⟩ is a vertex-transitive subgroup of Aut (Γ). By calculation, ω34 = R(an),
ω−13 R(a)ω3 = R(a)
1−l. So M2 = ⟨R(a)⟩⟨ω3⟩ and furthermore |M2| = 8n. Therefore
M2 acts regularly on V (Γ), and hence Γ is a Cayley graph of type (2) in the
theorem. □
J. Cao et al.: A classification of connected cubic vertex-transitive bi-Cayley graphs . . . 571
Proof of Theorem 1.4. Let Γ be a 2-type bi-Cayley graph and let R = {x1, x2}, L =
{y1, y2} and S = {1}.
Firstly, we assume that all of x1, x2, y1 and y2 belong to b⟨a⟩. By the structure of
SD4n, if their orders are the same, then SD4n ̸= ⟨x1, x2, y1, y2⟩. So without loss of gen-
erality, we can assume that x1, x2 have order 2 and y1, y2 have order 4. By Lemma 3.1 (2),
Aut (SD4n) acts transitively on set {ba2i} with 0 ⩽ i ⩽ n − 1, we can let x1 = b,
x2 = ba
2t, y1 = bas and y2 = ban+s. So (R,L, S) ≡ ({b, ba2t}, {bas, ban+s}, {1}) with
s an odd. It is easy to find that (11, (bas)1, (an)1, (ban+s)1) is the unique 4-cycle pass-
ing through 11. The vertex-transitivity of Γ implies that there is a unique 4-cycle passing
through 10. Considering possible 4-cycles containing 10, we have (a2t)0 = (a−2t)0, and
hence n = 2t. Noticing that ⟨a⟩ = ⟨as, an⟩, we get (s, 2n) = 1. So the map f : as 7→
a, b 7→ b can induce an automorphism of SD4n such that (R,L, S)f ≡ ({b, ban}, {ba,
ban+1}, {1}). Hence, we can assume that (R,L, S) = ({b, ban}, {ba, ban+1}, {1}).
Let Σ = Cay(D8n, {d, dc, dc2n}) where D8n = ⟨c, d | c4n = d2 = 1, dcd = c−1⟩.
Define a map from V (Γ) to V (Σ) as follows:
ϕ : (ar)0 7→ c2r, (ar)1 7→ dc2r+1,
(bar)0 7→ dc2r, (bar)1 7→ c2r−1,
with 0 ≤ r ≤ 2n− 1. Furthermore, for any r ∈ Z2n, we have
NΓ((a
r)0)
ϕ = {(ar)1, (bar)0, (ban+r)0}ϕ = {dc2r+1, dc2r, dc2n+2r} = NΣ(c2r),
NΓ((a
r)1)
ϕ = {(ar)0, (bar+1)1, (ban+r+1)1}ϕ = {c2r, c2r+1, c2n+2r+1}
= NΣ(dc
2r+1),
NΓ((ba
r)0)
ϕ = {(bar)1, (ar)0, (an+r)0}ϕ = {c2r−1, c2r, c2n+2r} = NΣ(dc2r),
NΓ((ba
r)1)
ϕ = {(bar)0, (an+r−1)1, (ar−1)1}ϕ = {dc2r, dc2n+2r−1, dc2r−1}
= NΣ(c
2r−1).
It follows that ϕ is an isomorphism from Γ to Σ. Then Γ is a Cayley graph over a dihedral
group. This is the graph of type (1) in the theorem.
In the following, we assume that there is at least one of x1, x2, y1, y2 belongs to ⟨a⟩.
Without loss of generality, let x1 ∈ ⟨a⟩. We divide the proof into two cases:
Case 1: |⟨x1⟩| = 2.
In this case, x1 = an. The condition R = R−1 implies that x2 is also an element of
order 2, and hence x2 = ba2i1 for some integer i1. Since there is an automorphism of SD4n
sending a and ba2i1 to a and b, respectively, we can assume x2 = b up to equivalence. If
y1 ∈ ⟨a⟩ and y2 /∈ ⟨a⟩, then y1 ̸= y2−1. So L = L−1 implies that y1 = an and y2 = ba2i2 .
Then SD4n ̸= ⟨b, ba2i2 , an⟩, contrary to the connectivity of Γ. Similarly, if y2 ∈ ⟨a⟩
and y1 /∈ ⟨a⟩, then one can get a similar contradiction. So we consider the following two
subcases:
Subcase 1.1: Both y1 and y2 belong to b⟨a⟩.
In this case, both y1 and y2 have order 4 by the connectivity of Γ and the structure
of SD4n. Let y1 = bal1 and y2 = ban+l1 for some odd integer l1. Now it holds that
SD4n = ⟨b, an, al1⟩. This implies that (l1, 2n) = 1. So the map d : al1 7→ a, b 7→ b
can induce an automorphism of SD4n such that (R,L, S)d ≡ ({an, b}, {ba, ban+1}, {1}).
Hence, we can assume that (R,L, S) ≡ ({an, b}, {ba, ban+1}, {1}) up to equivalence. Let
572 Ars Math. Contemp. 23 (2023) #P4.04 / 563–575
us consider the subgraphs of Γ induced by the vertices at distance from 10 and 11 at most 3,
respectively. There are two 7-cycles passing through 10 but not passing through 11 that is
(10, b0, b1, (a
n−1)1, (ba
n)1, (ba
n)0, (a
n)0) and (10, b0, b1, (a−1)1, (ban)1, (ban)0, (an)0).
On the other hand, (11, (ba)1, (ba)0, a0, (an+1)0, (ban+1)0, (ban+1)1) is the unique 7-
cycle passing through 11 but not passing through 10, contrary to the vertex-transitivity
of Γ.
Subcase 1.2: Both y1 and y2 belong to ⟨a⟩.
In this case, we can let y1 = ai3 , y2 = a−i3 . Now it holds that SD4n = ⟨b, an, ai3⟩.
This implies that (i3, 2n) = 1. So the map e : ai3 7→ a, b 7→ b can induce an automor-
phism of SD4n such that (R,L, S)e ≡ ({an, b}, {a, a−1}, {1}). Hence we can assume
that (R,L, S) ≡ ({an, b}, {a, a−1}, {1}). It is easy to find that there is a unique 4-cycle
(10, b0, (ba
n)0, (a
n)0) passing through 10. The vertex-transitivity of Γ implies that there is
a unique 4-cycle passing through 11. Considering possible 4-cycles containing 11, we have
(a2)1 = (a
−2)1, and hence n = 2, contrary to n ≥ 4.
Case 2: |⟨x1⟩| ≠ 2
In this case, we can let x1 = ai, x2 = a−i. By the connectivity of Γ and the structure
of SD4n, if y1 ∈ ⟨a⟩, then y2 /∈ ⟨a⟩ and y1 ̸= y2−1. By the condition L = L−1, we can
assume that L = {an, b} up to equivalence. In this case, Γ is isomorphic to a graph of
subcase 1.2, by Proposition 2.1(4). So we can assume that both y1 and y2 belong to b⟨a⟩.
We divide the proof into the following two subcases:
Subcase 2.1: The orders of y1 and y2 are 2.
By Lemma 3.1(2), Aut (SD4n) acts transitively on the set {ba2i} for 0 ⩽ i ⩽ n − 1,
we can let y1 = b, y2 = ba2j . So (R,L, S) ≡ ({ai, a−i}, {b, ba2j}, {1}). The vertex-
transitivity of Γ implies that there must exist an automorphism α in Aut(Γ) such that
1α0 = 11. We consider the subgraphs of Γ induced by the vertices at distance from 10
and 11 at most 5, respectively. There are six 10-cycles passing through edge {10, (ai)0},
{10, (a−i)0}, {11, b1} and {11, (ba2j)1} separately. In the same time, there are eight 10-
cycles passing through edge {10, 11}. This implies that {10, (ai)0}α ̸= {10, 11}. Hence
{10, (ai)0}α = {11, b1} or {11, (ba2j)1} and {10, (a−i)0}α = {11, b1} or {11, (ba2j)1}.
So (ai)0
α
= b1 or (ba2j)1 and (a−i)0
α
= b1 or (ba2j)1. Without loss of generality,
let (ai)0
α
= b1. We consider the subgraphs of Γ induced by the vertices at distance
from (ai)0 and b1 at most 5, respectively. Similarly, we can find that there exists six 10-
cycles passing through edge {(ai)0, (a2i)0} and {b1, (a−2j)1} separately. There are eight
10-cycles passing through edge {(ai)0, (ai)1} and {b1, b0} separately. So by the vertex-
transitivity of Γ, {(ai)0, (a2i)0}α = {b1, (a−2j)1}, and hence (a2i)0
α
= (a−2j)1. In a
similar way, one can see that the cycle (10, (ai)0, (a2i)0, (a3i)0, · · · , (a−i)0) is mapped
to the cycle (11, b1, (a−2j)1, (ba−2j)1, · · · , (ba2j)1) by α. So the lengthes of the cycles
(10, (a
i)0, (a
2i)0, (a
3i)0, · · · , (a−i)0) and (11, b1, (a−2j)1, (ba−2j)1, · · · , (ba2j)1) are the
same, and hence ⟨ai⟩ = ⟨aj⟩. Furthermore, SD4n = ⟨ai, a2j , b⟩ = ⟨ai, b⟩ implies that
(i, 2n) = (j, 2n) = 1. It is easy to find that the map g : ai 7→ a, b 7→ b can induce an
automorphism of SD4n such that (R,L, S)g ≡ ({a, a−1}, {b, ba2l}, {1}) with (l, 2n) = 1.
So we can assume that (R,L, S) = ({a, a−1}, {b, ba2l}, {1}) with (l, 2n) = 1 up to
equivalence.
Now the cycle (10, a0, (a2)0, (a3)0, · · · , (a−1)0) is mapped to the cycle (11, b1,
(a−2l)1, (ba
−2l)1, · · · , (a2l)1, (ba2l)1) by α. So we can get
(a2k)0
α
= (a−2kl)1, (a
2k+1)0
α
= (ba−2kl)1 (I)
J. Cao et al.: A classification of connected cubic vertex-transitive bi-Cayley graphs . . . 573
where 0 ≤ k ≤ n − 1. Note that (a2k)0 and (a2k+1)0 are adjacent with (a2k)1 and
(a2k+1)1, respectively, and (a−2kl)1 and (ba−2kl)1 are adjacent with (a−2kl)0 and
(ba−2kl)0, respectively. So we can get
(a2k)1
α
= (a−2kl)0, (a
2k+1)1
α
= (ba−2kl)0 (II)
where 0 ≤ k ≤ n− 1. By (I) and (II), it is easy to get
α : (a2k)0 7→ (a−2kl)1 7→ (a2kl
2
)0. (III)
Since (a2k)0 is adjacent with (a2k+1)0, we have (a2k+1)0
α2
= (a2kl
2±1)0. Similarly,
by {(a2k+1)0, (a2k+2)0}α
2
= {(a2kl2±1)0, (a2kl
2+2l2)0}, it holds that (a2kl
2+2l2)0 =
(a2kl
2±2)0 or (a2kl
2+2l2)0 = (a
2kl2)0. If (a2kl
2+2l2)0 = (a
2kl2)0, then 2l2 ≡ 0 (mod 2n),
which implies that n|l2, contrary to l is odd. So (a2kl2+2l2)0 = (a2kl
2±2)0, and hence,
2l2 ≡ ±2 (mod 2n).
If 2l2 ≡ 2 (mod 2n), then the map
β : (a2k)0 7→ (a−2kl)1, (a2k)1 7→ (a−2kl)0,
(a2k+1)0 7→ (ba−2kl)1, (a2k+1)1 7→ (ba−2kl)0,
(ba2k)0 7→ (a−2kl+1)1, (ba2k)1 7→ (a−2kl+1)0,
(ba2k+1)0 7→ (ban+1−2kl)1, (ba2k+1)1 7→ (ban+1−2kl)0,
with 0 ≤ k < n is a permutation on V (Γ) with order 2. Furthermore, for any 0 ≤ k < n
we have
NΓ((a
2k)0)
β = {(ba−2kl)1, (ba2(1−k)l)1, (a−2kl)0} = NΓ((a−2kl)1),
NΓ((a
2k)1)
β = {(a−2kl+1)0, (a−2kl−1)0, (a−2kl)1} = NΓ((a−2kl)0),
NΓ((a
2k+1)0)
β = {((a−2kl)1, (a−2(k+1)l)1, (ba−2kl)0} = NΓ((ba−2kl)1),
NΓ((a
2k+1)1)
β = {(ban−1−2kl)0, (ban+1−2kl)0, (ba−2kl)1} = NΓ((ba−2kl)0),
NΓ((ba
2k)0)
β = {(ba−2kl+1)1, (ba2(1−k)l+1)1, (a−2kl+1)0} = NΓ((a−2kl+1)1),
NΓ((ba
2k)1)
β = {(a−2kl+2)0, (a−2kl)0, (a−2kl+1)1} = NΓ((a−2kl+1)0)
NΓ((ba
2k+1)0)
β = {(an+1−2(k+1)l)1, (an+1−2kl)1, (ban+1−2kl)0} = NΓ((ban+1−2kl)1),
NΓ((ba
2k+1)1)
β = {(ba−2kl)0, (ba2−2kl)0, (ban+1−2kl)1} = NΓ((ban+1−2kl)0).
So β induces an automorphism of Γ of order 2. Denote H01 = {(a2i)0}, H02 =
{(a2i+1)0}, H03 = {(ba2i)0}, H04 = {(ba2i+1)0}, H11 = {(a2i)1}, H12 = {(a2i+1)1},
H13 = {(ba2i)1}, H14 = {(ba2i+1)1}, with 0 ≤ i < n. Let H0 = H01
⋃
H02
⋃
H03⋃
H04; H1 = H11
⋃
H12
⋃
H13
⋃
H14; We have the following:
H01
β H11
R(b) H13
β H02
R(b) H04
β H14
R(b) H12
β H03
R(b) H01.
R(a2) acts transitively on Hij where i = 0, 1; j = 1, 2, 3, 4; So M = ⟨R(a2), R(b), β⟩ is
a vertex-transitive subgroup of Γ.
574 Ars Math. Contemp. 23 (2023) #P4.04 / 563–575
By calculation, βR(a2)β = R(a−2l) ∈ ⟨R(a2)⟩, so β normalizes ⟨R(a2)⟩. Noticing
that R(b) also normalizes ⟨R(a2)⟩, we see that ⟨R(a2)⟩ ⊴ M . We consider the group
M/⟨R(a2)⟩ = ⟨β,R(b)⟩ = ⟨βR(b), R(b)⟩. By calculation (βR(b))4 = R(a−2l−2) ∈
⟨R(a2)⟩ andR(b)2 = 1 imply that βR(b)
4
= R(b)
2
= 1. It is easy to find thatR(b)
−1
βR(b)
R(b) = βR(b)
−1
. So M ≃ D8, and |M | = 8n. Therefore M acts regularly on V (Γ). So
Γ is a Cayley graph. This is the graph of type (2) in the theorem.
Let 2l2 ≡ −2 ( mod 2n). Now the map
γ : (a2k)0 7→ (a−2kl)1, (a2k)1 7→ (a−2kl)0,
(a2k+1)0 7→ (ba−2kl)1, (a2k+1)1 7→ (ba−2kl)0,
(ba2k)0 7→ (a−2kl−1)1, (ba2k)1 7→ (a−2kl−1)0,
(ba2k+1)0 7→ (ban−1−2kl)1, (ba2k+1)1 7→ (ban−1−2kl)0,
with 0 ≤ k < n is a permutation on V (Γ). Furthermore, for any 0 ≤ k < n we have
NΓ((a
2k)0)
γ = {(ba−2kl)1, (ba2(1−k)l)1, (a−2kl)0} = NΓ((a−2kl)1),
NΓ((a
2k)1)
γ = {(a−2kl+1)0, (a−2kl−1)0, (a−2kl)1} = NΓ((a−2kl)0),
NΓ((a
2k+1)0)
γ = {((a−2kl)1, (a−2(k+1)l)1, (ba−2kl)0} = NΓ((ba−2kl)1),
NΓ((a
2k+1)1)
γ = {(ban−1−2kl)0, (ban+1−2kl)0, (ba−2kl)1} = NΓ((ba−2kl)0),
NΓ((ba
2k)0)
γ = {(ba−2kl−1)1, (ba2(1−k)l−1)1, (a−2kl−1)0} = NΓ((a−2kl−1)1),
NΓ((ba
2k)1)
γ = {(a−2kl−2)0, (a−2kl)0, (a−2kl−1)1} = NΓ((a−2kl−1)0)
NΓ((ba
2k+1)0)
γ = {(an−1−2kl)1, (an−1−2(k+1)l)1, (ban−1−2kl)0} = NΓ((ban−1−2kl)1),
NΓ((ba
2k+1)1)
γ = {(ba−2kl)0, (ba−2−2kl)0, (ban−1−2kl)1} = NΓ((ban−1−2kl)0).
Therefore γ induces an automorphism of Γ mapping 10 to 11. So Γ is a vertex-transitive
graph. This is the graph of type (3) in the theorem. Now we aim to show that Γ is a non-
Cayley graph. Let H be a vertex-transitive subgroup of Aut (Γ) on V (Γ). Then there
exists an automorphism φ ∈ H such that 10φ = 11. It is easy to find that 11φ = 10. So
φ2 ∈ H10 . Similar with proof of (III), we can get (a2k)0
φ2
= (a2kl
2
)0 = (a
−2k)0. Since
(a2k+1)0 is adjacent with (a2k)0 and (a2k+2)0, we have (a2k+1)0
φ2
= (a−2k−1)0, where
0 ≤ k ≤ n − 1. The condition (a2k+1)0
φ2
= (a−2k−1)0 implies that a0φ
2
= a−10 . So
φ2 ̸= 1 and |H10 | ≥ 2. This implies that H does not act regularly on V (Γ). Since we
choose an arbitrary vertex-transitive subgroup H , Γ is a non-Cayley graph.
Subcase 2.2: The orders of y1 and y2 are 4.
By Lemma 3.1(2), Aut (SD4n) acts transitively on the set {ba2i+1} for 0 ⩽ i ⩽ n− 1,
we can let y1 = ba, y2 = ban+1. The condition SD4n = ⟨ai, ba⟩ implies that (i, 2n) = 1.
So we can assume (R,L, S) = ({a, a−1}, {ba, ban+1}, {1}) up to equivalence. It is easy to
find that (11, (ba)1, (an)1, (ban+1)1) is the unique 4-cycle passing through 11. The vertex-
transitivity of Γ implies that there is a unique 4-cycle through 10. Considering possible
4-cycles containing 10, we have (a2)0 = (a−2)0, and hence n = 2, contrary to n ≥ 4. The
proof is now complete. □
ORCID iDs
Young Soo Kwon https://orcid.org/0000-0002-1765-0806
J. Cao et al.: A classification of connected cubic vertex-transitive bi-Cayley graphs . . . 575
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P4.05 / 577–590
https://doi.org/10.26493/1855-3974.2853.b51
(Also available at http://amc-journal.eu)
Domination and independence numbers of large
2-crossing-critical graphs*
Vesna Iršič † , Maruša Lekše
Faculty of Mathematics and Physics, University of Ljubljana, Slovenia and
Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
Mihael Pačnik, Petra Podlogar, Martin Praček
Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
Received 23 March 2022, accepted 10 January 2023, published online 20 March 2023
Abstract
After 2-crossing-critical graphs were characterized in 2016, their most general subfam-
ily, large 3-connected 2-crossing-critical graphs, has attracted separate attention. This paper
presents sharp upper and lower bounds for their domination and independence numbers.
Keywords: Crossing-critical graphs, domination number, independence number.
Math. Subj. Class. (2020): 05C10, 05C62, 05C69
1 Introduction
The crossing number cr(G) of a graph G is the smallest number of edge crossings in a
drawing of G in the plane. The topic has been widely studied, see for example [7, 8, 17, 18,
20]. A graph G is k-crossing-critical if cr(G) ≥ k, but every proper subgraph H of G has
cr(H) < k. Note that subdividing an edge or its inverse operation (suppressing a vertex)
*The authors were introduced to the structure of 2-crossing-critical graphs in a workshop at the University of
Ljubljana, organized by prof. dr. Drago Bokal. We thank him for several illuminating conversations and ideas.
We would also like to thank Sandi Klavžar, Alen Vegi Kalamar, and Simon Brezovnik for co-organizing the
workshop. We would also like to thank the anonymous referees for valuable suggestions, especially for the idea
of how to simplify the proof of Theorem 3.4.
†Corresponding author. V. I. was supported by a postdoctoral fellowship at the Simon Fraser University
(Canada) and by the Slovenian Research Agency (research core funding P1-0297 and projects J1-2452, J1-1693,
N1-0095, N1-0218).
E-mail addresses: vesna.irsic@fmf.uni-lj.si (Vesna Iršič), marusa.lekse@imfm.si (Maruša Lekše),
mihapac@gmail.com (Mihael Pačnik), petra.podlogar@hotmail.com (Petra Podlogar), mpracek299@gmail.com
(Martin Praček)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
578 Ars Math. Contemp. 23 (2023) #P4.05 / 577–590
do not affect the crossing number of a graph. Thus we can restrict our studies to graphs
without degree 2 vertices. Under this restriction, Kuratowski’s Theorem tells us that the
only 1-crossing-critical graphs are K5 and K3,3. The classification of 2-crossing-critical
graphs has been of interest since the 1980s. Partial results on the topic have been reported
in [2, 9, 12, 16, 19], and some related results can be found in [1, 10, 13]. Crossing numbers
of graphs with a tile structure have been studied in [14, 15]. Finally, Bokal, Oporewski,
Richter, and Salazar [6] provided an almost complete characterization of 2-crossing-critical
graphs. In particular, they describe a tile structure of large 3-connected 2-crossing-critical
graphs (i.e., all but finitely many 3-connected 2-crossing-critical graphs). Recently, the
degree properties of crossing-critical graphs have been studied in [3, 5, 11].
The above-mentioned large 3-connected 2-crossing-critical graphs have since attracted
separate attention, see [4, 21, 22]. In [21, 22], the Hamiltonicity of these graphs is dis-
cussed, and the number of all Hamiltonian cycles is determined. In [4], several additional
properties of large 3-connected 2-crossing-critical graphs have been studied. In particular,
the number of vertices and edges can be determined from the signature of a graph, and
several results regarding their chromatic number, chromatic index, and tree-width are pre-
sented. In the present paper, we extend the studies of large 3-connected 2-crossing-critical
graphs to their domination and independence numbers.
The rest of the paper is organized as follows. In the next section, necessary definitions
and known results are listed. In Section 3, the sharp upper and lower bounds for the domi-
nation number of large 3-connected 2-crossing-critical graphs are given, while in Section 4
analogous results are proved for their independence number.
2 Preliminaries
Let G be a graph. Its vertex set is denoted by V (G) and its edge set by E(G). The (open)
neighborhood of a vertex v ∈ V (G) is N(v) = {u ∈ V (G); uv ∈ E(G)} and the closed
neighborhood of v is N [v] = {v}∪N(v). Similarly, for D ⊆ V (G), N [D] =
⋃
v∈D N [v]
is the closed neighborhood of D. Note also that [n] = {1, . . . , n} and that the reversed
sequence of a sequence a is denoted by a.
We now recall the definitions of the domination number and the independence number.
Definition 2.1. Let G be a graph. A subset D ⊆ V (G) dominates the set of vertices
X ⊆ V (G) if X ⊆ N [D]. If N [D] = V (G), then D is a dominating set of G. The
domination number γ(G) of a graph G is the size of a smallest dominating set in G.
Definition 2.2. Let G be a graph. A subset X ⊆ V (G) is independent if none of the
vertices from X are adjacent. The independence number α(G) of the graph G is the size
of a largest independent set.
In the rest of the section, we recall the characterization of 2-crossing-critical graphs
and provide the necessary definitions which help us describe large 3-connected 2-crossing-
critical graphs, i.e., graphs studied in this paper. Note that vertices of degrees 1 and 2 do
not affect the crossing number, thus the assumption that the minimum degree is at least
3 is reasonable. Note also that V10 is the graph obtained from C10 by adding the five
diagonal edges. We quote the following theorem from [6]. The detailed explanation of the
terminology used in this theorem is given afterwards.
Theorem 2.3 ([6, Theorem 1.1]). Let G be a 2-crossing-critical graph with a minimum
degree of at least 3. Then one of the following holds.
V. Iršič et al.: Domination and independence numbers of large 2-crossing-critical graphs 579
(i) G is 3-connected, contains a subdivision of V10, and has a very particular twisted
Möbius band tile structure, with each tile isomorphic to one of 42 possibilities.
(ii) G is 3-connected, does not have a subdivision of V10, and has at most 3 million
vertices.
(iii) G is not 3-connected and is one of 49 particular examples.
(iv) G is 2- but not 3-connected and is obtained from a 3-connected 2-crossing-critical
graph by replacing digons with digonal paths.
In the present paper, we study graphs from (i), i.e., 3-connected 2-crossing-critical
graphs that contain a subdivision of V10. Since 3-connected 2-crossing-critical graphs that
do not contain a subdivision of V10 have at most 3 million vertices, we may call graphs from
Theorem 2.3(i) large 3-connected 2-crossing-critical graphs or large 3-con 2-cc graphs for
short. This abbreviation is used throughout the paper. Note that it would also be interesting
to study other subclasses of graphs, especially graphs from (iv). However, like in [4], we
restrict our studies to graphs from (i).
To understand the tile structure of large 3-con 2-cc graphs, we need the following defi-
nitions that first appeared in [14, 15].
Definition 2.4. 1. A tile is a triplet T = (G,λ, ρ), where G is a graph and λ, ρ are
sequences of pairwise distinct vertices of G, where no vertex of G appears in both λ
and ρ. The left wall of T is λ, and the right wall of T is ρ.
2. A tile drawing is a drawing D of G in the unit square [0, 1] × [0, 1] for which the
intersection of the boundary of the square with D contains precisely the images of
the left wall λ and the right wall ρ, and these are drawn in {0}×[0, 1] and {1}×[0, 1],
respectively, such that the y-coordinates of the vertices are increasing with respect to
their orders in the sequences λ and ρ.
3. The tiles T = (G,λ, ρ) and T ′ = (G′, λ′, ρ′) are compatible if |ρ| = |λ′|.
4. A sequence (T0, . . . , Tm) of tiles is compatible if Ti−1 is compatible with Ti for
every i ∈ [m].
5. The join of compatible tiles (G,λ, ρ) and (G′, λ′, ρ′) is the tile, denoted as (G,λ, ρ)⊗
(G′, λ′, ρ′), whose graph is obtained from G and G′ by identifying the sequence ρ
term by term with the sequence λ′. The left wall of the obtained tile is λ and the right
wall is ρ′.
6. The join ⊗T of a compatible sequence T = (T0, . . . , Tm) of tiles is defined as
T0 ⊗ · · · ⊗ Tm.
7. A tile T is cyclically-compatible if T is compatible with itself. For a cyclically-
compatible tile T , the cyclization of T is the graph ◦T obtained by identifying the
respective vertices of the left wall with the right wall. Cyclization of a cyclically-
compatible sequence of tiles is ◦T = ◦(⊗T ).
8. Let T = (G,λ, ρ) be a tile. The right-inverted tile of T is T ↕ = (G,λ, ρ). The
left-inverted tile of T is ↕T = (G,λ, ρ). The inverted tile is ↕T ↕ = (G,λ, ρ). The
reversed tile is T↔ = (G, ρ, λ).
580 Ars Math. Contemp. 23 (2023) #P4.05 / 577–590
Note that ⊗T in Definition 2.4, 6. is well-defined since ⊗ is associative.
Definition 2.5. The set S of tiles consists of tiles obtained as a combination of one of the
two frames shown in Figure 1 and one of the 13 pictures shown in Figure 2 in such a way
that a picture is inserted into a frame by identifying the two geometric squares. (This can
mean subdividing the frame’s square.) A given picture can be inserted into a frame either
with the given orientation or with a 180◦ rotation.
Figure 1: Both possible frames.
Figure 2: All possible pictures. For later need, the red vertices mark a dominating set of
each of them.
Note that each picture yields either two or four tiles in S. Altogether the set S contains
42 different tiles. For example, in Figure 3 we see that picture V IA yields four different
tiles.
We can now define the tile structure of graphs that are of our interest. Their definition
first appeared in [6].
Definition 2.6. The set T (S) consists of all graphs of the form ◦((⊗T )↕), where T is
a sequence (T0,↕ T
↕
1 , T2, . . . ,
↕ T
↕
2m−1, T2m), where m ≥ 1 and Ti ∈ S for every i ∈
{0, . . . , 2m}. The obtained vertices of degree 2 are suppressed.
Suppressing a vertex of degree 2 is the inverse operation to subdividing an edge, and
it does not affect the crossing number of a graph. Note that for the case of calculating the
domination and independence numbers of graphs, double edges can be replaced with single
ones without changing the invariant.
V. Iršič et al.: Domination and independence numbers of large 2-crossing-critical graphs 581
Figure 3: All possible tiles that can be obtained from picture V IA.
Theorem 2.7 ([6, Theorems 2.18 and 2.19]). Each graph from T (S) is 3-connected and
2-crossing-critical. Moreover, all but finitely many 3-connected 2-crossing-critical graphs
are contained in T (S).
Theorem 2.7 gives a nice representation of large 3-con 2-cc graphs, i.e., graphs from
Theorem 2.3(i).
Graphs from the set T (S) can be described as sequences over the alphabet Σ =
{L, d,A,B,D,H, I, V } (see [22]). A signature of a tile T is
sig(T ) = Pt IdPb Fr,
where Pt ∈ {A,B,D,H, V } describes the top path of the picture, Id ∈ {I, ∅} indicates
a possible identifier of the picture, Pb ∈ {A,B,D, V, ∅} describes the bottom path of the
picture, and Fr ∈ {L, dL} describes the frame. Here, ∅ labels the empty word. See
Figure 1 for possible signatures of frames (Fr), Figure 2 for all possible signatures of
pictures (Pt IdPb), and Figure 3 for an additional example of how to describe a tile with
its signature.
For a graph G ∈ T (S), G = ◦((⊗T )↕) = (T0,↕ T ↕1 , T2, . . . ,↕ T
↕
2m−1, T2m), a signa-
ture is defined as
sig(G) = sig(T0) sig(T1) · · · sig(T2m).
Additionally, #X denotes the number of occurrences of X in sig(G), where X ∈ Σ. Given
a tile T , the join of a sequence of k tiles, starting with T and then alternating between T
and T ↕, is denoted by k · T .
3 Domination number
In this section, we present an upper bound and a lower bound for the domination number
of large 3-con 2-cc graphs, including equality cases for both bounds.
3.1 Upper bound
Theorem 3.1. If G is a large 3-con 2-cc graph, then
γ(G) ≤ #A+#B +#D +#V + 2 ·#H −#AIV −#V IA.
582 Ars Math. Contemp. 23 (2023) #P4.05 / 577–590
Proof. Each vertex lies on at least one picture. Thus, if D ⊆ V (G) dominates all vertices
in each picture, then D is a dominating set of G. Inside each picture we have at least one
path (by path we mean the top and the bottom path as in the definition of the signature of
a tile). We can see that domination of A, B, D, and V requires at least one vertex, while
domination of H requires two vertices. The only exceptions are pictures AIV and V IA,
where domination of the picture only requires one vertex and not two, which would be
the result of the summation of domination numbers of paths A and V . Figure 2 shows all
possible pictures with marked smallest dominating sets.
Edges between pictures only add edges between vertices and lower the domination
number. This means that the domination number has an upper bound of the sum of domi-
nation numbers for individual paths.
The upper bound from Theorem 3.1 is sharp, which can be seen in the following two
examples. They also show that the number of frames L and dL does not affect the upper
bound.
Example 3.2. Let G1 = n · V BdL, where n ≥ 3 is an odd number. Figure 4 shows a
dominating set of size 2n, meaning γ(G1) ≤ 2n. The formula from Theorem 3.1 shows
the same, as #A+#B +#D +#V + 2 ·#H −#AIV −#V IA = 2n.
Figure 4: Graph G1 with a marked dominating set of size 2n. Note that to obtain the desired
graph, vertices a are identified, vertices b are identified, and after this vertices of degree 2
are suppressed. The same simplification of drawings is used for the rest of the paper.
Assume γ(G1) < 2n. The Pigeonhole principle says that there exists at least one
picture, which is dominated by at most one vertex. Vertices in the corners of the picture can
be dominated by vertices from neighboring pictures. The remaining three inner vertices,
which we get from B and V and are painted orange in Figure 5, are yet to be dominated.
Since these three vertices cannot be dominated by one vertex, we need at least two vertices
to dominate this picture, which leads to a contradiction. Therefore γ(G1) ≥ 2n.
Figure 5: Picture V B, where we require that the inner vertices, marked orange, are domi-
nated by one vertex.
From this, it follows that γ(G1) = 2n.
Example 3.3. Let G2 = n · AIV L, where n ≥ 3 is an odd number. We can find a
dominating set of size n (see Figure 6), thus γ(G2) ≤ n. This also follows from the
formula in Theorem 3.1, as #A+#B +#D +#V + 2 ·#H −#AIV −#V IA = n.
V. Iršič et al.: Domination and independence numbers of large 2-crossing-critical graphs 583
Figure 6: Graph G2 with a marked dominating set of size n.
We next show that γ(G2) ≥ n. Divide the graph G2 into n disjoint subgraphs, as
shown in Figure 7. Each subgraph is induced on the closed neighborhood of the degree 3
vertex and is isomorphic to the paw graph. The position of degree 3 vertices in G2 ensures
that the obtained n subgraphs are all pairwise disjoint. We notice that the middle vertex of
each subgraph (the vertex of degree 3) can only be dominated by one of the vertices in the
same subgraph. Hence we must choose at least one vertex from each one of the n disjoint
subgraphs, which means that γ(G2) ≥ n.
Figure 7: Graph G2 with disjoint subgraphs marked orange and the middle vertex of each
subgraph marked blue. Recall that when vertex a is identified, the obtained vertex of degree
2 is suppressed.
It follows that γ(G2) = n.
3.2 Lower bound
Theorem 3.4. If G is a large 3-con 2-cc graph, then
γ(G) ≥
⌈
2
3
·#L
⌉
.
Before proving the result, we list two useful observations. Let G be a large 3-con 2-cc
graph.
1. Every vertex of G lies on at least one and at most two tiles.
2. All vertices of a picture P can be dominated by a single vertex v only if the picture
is V IA. Moreover, in this case, the vertex v only dominates vertices within picture
P .
Proof of Theorem 3.4. Let G be a 3-con 2-cc graph. We can assume that G only has
L frames, since replacing dL frames with L frames means that we contract some edges,
which can only decrease the domination number. Replacing dL frames with L frames
doesn’t change the number #L.
From Observation 1 we know that every vertex of G lies either on only one tile or on
two consecutive tiles. If it lies on only one tile, we say that it belongs to that tile. If it lies
on two tiles, we say that it belongs to the tile on the right. Hence every vertex belongs to
exactly one tile.
584 Ars Math. Contemp. 23 (2023) #P4.05 / 577–590
Let D be a smallest dominating set of G. First we show that for every trinity of consec-
utive tiles, there exist at least two vertices from the set D that belong to one of the tiles in
the trinity. Figure 8 shows frames of a trinity of tiles (without the pictures). Vertices that
are marked red belong to one of the tiles in the trinity. Vertices that are marked with a green
circle can only be dominated from one of the vertices that belong to the trinity of tiles.
From Observation 2 it follows that we need at least two elements from the set D to
dominate all green vertices. Hence there exist at least two vertices from the set D that
belong to the trinity of tiles.
Figure 8: A trinity of consecutive tiles. Vertices that are marked with a green circle can
only be dominated from one of the vertices that belong to this trinity of tiles.
Now we can prove that |D| ≥ 23#L. There are #L trinities of consecutive tiles in the
graph G, we denote them 1, 2, . . . ,#L. Let
D′ = {(d, k) | d ∈ D, d belongs to one of the tiles in the trinity k}.
Since each vertex belongs to exactly three trinities of consecutive tiles, it follows that
|D′| = 3 · |D|. For every trinity of tiles k, there exist at least two vertices from D that
belong to tiles in the trinity k. Hence there exist at least two elements in D′ with sec-
ond component k for every k = 1, 2, . . . ,#L. Therefore |D′| ≥ 2 · #L, hence it holds
that 3 · |D| ≥ 2 · #L. Because the domination number of G is an integer, it follows that
γ(G) ≥
⌈
2
3 ·#L
⌉
.
The lower bound from Theorem 3.4 is sharp, which can be seen in the following exam-
ple.
Example 3.5. Let G3 = n ·DDLDDLAIV L, where n ≥ 1 is an odd number. Figure 9
shows the dominating set of size 23 · 3 · n, meaning γ(G3) ≤ 2n. Our formula from
Theorem 3.4 shows the same, as
⌈
2
3 ·#L
⌉
=
⌈
2
3 · 3 · n
⌉
= 2n.
Figure 9: Graph G3 with a marked dominating set of size 2n.
Every trinity of consecutive pictures DDLDDLAIV L requires at least two vertices
from the dominating set to dominate all the inner vertices of the trinity, which are marked
orange in Figure 10. This means that at least two vertices are needed to dominate this trinity
of pictures. Therefore γ(G3) ≥ 2n.
From this follows that γ(G3) = 2n.
V. Iršič et al.: Domination and independence numbers of large 2-crossing-critical graphs 585
Figure 10: Trinity of consecutive pictures DDLDDLAIV L, where we want the inner
vertices, marked orange, to be dominated by one vertex. Note that none of these vertices
can be dominated by a vertex outside of this trinity of tiles.
4 Independence number
In this section, we present sharp upper and lower bounds for the independence number of
large 3-con 2-cc graphs.
4.1 Upper bound
Theorem 4.1. If G is a large 3-con 2-cc graph, then
α(G) ≤
⌊
|V(G)|
2
⌋
.
Proof. Since all large 3-con 2-cc graphs are Hamiltonian [22], and the independence num-
ber of Hamiltonian graphs is at most 12 |V(G)|, we obtain the desired upper bound.
The following example shows that the upper bound from Theorem 4.1 is sharp.
Example 4.2. Let G4 = n ·HdL, where n ≥ 3 is an odd number. Then |V(G4)| = 6n.
Figure 11 shows that we can choose 3n independent vertices from the graph G4, meaning
α(G4) ≥ 3n.
Figure 11: Graph G4 with a marked independent set of size 3n.
Every vertex of graph G4 lies in exactly one picture. Since we can choose at most three
independent vertices in each of the n pictures, α(G4) ≤ 3n, which is also the result of
Theorem 4.1. Therefore α(G4) = 3n =
⌊
|V(G4)|
2
⌋
.
4.2 Lower bound
Theorem 4.3. If G is a large 3-con 2-cc graph, then
α(G) ≥ min{#L+#d, 2 ·#L− 1}.
Proof. For every large 3-con 2-cc graph G we can construct the graph G′ from the same
frames used for G, without using the pictures. We notice that if we add pictures into the
frames in G′ to get the original graph G, we only add vertices and do not connect any
586 Ars Math. Contemp. 23 (2023) #P4.05 / 577–590
vertices that were previously not connected, thus any picture we add can only increase the
independence number, therefore α(G) ≥ α(G′). Note that the graph G′ will have the
same number of frames L and dL as the initial graph G. Therefore it suffices to prove the
proposed lower bound for the graph G′.
We distinguish two cases, the first case is if there are only dL frames and the second if
there is at least one L frame.
Case 1 If there are only dL frames, we can find 2 · #L − 1 independent vertices, as is
shown in Figure 12. Note that in this case min{#L+#d, 2 ·#L−1} = 2 ·#L−1.
Figure 12: Graph G′ from Case 1 with a marked independent set of size 2 ·#L− 1.
Case 2 If there is at least one L frame, then we can choose the independent set based on
the following method. Note that double edges can be ignored when studying the
independence number. The graph G′ is then composed of 3- and 4-cycles, which
are connected with additional edges (marked orange in Figure 13). These additional
edges come from where the top and bottom paths of the pictures were in G. To
obtain an independent set of appropriate size, we select one vertex from each 3-
cycle and two vertices from each 4-cycle. For every 3-cycle, we select the vertex of
degree 3 on its left side. If we have two consecutive 3-cycles, the vertices we chose
from them are independent. When selecting vertices in the 4-cycles, we consider all
consecutive 4-cycles between two 3-cycles and select vertices for the independent
set in these 4-cycles from right to left. The 3-cycle on the right of the consecutive
4-cycles determines how we choose the independent set in the right-most 4-cycle,
which in turn uniquely determines how we select two independent vertices in each
of these 4-cycles (in the same manner as in Figure 12). Notice that the 3-cycle on the
left of these 4-cycles gives no restriction on the selected vertices.
Figure 13: An example of the graph G′ from Case 2 with a marked independent set of size
#L+#d. The edges that connect the 3- and 4-cycles are marked orange.
We have thus chosen two vertices in each 4-cycle and one vertex in each 3-cycle.
Since the number of 3-cycles is #L − #d and the number of 4-cycles is #d, we
have found an independent set of size (#L−#d) + 2 ·#d = #L+#d. Note that
in this case min{#L+#d, 2 ·#L− 1} = #L+#d.
V. Iršič et al.: Domination and independence numbers of large 2-crossing-critical graphs 587
The following two examples show that the lower bound from Theorem 4.3 is sharp. The
first example naturally follows from the proof of Theorem 4.3, while the second example
provides a non-trivial family of sharpness examples. Additionally, examples are selected
in such a way that different parts of the minimum are attained.
Example 4.4. Let G5 be a large 3-con 2-cc graph built from tiles DDdL and DDL, so
that not all of the tiles are DDdL.
From Theorem 4.3 we know that α(G5) ≥ #L+#d. Similarly as in the proof, we can
find #d 4-cycles and #L−#d 3-cycles in G5, so that every vertex lies on exactly one of
them. Every 4-cycle is formed by the two vertices on the right of a DDdL tile and the two
vertices on the left of the next tile to the right. Every 3-cycle is formed by the two vertices
on the right of a DDL tile and the two vertices on the left of the next tile. Two of those
vertices are identified, thus giving us a 3-cycle. The 3-cycles and 4-cycles are marked in
Figure 14.
Figure 14: Graph G5 with marked 3-cycles and 4-cycles.
We can choose at most one independent vertex from every 3-cycle and at most two
independent vertices from every 4-cycle, therefore α(G5) ≤ 2 · #d + 1 · (#L − #d) =
#L+#d.
From this it follows that α(G5) = #L+#d.
Example 4.5. Let G6 be a large 3-con 2-cc graph that is built from DDdL, V IAdL, and
AIV dL tiles, but not all tiles are V IAdL, and not all tiles are AIV dL.
From Theorem 4.3 we know that α(G6) ≥ 2 · #L − 1. We can find at most two
independent vertices in each of the tiles DDdL, V IAdL, and AIV dL, therefore we can
find at most 2 ·#L independent vertices in G6.
For contradiction suppose that α(G6) ̸= 2 · #L − 1, meaning α(G6) = 2 · #L. We
try to construct an independent set A with 2 ·#L vertices. Set A must include exactly two
vertices from every tile because otherwise set A would have to include at least 3 vertices
from one tile, which is impossible.
There are two different ways in which we can choose two independent vertices from a
DDdL tile, and three different ways for tiles V IAdL and AIV dL. All options are shown
in Figure 15.
Even though tiles V IAdL and AIV dL have a third option for the choice of two inde-
pendent vertices (where the selected vertices are not diagonal), we can’t choose the vertices
in set A in this way, since we know that we have to choose two independent vertices from
every tile. If we choose the top and bottom right vertex in a V IAdL tile, then the only way
to choose two vertices in the next tile is if that tile is also a V IAdL tile and we choose
the top and bottom right vertices. We continue this for all tiles, but since not all tiles are
V IAdL, at some point we are not able to choose two independent vertices in the next tile.
For the same reason, we also cannot choose the two vertices on the left of an AIV dL tile.
588 Ars Math. Contemp. 23 (2023) #P4.05 / 577–590
Figure 15: Tiles DDdL, V IAdL, and AIV dL with two independent vertices marked.
This means that for all tiles, the two vertices that are included in set A are the diagonal
ones, without loss of generality we can assume that those diagonal vertices in the first tile
are the bottom left and the top-right vertex. This choice determines which vertices we must
choose in the tile to the right and so on, as is shown in Figure 16.
Figure 16: Graph G′6 was constructed from the same frames used for G6, without using the
pictures. The first tile of graph G′6 determines which two vertices are included in set A for
all other tiles. When we get to the last tile, we get a contradiction (marked orange).
When we get to the last tile we get a contradiction. Because of the tile to the left, the
only possible vertices from the last tile that can be included in A are the bottom left and
the top-right vertex. But the top-right vertex is connected to a vertex in the first tile that is
already included in set A, therefore set A cannot include two vertices from the last tile.
This means that the independent set A that has 2 · #L elements cannot exist and
α(G6) = 2 ·#L− 1.
ORCID iDs
Vesna Iršič https://orcid.org/0000-0001-5302-5250
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P4.06 / 591–603
https://doi.org/10.26493/1855-3974.3061.5bf
(Also available at http://amc-journal.eu)
Enumerating symmetric pyramids in
Motzkin paths*
Rigoberto Flórez †
Department of Mathematical Sciences, The Citadel, Charleston, SC, U.S.A.
José L. Ramı́rez ‡
Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
Received 4 February 2023, accepted 29 March 2023, published online 7 April 2023
Abstract
A path in the first quadrant of the xy-plane that starts at the origin having North-
East steps (X), Horizontal steps (Z), South-East steps (Y ), and that ends on the x-axis
is called Motzkin. A maximal pyramid is a subpath of the form XhZmY h that cannot
be extended to Xh+1ZmY h+1. It is symmetric if it cannot be extended to any of these
subpaths: Xh+1ZmY h or XhZmY h+1. We use generating functions to enumerate sym-
metric pyramids and give the asymptotic behavior of the number of symmetric pyramids.
Additionally, we give combinatorial arguments to count some of the mentioned aspects.
Keywords: Motzkin path, generating function, symmetric pyramid.
Math. Subj. Class. (2020): 05A15, 05A19
1 Introduction
A path in the first quadrant of the xy-plane that starts at the origin having North-East
steps (X), Horizontal steps (Z), South-East steps (Y ), and that ends on the x-axis is called
Motzkin. Whoever is familiar with Dyck paths may verify that Motzkin paths are a gener-
alization of them.
The set of all Motzkin paths of length n is denoted by Mn and the set ∪∞n=0Mn is
denoted by M. A maximal pyramid of weight h in a Motzkin path is a subpath of the
*The authors are grateful to an anonymous referee for helpful comments.
†The first author was partially supported by the Citadel Foundation, Charleston, SC.
‡The second author was partially supported by Universidad Nacional de Colombia, Project No. 53490. He
started working on this project when he was in a short research visit at The Citadel.
E-mail addresses: rigo.florez@citadel.edu (Rigoberto Flórez), jlramirezr@unal.edu.co (José L. Ramı́rez)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
592 Ars Math. Contemp. 23 (2023) #P4.06 / 591–603
form XhZmY h that cannot be extended to Xh+1ZmY h+1, with h ≥ 1 and m ≥ 0. It is
symmetric if it can not be extended to any of these subpaths: Xh+1ZmY h or XhZmY h+1.
There are two types of distinguishable maximal pyramids: if m = 0, the pyramid is called
a triangular pyramid, denoted by h or by if there is no ambiguity, see the left-hand
side of Figure 1; and if m > 0, the pyramid is called truncated, denoted by h or by
if there is no ambiguity, see the right-hand side of Figure 1. The symmetric weight of
a path is the sum of the weights of its symmetric pyramids. The height of a pyramid is
the y-coordinate of its highest point. That is, the height is measured from the x-axis to the
highest point. The symmetric height of a path is the sum of the heights of its symmetric
pyramids. A hump of a Motzkin path is a subpath of the form XZkY for any non-negative
integer k. If k = 0, it is called sharp hump (or sharp peak). Notice that every pyramid has
a hump.
Motivated in part by the work done by Asakly [1], Flórez and Ramı́rez [8] introduced
the concept of symmetric and asymmetric peaks in Dyck paths, see also Elizalde et al. [5],
Flórez et al. [7], and Sun et al. [14]. Following up these works we generalize the symmetric
concept to Motzkin paths. Using generating functions we enumerate symmetric pyramids
and give the asymptotic behavior of the number of symmetric pyramids. Additionally, we
give combinatorial arguments to count some of the mentioned aspects. Our results here, in
particular, apply to recover the known results for Dyck paths.
Figure 1: Triangular symmetric pyramid and truncated symmetric pyramid.
2 Motzkin paths, humps, and valleys
If ck = 1k+1
(
2k
k
)
is the kth Catalan number, then the number of Motzkin paths of length n,
|Mn|, is given by the nth Motzkin number defined as
mn :=
⌊n/2⌋∑
k=0
1
k + 1
(
n
2k
)(
2k
k
)
=
⌊n/2⌋∑
k=0
ck
(
n
2k
)
. (2.1)
See, for example, the sequence A001006 of [12].
In this section we give a bivariate generating function enumerating Motzkin paths with
respect to the length and the number of humps. The results here are, in fact, lemmas for
the main results studied in the upcoming sections. For example, in Section 3 we compare
the number of symmetric pyramids with respect to the number of humps. In particular, we
prove that these two quantities are in a proportion of 1/2 (see Corollary 3.8).
We denote by humps(M) the number of humps in a Motzkin path M . We now in-
troduce a bivariate generating function to count the number of humps with respect to the
length of a Motzin path:
H(q, x) :=
∑
M∈M
qhumps(M)x|M |.
R. Flórez et al.: Enumerating symmetric pyramids in Motzkin paths 593
Proposition 2.1. The generating function for Motzkin paths with respect to the number of
humps is given by
H(q, x) =
1− 2x+ 2x2 − qx2 −
√
(1− 4x+ (4− q)x2)(1− qx2)
2x2(1− x)
.
Proof. A Motzkin path M can be decomposed as —using first return decomposition—
λ (the empty path), ZM , or XM1YM2, where M1 and M2 are Motzkin paths, possibly
empty. Notice that for the last decomposition, humps(M) = humps(M1)+humps(M2)
—unless M1 is the path Zk, for any k ≥ 0, in which case humps(M) = 1+humps(M2).
Therefore, the generating function H(q, x) satisfies this functional equation
H(q, x) = 1 + xH(q, x) + x2
(
H(q, x)− 1
1− x
+
q
1− x
)
H(q, x).
Solving this functional equation for H(q, x) we obtain the desired result.
Let us denote by hmn the total number of humps in Mn. Then we have
∂
∂q
H(q, x) |q=1=
∑
n≥0
hmnx
n =
1− 2x+ x2 − (1− x)
√
1− 2x− 3x2
2(1− x)2
√
1− 2x− 3x2
.
The sequence hmn is given by the combinatorial sum (cf. [13, 3])
hmn =
1
2
∑
j≥0
(
n
j
)(
n− j
j
)
− 1
2
.
Notice that if
(
n,3
n
)
denotes the central trinomial coefficient, that is, the nth coefficient of
the expansion of (x2 + x+ 1)n, then hmn = 12
(
n,3
n
)
− 1. The central trinomial coefficient
has the asymptotic approximation (cf. [11])(
n, 3
n
)
∼ 3
n
2
√
3
πn
.
Therefore, the sequence hmn has this asymptotic approximation
hmn ∼
3n
4
√
3
πn
. (2.2)
A weak valley of a Motzkin path is any subpath of the form ZZ, Y X,ZX , or Y Z,
see Figure 2. Let wv(M) be the number of weak valleys in a Motzkin path M . Let us
denote by e(M) the number of horizontal steps in M . We now introduce a multivariate
generating function to count the number of weak valleys and horizontal steps, with respect
to the length of a Motzin path:
Mwv(q, t, x) :=
∑
M∈M
qwv(M)te(M)x|M |.
Figure 2: Weak valleys in a Motzkin path.
594 Ars Math. Contemp. 23 (2023) #P4.06 / 591–603
We need this proposition for our main theorem of Section 3. See [2] for more results
about valleys of Motzkin paths.
Proposition 2.2. The generating function for Motzkin paths with respect to the number of
weak valleys and horizontal steps is given by:
Mwv(q, t, x) =
1− tqx− x2 + qx2 −
√
(1− tqx− x2 + qx2)2 − 4qx2(1 + tx− tqx)
2qx2
.
Proof. From the first return decomposition, a Motzkin path M can be decomposed as λ,
ZM1 or XM2YM3, where Mi ∈ M, for i = 1, 2, 3. Notice that
humps(M) = humps(M1)+1 and humps(M) = humps(M2)+humps(M3)+1,
unless M1 and M3 are the empty paths, in this case, we have that humps(M) = 0 and
humps(M) = humps(M1), respectively. Therefore, the generating function Mwv(q, t, x)
satisfies the functional equation
Mwv(q, t, x) = 1 + tx(qMwv(q, t, x)− q + 1) + x2Mwv(q, t, x)(qMwv(q, t, x)− q + 1).
Solving this functional equation for Mwv(q, t, x) we obtain the desired result.
The series expansion of the generating function Mwv(q, 1, x) is
1 + x+ (1 + q)x2 + (1 + 2q + q2)x3 + (1 + 4q + 3q2 + q3)x4
+ (1 + 6q + 9q2 + 4q3 + q4)x5 + (1 + 9q + 19q2 + 16q3 + 5q4 + q5)x6 +O(x7).
The array coefficients of the above series correspond to the sequence A110470.
3 Symmetric pyramids of Motzkin paths
In this section we give a multivariate generating function to count the number of symmetric
pyramids, the number of triangular, and truncated symmetric pyramids with respect to the
length of a Motzkin path. We give an asymptotic behavior of symmetric pyramids. In
the end of the section we analyze the proportion existing between the total number of
symmetric pyramids and the total number of humps in Mn, when n → ∞.
Let us denote by sp(M), stp(M), and strp(M) the number of symmetric pyramids,
the number of symmetric triangular pyramids, and the number symmetric truncated pyra-
mids, respectively, of a Motzkin path M . It is clear that sp(M) = stp(M) + strp(M).
We now introduce a multivariate generating function with respect to the defined parameters:
Msp(q, p, t, x) :=
∑
M∈M
qstp(M)pstrp(M)te(M)x|M |.
Theorem 3.1 gives an expression for this multivariate generating function. The result of this
theorem is based on marking or distinguishing elements within the generating function. For
more details about the method used in the theorem see, for example, these books: Goulden
and Jackson [9, page 128] or Flajolet and Sedgewick [6, page 209].
R. Flórez et al.: Enumerating symmetric pyramids in Motzkin paths 595
Theorem 3.1. The generating function for Motzkin paths with respect to the number of
symmetric triangular pyramids, the number symmetric truncated pyramids, and the number
of horizontal steps is given by:
Msp(q, p, t, x) =
(1− x2)(1− tx)
(
p1 −
√
p2
)
2x2(1− tx− qx2 + (1− p+ q)tx3)2
,
where p1 and p2 are these multivariate polynomials
p1 = 1− 2tx− (q − t2)x2 + (2− p+ q)tx3 + (1− q − t2)x4 − (p− q)tx5
and
p2 = (1−x2)(1−(1+2t)x−(1+q−t2)x2−(1−q−2t+pt−qt−t2)x3+(p−q)tx4)
× (1 + (1− 2t)x− (1 + q − t2)x2 + (1− q + 2t− pt+ qt− t2)x3 − (p− q)tx4).
Proof. Consider the set G of all Motzkin paths in which an arbitrary number of symmetric
pyramids (some, possibly none, possibly all) have been marked. Any path in G can be
obtained from a Motzkin path M by inserting pyramids (triangular or truncated) with a
marked hump. These pyramids can be inserted in the node of a weak valley, in the initial
or final node of M . These kinds of nodes are called insertion nodes (see [4]). Let ins(M)
be the number of insertion nodes of M . It is clear that ins(M) = wv(M) + 2 —unless
M is the empty path, in which case ins(M) = 1. Consider the generating function
Mins(q, t, x) =
∑
M∈M
qins(M)te(M)x|x|.
Then, it is clear that
Mins(q, t, x) = q
2(Mwv(q, t, x)− 1) + q.
Let M be a marked Motzkin path in G. We denote by mark1(M) and mark2(M) the num-
ber of marked triangular pyramids and truncated pyramids of M , respectively. Consider the
generating function
G(q, p, t, x) =
∑
M∈G
qmark1(M)pmark2(M)te(M)x|M |.
Since the elements of G are obtained from elements of M by inserting marked pyramids in
the insertions nodes, it translates in terms of generating functions as
G(q, p, t, x) = Mins
( 1
1− qx2/(1− x2)− ptx3/((1− x2)(1− tx))
, t, x
)
.
Finally, the Motzkin paths with marked symmetric pyramids are generated by these substi-
tutions: q 7→ q + 1 and p 7→ p+ 1 within Msp(q, p, t, x). That is, we obtain the equality
G(q, p, t, x) = Msp(q + 1, p+ 1, t, x)
or equivalently
Msp(q, p, t, x) = G(q − 1, p− 1, t, x).
Using the above equality and Proposition 2.2 we obtain the desired result.
596 Ars Math. Contemp. 23 (2023) #P4.06 / 591–603
The series expansion of the generating function Msp(q, p, t, x) is
1 + tx+ (q + t2)x2 + (pt+ 2qt+ t3)x3 + (q + q2 + 3pt2 + 3qt2 + t4)x4
+ (2t+ pt+ 2qt+ 2pqt+ 3q2t+ 6pt3 + 4qt3 + t5)x5 +O(x6).
Figure 3 shows some Motzkin paths. Their corresponding weights are shown in the previ-
ous expansion in boldface.
qx4 pt2x4 pt2x4 q2x4 qt2x4
pt2x4 qt2x4 qt2x4 t4x4
Figure 3: Motzkin paths and the symmetric pyramid statistic.
We use sn to denote the total number of symmetric pyramids in Mn.
Corollary 3.2. The generating function for Motzkin paths with respect to the number of
symmetric pyramids is given by:
(1− x)2(1 + x)
(
1− 2x+ (1− q)x2 + 2x3 − qx4 − (1− x2)
√
p(x, q)
)
2x2(1− x− qx2 + x3)2
,
where p(x, q) = 1 − 4x + 2(2 − q)x2 + 4qx3 − (4 − q2)x4. Moreover, the sequence sn
has the generating function
∂
∂q
Msp(q, q, 1, x) |q=1=
1− 4x3 + 4x+ x4 + (1− x2 − 2x)(1− x)
√
1− 2x− 3x2
2(1− x)3(1 + x)
√
1− 2x− 3x2
.
These are the first 11 values of the sequence sn for n = 1, 2, . . . , 11:
0, 1, 3, 9, 23, 60, 156, 415, 1121, 3076, 8540.
We use s∗k, g
∗
k, and t
∗
k to denote the number of all first symmetric pyramids, all first
symmetric triangular pyramids, and all first truncated symmetric pyramids at a ground
level, respectively, in Mk. Note that given a path P ∈ Mk, there is another path Q ∈
Mk that is symmetric to P —the first seen from left-to-right and the second seen from
from right-to-left. Therefore, using this symmetry of paths in Mk we have that s∗k, g∗k,
and t∗k also count the number of all last symmetric pyramids, all last symmetric triangular
pyramids, and all last truncated symmetric pyramids at ground level, respectively.
Lemma 3.3. If ck is the kth Catalan number, then s∗k = g∗k + t∗k, where
g∗k = 2
k−1∑
i=1
(−1)i+1
(
i⌊
i−1
2
⌋)(k − 1
i+ 1
)
and
t∗k = 2
k−1∑
i=1
( ⌊ k−i2 ⌋∑
j=0
cj
⌊
i− 1
2
⌋(
k − i
2j
))
.
R. Flórez et al.: Enumerating symmetric pyramids in Motzkin paths 597
Proof. Let us find g∗k. A Motzkin path µk can be decomposed as iµk−i for i = 0, 1, . . . ,
k − 1, where 0 = ∅. This implies that the number of that type of pyramid is given by
g∗k =

⌊ k−12 ⌋∑
i=1
m2i, if k is even;
⌊ k−12 ⌋∑
i=1
m2i−1, if k is odd.
This formula coincides with the sequence A082397 in OEIS multiplied by 2. Therefore,
the expression given in the statement is from A082397.
We now find t∗k. A Motzkin path of the form µk can be decomposed as iµk−i, for
i = 0, 1, . . . , k− 1, where 0 = ∅. This implies that the number of that type of pyramids,
in the set of all paths of the form iµk−i, is given by t∗k =
∑k−1
i=1
⌊
i−1
2
⌋
mk−i. This and
(2.1) give the result in the statement of this lemma.
Clearly, the number of first symmetric pyramids is the sum of these two quantities.
From the first return decomposition of a Motzkin path we have this particular expres-
sion
Zµn−1 and XµkY µn−2−k, with µi ∈ Mi and µ0 = λ. (3.1)
Here we say that XµkY is the first primitive subpath. We need this decomposition again
for the proof of Theorem 4.4.
Theorem 3.4. Let sn be the total number of symmetric pyramids in Mn. Then sn satisfies
s1 = 0, s2 = 1, s3 = 3, and for n > 3
sn = sn−1 + sn−2 + sn−3 +
n−2∑
k=0
mn−2 +
n−2∑
k=2
(
(sk − 2s∗k)mn−k−2 + sn−k−2mk
)
.
Proof. From the decomposition given in (3.1) the number of symmetric pyramids in a path
of the form Zµn−1 is given by the number of symmetric pyramids in µn−1, that is counted
by sn−1.
We now analyze XµkY µn−1−k, for k = 0, 1, . . . , n − 1. If k = 0 or 1, then 1 or
1 becomes the first symmetric pyramid in the path. These give that the total number of
symmetric pyramids in the set of all paths of the form XY µn−2 and XZY µn−3 is given
by mn−2 + sn−2 +mn−3 + sn−3.
Let 0 < k < n be a fixed number. Then the number of symmetric pyramids in the set
of paths of the form XµkY µn−1−k is given by (sk − 2s∗k)mn−2−k + sn−2−kmk. Thus,
sn−2−kmk counts all symmetric pyramids after the first primitive subpath and
(sk − s∗k)mn−2−k counts all symmetric pyramids in the first primitive subpath. Note that
skmn−2−k counts all symmetric pyramids in µk. However, the first and the last pyra-
mids at the ground level in µk are counted by sk as symmetric, but in XµkY they are not
symmetric. So, we must subtract all of them; and that is what s∗k does (see Lemma 3.3).
Therefore, for k = 1, 2, . . . , n − 1 we have that the number of symmetric pyramids in
paths of the form XµkY µn−1−k is given by
∑n−1
k=1((sk − 2s∗k)mn−k−2 + sn−k−2mk).
Now adding the results from each decomposition and varying k from 0 to n− 1, we obtain
the desired result.
598 Ars Math. Contemp. 23 (2023) #P4.06 / 591–603
Remark 3.5. Notice that the generating function of the number of symmetric pyramids in
Mn is algebraic, then the counting sequence sn satisfies a recurrence relation with poly-
nomial coefficients. This can be automatically solved with Kauers’s algorithm [10]. In
particular we obtain that sn satisfies the recurrence relation
p0(n)sn + p1(n)sn+1 + p2(n)sn+2 + p3(n)sn+3 + p4(n)sn+4 + p5(n)sn+5 = 0,
for n ≥ 5, where
p0(n) = n
3 − 3n2 − 3n; p1(n) = −3n3 + 9n2 + 15n− 26;
p2(n) = −2n3 + 12n2 − 6n− 34; p3(n) = 6n3 − 24n2 − 18n+ 86;
p4(n) = n
3 − 9n2 + 9n+ 34; p5(n) = −3n3 + 15n2 + 3n− 60.
Theorem 3.6. An asymptotic approximation for sn is given by:
sn ∼
3n
8
√
3
πn
.
Proof. The generating function of the sequence sn (see Corollary 3.2) can be written as
A(x) +B(x), where
A(x) =
∑
n≥0
anx
n =
1− 2x− x2
2(1− x)2(1 + x)
and
B(x) =
∑
n≥0
bnx
n =
−1 + 3x+ 3x2 − x3
2(1− x)2(1 + x)
√
1− 2x− 3x2
.
The sequence an satisfies a0 = 1 and an = −2⌊(n − 1)/2⌋ + 1, for all n ≥ 1. Then,
an ∼ −n/2. On the other hand, the main singularity of B(x) is 1/3. From the singularity
analysis described in [6] we obtain that bn ∼ 3
n
8
√
3
πn . Therefore, sn ∼ bn.
Comparing the nth coefficients of the generating functions A(x) and B(x) defined in
Theorem 3.6 we obtain this corollary.
Corollary 3.7. For all n ≥ 0, we have
sn =
1
2
n∑
i=0
(
n, 3
i
)
bn−i −
⌊
n− 1
2
⌋
− 1
2
,
where b0 = −1 and bn = (n((n+ 1) mod 2) + 2(n mod 2))(−1)n(n−1)/2, for all n ≥ 1.
The trinomial coefficient
(
n,3
i
)
can be calculated using the established combinatorial
sum
(
n,3
i
)
=
∑n
k=0
(
n
k
)(
k
i−k
)
. From Theorem 3.6 and (2.2) we can also conclude the
following asymptotic expression.
Corollary 3.8. The asymptotic proportion between the number of symmetric pyramids and
the number of humps is
lim
n→∞
sn
hmn
=
1
2
.
R. Flórez et al.: Enumerating symmetric pyramids in Motzkin paths 599
From Theorem 3.1, setting t = 0, we obtain the distribution of symmetric pyramids
(symmetric peaks) over the Dyck lattice paths. A Dyck path is a Motzkin path without
horizontal steps. The number of Dyck paths of semi-length n is given by the Catalan
number cn = 1n+1
(
2n
n
)
.
The generating function of the Dyck paths with respect to the number of symmetric
peaks and semi-length n is given by Msp(q, 0, 0, x1/2). The explicit expression is
(1− x)
(
1 + x2 − qx(1 + x)−
√
t(x, q)
)
2x(1− qx)2
,
where t(x, q) = (1− x)(1− (3+ 2q)x− (1− 4q− q2)x2 − (1− q)2x3), see [4, Theorem
2.1]. Moreover, the generating function of the total number of symmetric peaks over all
Dyck paths of semi-length n is (see [8, Theorem 2.3])
∂
∂q
Msp(q, 0, 0, x) |q=1=
−1 + 5x+ (1− x)
√
1− 4x
2(1− x)
√
1− 4x
.
4 Symmetric triangular pyramids
In this section we give asymptotic approximations for the proportions between the different
aspects that we have been studying in this paper. Thus, we give the proportion between the
number of symmetric triangular pyramids and these quantities: the number of pyramids,
the number of humps, and the number of sharp peaks. We also discuss the combinatorial
behavior of triangular pyramids —symmetric pyramids, symmetric weight, and symmetric
height.
We use gn to denote the number of symmetric triangular pyramids in Mn.
Theorem 4.1. The generating function for Motzkin paths with respect to the number of
symmetric triangular pyramids is
G(x) :=
∂
∂q
Msp(q, 1, 1, x) |q=1 .
More precisely, G(x) is given by
1− 6x+ 9x2 + 4x3 − 13x4 + 2x5 + 3x6 − (1− 4x+ x2)(1− x)2(1 + x)
√
p(x)
2(1− x)3(1 + x)2(1− 3x)
,
where p(x) = 1− 2x− 3x2.
These are the first 11 values of the sequence gn for n = 1, 2, . . . , 11:
0, 1, 2, 6, 14, 37, 96, 259, 706, 1955, 5464.
Theorem 4.2. The sequence gn has this asymptotic approximation
gn ∼
3n
4
√
3
πn
.
Let spn be the number of sharp humps in Mn. Brennan and Mavhung [2] proved that
spn ∼ 3
n
6
√
3
πn .
600 Ars Math. Contemp. 23 (2023) #P4.06 / 591–603
Corollary 4.3. If sn, gn, hmn, and spn represent the number of symmetric pyramids,
triangular pyramids, humps, and sharp humps of Mn, respectively, then these hold:
lim
n→∞
gn
sn
=
2
3
; lim
n→∞
gn
hmn
=
1
3
; and lim
n→∞
gn
spn
=
1
2
.
We recall that mi is the number of Motzkin paths of length i. This theorem gives a
recursive relation for the sequence gn.
Theorem 4.4. If g∗k is as in Lemma 3.4, then the sequence gn satisfies g1 = 0, g2 = 1, and
for n ≥ 3
gn = gn−1 + gn−2 +mn−2 +
n−1∑
k=1
(
(gk − 2g∗k)mn−k−2 + gn−k−2mk
)
.
Proof. This proof is similar to the proof of Theorem 3.4. Replace sk by gk and instead of
s∗k use Lemma 3.4 with g
∗
k.
4.1 Weight and height of the symmetric triangular pyramids
We use wn to denote the total weight of symmetric triangular pyramids in Mn. In Proposi-
tion 4.5 we give a recursive relation to find the values of wn. Its proof is based on a similar
argument as in Theorem 4.4. So, we omit it.
Proposition 4.5. The sequence wn satisfies that w1 = 0, w2 = 1, and for n ≥ 3
wn = wn−1 + wn−2 +mn−2
+
n−2∑
k=1
((
wk + (k + 1 mod 2)− 2
⌊ k−12 ⌋∑
i=1
imk−2i
)
mn−k−2 +mkwn−k−2
)
.
These are the first 11 values of the sequence wn, for n = 1, 2, . . . , 11:
0, 1, 2, 7, 16, 44, 112, 303, 818, 2258, 6282.
We use hn to denote the symmetric height of triangular pyramids in Mn.
Proposition 4.6. Let gk be number of symmetric triangular pyramids in Mk. The sequence
hn satisfies h1 = 0, h2 = 1, h3 = 2, and for n ≥ 4
hn = hn−1 + hn−2 +mn−2
+
n−2∑
k=1
((
hk + gk − 2
⌊ k−12 ⌋∑
i=1
(i+ 1)mk−2i
)
mn−k−2 +mkhn−k−2
)
.
These are the first 11 values of the sequence hn, for n = 1, 2, . . . , 11:
0, 1, 2, 7, 16, 45, 118, 331, 930, 2673, 7744.
R. Flórez et al.: Enumerating symmetric pyramids in Motzkin paths 601
5 Symmetric truncated pyramids
In this section we give asymptotic approximations for the proportions between the different
aspects that we have been studying in this paper. Thus, we give the proportion between the
number of symmetric truncated pyramids and these quantities: the number of pyramids and
the number of humps. We also discuss the combinatorial behavior of truncated pyramids
—symmetric pyramids, symmetric weight, and symmetric height.
We use tn to denote the number of symmetric truncated pyramids in the family of
Motzkin paths of length n.
Theorem 5.1. The generating function for Motzkin paths with respect to the number of
symmetric truncated pyramids is
T (x) :=
∂
∂p
Msp(1, p, 1, x) |p=1 .
In particular, if G(x) denotes the generating function of the symmetric triangular pyramids,
see Theorem 4.1, then
T (x) =
x
1− x
G(x).
These are the first 11 values of the sequence tn for n = 1, 2, . . . , 11:
0, 0, 1, 3, 9, 23, 60, 156, 415, 1121, 3076.
From the equality given in Theorem 5.1 we have the relation tn =
∑n−1
i=0 gi.
Theorem 5.2. The sequence tn has this asymptotic approximation
tn ∼
3n
8
√
3
πn
.
Corollary 5.3. If sn, tn, and hmn represent the number of symmetric pyramids, symmetric
truncated pyramids, and humps of Mn, respectively, then these hold:
lim
n→∞
tn
sn
=
1
3
and lim
n→∞
tn
hmn
=
1
6
.
In Theorem 5.4 we give a recursive relation to calculate the values of the sequence tn.
Its proof uses the same technique used in the proof of Theorem 4.4.
Theorem 5.4. If t∗k is as in Lemma 3.4, then the sequence tn satisfies t0 = t1 = t2 = 0,
t3 = 1, and for n ≥ 4
tn = 2tn−1 − tn−4 +mn−3
+
n−2∑
k=2
((tk − 2t∗k)(mn−k−2 −mn−k−3) +mk(tn−k−2 − tn−k−3)).
We use vn to denote the total weight of symmetric truncated pyramids in Mn. We
define Ob(i, j) :=
⌊
i−1
2
⌋ (⌊
i−1
2
⌋
+ j
)
. The proofs of the following two results are similar
to the proofs of Theorems 4.4 and 5.4. Therefore, we omit them.
602 Ars Math. Contemp. 23 (2023) #P4.06 / 591–603
Proposition 5.5. The sequence vn satisfies v1 = v2 = 0, v3 = 1, and for n ≥ 4
vn = vn−3 + vn−2 + vn−1 +
n−3∑
k=0
mk
+
n−2∑
k=2
((
vk +
⌊
k − 1
2
⌋
−
k−1∑
i=2
Ob(i, 1)mk−i
)
mn−k−2 + vn−k−2mk
)
.
These are the first 11 values of the sequence wn for n = 1, 2, . . . , 11:
0, 1, 3, 10, 26, 70, 182, 485, 1303, 3561, 9843.
The symmetric truncated height in Mn is the sum of the heights of all symmetric
truncated pyramids in Mn.
Proposition 5.6. Let kn be the symmetric truncated height in Mn and let ti be the number
of symmetric truncated pyramids in Mi. Then the sequence kn satisfies k1 = k2 = 0,
k3 = 1, and for n ≥ 4
kn = kn−1 + kn−2 + kn−3 +
n−3∑
j=0
mj
+
n−2∑
j=2
((
kj + tj −
j−1∑
i=2
Ob(i, 3)mj−i
)
mn−j−2 +mjkn−j−2
)
.
These are the first 11 values of the sequence kn for n = 1, 2, . . . , 11:
0, 0, 1, 3, 10, 26, 71, 189, 520, 1450, 4123.
ORCID iDs
Rigoberto Flórez https://orcid.org/0000-0002-3644-9358
José L. Ramı́rez https://orcid.org/0000-0002-8028-9312
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P4.07 / 605–613
https://doi.org/10.26493/1855-3974.3072.3ec
(Also available at http://amc-journal.eu)
The core of a vertex-transitive complementary
prism
Marko Orel *
University of Primorska, FAMNIT, Glagoljaška 8, Koper, Slovenia and
University of Primorska, IAM, Muzejski trg 2, Koper, Slovenia and
IMFM, Jadranska 19, Ljubljana, Slovenia
Received 20 February 2023, accepted 15 March 2023, published online 12 May 2023
Abstract
The complementary prism ΓΓ̄ is obtained from the union of a graph Γ and its comple-
ment Γ̄ where each pair of identical vertices in Γ and Γ̄ is joined by an edge. It generalizes
the Petersen graph, which is the complementary prism of the pentagon. The core of a
vertex-transitive complementary prism is studied. In particular, it is shown that a vertex-
transitive complementary prism ΓΓ̄ is a core, i.e. all its endomorphisms are automorphisms,
whenever Γ is a core or its core is a complete graph.
Keywords: Graph homomorphism, core, complementary prism, self-complementary graph, vertex-
transitive graph.
Math. Subj. Class. (2020): 05C60, 05C76
1 Introduction
In the study of graph homomorphism a basic object is a core (a.k.a. unretractive graph),
which is a graph such that all its endomorphisms are automorphisms. A subgraph Γ′ of a
graph Γ is its core, if there exists some graph homomorphism φ : Γ → Γ′ and Γ′ is a core.
Equivalently, Γ′ is the minimal retract of Γ (cf. [17]). Despite that each graph has its core,
which is unique up to isomorphism, it can be often very difficult to determine if a given
graph is a core or not (cf. [6, 16, 30]). From this point of view, graphs that have either high
degree of symmetry (i.e. ‘large’ automorphism group) or some ‘nice’ combinatorial prop-
erties are the most interesting. Many of such classes of graphs are core-complete, which
*The author thanks the anonymous referees for the helpful comments that improved the presentation of the
paper. This work is supported in part by the Slovenian Research Agency (research program P1-0285 and research
projects N1-0140, N1-0208, N1-0210, J1-4084, and N1-0296).
E-mail address: marko.orel@upr.si (Marko Orel)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
606 Ars Math. Contemp. 23 (2023) #P4.07 / 605–613
means that they are either cores or their cores are complete graphs. Among them we can
find all non-edge-transitive graphs [6, Corollary 2.2], connected regular graphs with the au-
tomorphism group that acts transitively on unordered pairs of vertices at distance two [16,
Theorem 4.1], and all primitive strongly regular graphs [36, Corollary 3.6]. Given a core-
complete graph, it can be extremely complicated to decide if the graph is a core or its core is
complete. For some graphs, this task is equivalent to some of the longstanding open prob-
lems in finite geometry (see [6, 30]). A well known core is the Petersen graph, which has
both ‘large’ automorphism group and ‘nice’ combinatorial properties. Given a family of
graphs that (naturally) generalize the Petersen graph it is interesting to study if its members
are cores or not. Kneser graphs K(n, r), with 2r < n, are all cores [15, Theorem 7.9.1].
The graph HGLn(F4), whose vertex set is formed by all n× n invertible hermitian matri-
ces over the field with four elements and with the edge set {{A,B} : rank(A − B) = 1},
is a core whenever n ≥ 2 [29]. The core of a generalized Petersen graph G(n, k) was stud-
ied very recently in [13]. The complementary prism ΓΓ̄, whose definition in full details
is given in Section 2, is another generalization of the Petersen graph, which is obtained if
Γ is the 5-cycle C51. Graph ΓΓ̄ was introduced in [18] and is the main mater of research
in several papers (see for example [1, 3, 7, 10, 19, 26]). Recall that C5 is strongly reg-
ular vertex-transitive self-complementary graph. The core of ΓΓ̄ was recently studied by
the author [28] (see also the arXiv version [31]). In particular, it was shown that ΓΓ̄ is
a core whenever Γ is strongly regular and self-complementary. In this paper we build on
a result from [28] and investigate vertex-transitive self-complementary graphs Γ. These
are precisely the graphs that provide vertex-transitive complementary prisms [31, Corol-
lary 3.8]. For such graphs we prove that ΓΓ̄ is a core whenever Γ is core-complete (see
Theorem 3.3), and state an open problem, which asks if there exists a vertex-transitive self-
complementary graph Γ such that ΓΓ̄ is not a core (Problem 3.6). The main results are
presented in Section 3. In Section 2 we recall some tools and definitions that we need in
what follows.
2 Preliminaries
All graphs in this paper are finite and simple. The vertex set and the edge set of a graph Γ
are denoted by V (Γ) and E(Γ), respectively. A subset of pairwise adjacent vertices in
V (Γ) is a clique, while a set of pairwise nonadjacent vertices in V (Γ) is an independent
set. The clique number ω(Γ) and the independence number α(Γ) are the orders of the
largest clique and the largest independent set in Γ, respectively. In particular, α(Γ) = ω(Γ̄),
where Γ̄ is the complement of the graph Γ. The chromatic number of a graph is denoted
by χ(Γ). It is well known that χ(Γ) ≥ ω(Γ) and χ(Γ) ≥ nα(Γ) , where n = |V (Γ)|
(cf. [5]). A graph homomorphism between graphs Γ1,Γ2 is a map φ : V (Γ1) → V (Γ2)
such that {φ(u), φ(v)} ∈ E(Γ2) whenever {u, v} ∈ E(Γ1). If in addition φ is bijective
and {u, v} ∈ E(Γ1) ⇐⇒ {φ(u), φ(v)} ∈ E(Γ2), then φ is a graph isomorphism and
graphs Γ1,Γ2 are isomorphic, which we denote by Γ1 ∼= Γ2. If Γ1 = Γ2, then a graph
homomorphism/graph isomorphism is a graph endomorphism/automorphism, respectively.
A graph Γ is self-complementary if there exists a graph isomorphism σ : Γ → Γ̄, which
is referred to as antimorphism or a complementing permutation. Observe that in this case
σ is also an antimorphism as a map Γ̄ → Γ. A graph Γ is regular if each vertex has
the same number of neighbors. If this number equals k, then we say that it is k-regular.
1Graphs K(5, 2), HGL2(F4), G(5, 2), C5C5 are all isomorphic to the Petersen graph.
M. Orel: The core of a vertex-transitive complementary prism 607
If for each pair of vertices u, v ∈ V (Γ) there exists an automorphism φ of Γ such that
u = φ(v), then Γ is a vertex-transitive graph. Clearly, each vertex-transitive graph is
regular. Lemma 2.1 can be found in [14, Corollaries 2.1.2, 2.1.3], where it is stated in a
more general settings.
Lemma 2.1. If a graph Γ is vertex-transitive, then α(Γ)ω(Γ) ≤ |V (Γ)|. If the equality
holds, then |C ∩ I| = 1 for each clique C and each independent set I that provide the
equality.
If a graph on n vertices is (n−12 )-regular, then n must be odd. Moreover, it follows
from the hand-shaking lemma that n = 4m+ 1 for some integer m ≥ 0. In particular, this
is true for all regular self-complementary graphs. By a result of Sachs [37] or Ringel [35],
each cycle in an antimorphism of a self-complementary graph has the length divisible by
four, except for one cycle of length one, in the case the order of the graph equals 1 modulo
4 (a proof in English can be found in [11, page 12]). Lemma 2.2 is a special case of this
fact, and is crucial in the proof of Theorem 3.3.
Lemma 2.2. If σ is an antimorphism of a regular self-complementary graph Γ, then there
exists a unique vertex v ∈ V (Γ) such that σ(v) = v.
A graph Γ is a core if each its endomorphism is an automorphism. Given a graph Γ, we
use core(Γ) to denote any subgraph of Γ that is a core and such that there exists some graph
homomorphism φ : Γ → core(Γ). Graph core(Γ) is referred to as the core of Γ. It is always
an induced subgraph and unique up to isomorphism [15, Lemma 6.2.2]. Clearly, a graph Γ
is a core if and only if Γ = core(Γ). On the other hand, core(Γ) is a complete graph if and
only if χ(Γ) = ω(Γ). We remark that there always exists a retraction ψ : Γ → core(Γ), i.e.
a graph homomorphism that fixes each vertex in core(Γ). In fact, if φ : Γ → core(Γ) is any
graph homomorphism, then the restriction φ|V (core(Γ)) is invertible and the composition
(φ|V (core(Γ)))−1 ◦ φ is the required retraction. Lemma 2.3 is proved in [39, Theorem 3.2],
where it is stated in an old terminology. Its proof can be found also in [15, 17].
Lemma 2.3. If graph Γ is vertex-transitive, then core(Γ) is vertex-transitive.
Let Γ be a graph with the vertex set V (Γ) = {v1, . . . , vn}. The complementary prism
of Γ is the graph ΓΓ̄, which is obtained from the disjoint union of Γ and its complement Γ̄,
by adding an edge between each vertex in Γ and its copy in Γ̄. In this paper we use
the following notation. The vertex set of the complementary prism of graph Γ is the set
V (ΓΓ̄) =W1 ∪W2, where
W1 =W1(ΓΓ̄) = {(v1, 1), . . . , (vn, 1)} and W2 =W2(ΓΓ̄) = {(v1, 2), . . . , (vn, 2)}.
The edge set E(ΓΓ̄) is the union of the sets{
{(u, 1), (v, 1)} : {u, v} ∈ E(Γ)
}
,{
{(u, 2), (v, 2)} : {u, v} ∈ E(Γ̄)
}
,{
{(u, 1), (u, 2)} : u ∈ V (Γ)
}
.
It follows from the definition that a complementary prism ΓΓ̄ is regular if and only if Γ is(
n−1
2
)
-regular (see also [7, Theorem 3.6]). The core of a complementary prism for general
graph Γ was recently studied in [28] (see also the arXiv version [31]). For regular case, the
following result was proved.
608 Ars Math. Contemp. 23 (2023) #P4.07 / 605–613
Lemma 2.4 ([28, Corollary 3.4]). Let Γ be any graph on n vertices that is
(
n−1
2
)
-regular.
Then one of the following three possibilities is true.
(i) Graph ΓΓ̄ is a core.
(ii) All vertices of core(ΓΓ̄) are contained in W1, in which case
core(ΓΓ̄) ∼= core(Γ).
(iii) All vertices of core(ΓΓ̄) are contained in W2, in which case
core(ΓΓ̄) ∼= core(Γ̄).
The same conclusion can be obtained also if the core of ΓΓ̄ is regular and we exclude
some small graphs. Below, K2 is a complete graph on two vertices, and P3 is a path on
three vertices.
Lemma 2.5 ([28, Corollary 3.6]). Let Γ be any graph, which is not isomorphic to K2, K2,
P3, or P3. If core(ΓΓ̄) is regular, then one of the three possibilities in Lemma 2.4 is true.
Clearly, Lemma 2.4 is valid for each regular self-complementary graph Γ. The study of
such graphs and their vertex-transitive counterparts has origins in the papers [21, 37, 40],
which influenced a lot of research related to vertex-transitive self-complementary graphs
(see for example [2, 4, 8, 9, 12, 20, 22, 23, 24, 25, 27, 33, 34, 38, 41] and the references
therein). In this paper the aim is to to study the core of a complementary prism ΓΓ̄, where Γ
is vertex-transitive and self-complementary graph. The following observation is obtained
for free, with a double proof.
Corollary 2.6. If graph Γ is vertex-transitive and self-complementary graph, then one of
the three possibilities in Lemma 2.4 is true.
Proof. The claim follows directly from Lemma 2.4. The same claim is deduced also if we
combine Lemmas 2.5 and 2.3.
In [28] some examples of regular self-complementary graphs Γ are provided such that
the statement (ii) or (iii) in Lemma 2.4 is true. In this paper we show that this is not possi-
ble for a large class of vertex-transitive self-complementary graphs. It should be mentioned
that it was recently proved that a complementary prism ΓΓ̄ is vertex-transitive if and only
if Γ is vertex-transitive and self-complementary [31, Corollary 3.8]. Despite our proofs do
not rely on this result, it means that this paper studies the core of vertex-transitive comple-
mentary prisms.
3 Main results
The main result of this paper is Theorem 3.3. Propositions 3.1 and 3.2 are the stepping
stones towards its proof.
Proposition 3.1. Let Γ be a regular self-complementary graph. If Γ is a core, then ΓΓ̄ is a
core.
M. Orel: The core of a vertex-transitive complementary prism 609
Proof. Let core(ΓΓ̄) be any core of Γ and let n = |V (Γ)|. Since Γ is
(
n−1
2
)
-regular, one
of the statements (i), (ii), (iii) in Lemma 2.4 is true. Suppose that (iii) is correct, that is,
V (core(ΓΓ̄)) ⊆W2 (3.1)
and
core(ΓΓ̄) ∼= core(Γ̄). (3.2)
Since Γ̄ is a core, (3.2) implies that core(ΓΓ̄) ∼= Γ̄. Hence, (3.1) yields
V (core(ΓΓ̄)) =W2. (3.3)
Let ψ1(v) = (v, 1), for v ∈ V (Γ), be the canonical isomorphism between Γ and the
subgraph of ΓΓ̄, which is induced by the set W1. Similarly, let ψ2(v) = (v, 2), for v ∈
V (Γ), be the canonical isomorphism between Γ̄ and the subgraph induced by W2. If Ψ
is any retraction from ΓΓ̄ onto core(ΓΓ̄), and σ is any antimorphism between Γ̄ and Γ,
then the composition σ ◦ ψ−12 ◦ (Ψ|W1) ◦ ψ1 is an endomorphism of Γ. Since Γ is a
core, the restriction Ψ|W1 is an isomorphism between the subgraphs in ΓΓ̄ that are induced
by the sets W1 and W2, respectively. Consequently ψ−12 ◦ (Ψ|W1) ◦ ψ1 : Γ → Γ̄ is an
antimorphism. By Lemma 2.2, there exists v ∈ V (Γ) such that
(
ψ−12 ◦(Ψ|W1)◦ψ1
)
(v) = v.
Consequently, Ψ(v, 1) = (Ψ|W1)(v, 1) = (v, 2). Since Ψ is a retraction, (3.3) implies that
Ψ(v, 2) = (v, 2). Since {(v, 1), (v, 2)} is an edge in ΓΓ̄, we have a contradiction.
In the same way we see that (ii) in Lemma 2.4 is not possible, which means that ΓΓ̄ is
a core.
Proposition 3.2. If Γ is a vertex-transitive self-complementary graph on n > 1 vertices,
then core(ΓΓ̄) is not a complete graph.
Proof. We need to prove that χ(ΓΓ̄) > ω(ΓΓ̄). Since n > 1 and Γ is both self-complement-
ary and vertex-transitive, Lemma 2.1 implies that
ω(ΓΓ̄) = max{α(Γ), ω(Γ)} = ω(Γ) ≤
√
n.
Let I be any independent set in ΓΓ̄. Then I is a disjoint union of some sets I1 ⊆ W1 and
I2 ⊆W2. If we write Ii = {(u, i) : u ∈ Ji} for i ∈ {1, 2}, where J1, J2 ⊆ V (Γ), then
J1 ∩ J2 = ∅. (3.4)
Since J1 and J2 are an independent set and a clique in Γ, respectively, we have |J1| ≤
α(Γ) = ω(Γ) ≤
√
n and |J2| ≤ ω(Γ) =
√
n, while Lemma 2.1 and (3.4) imply that
|J1| · |J2| < n. Hence, |I| = |J1|+ |J2| < 2
√
n, and therefore
χ(ΓΓ̄) ≥ |V (ΓΓ̄)|
α(ΓΓ̄)
>
2n
2
√
n
=
√
n ≥ ω(ΓΓ̄).
Theorem 3.3. Let Γ be a vertex-transitive self-complementary graph. If Γ is either a core
or its core is a complete graph, then ΓΓ̄ is a core.
610 Ars Math. Contemp. 23 (2023) #P4.07 / 605–613
Proof. If Γ is a core, then the claim follows from Proposition 3.1. Hence, we may as-
sume that Γ has a complete core and more than one vertex. By Corollary 2.6, one of the
statements (i), (ii), (iii) in Lemma 2.4 is true for core(ΓΓ̄). If (ii) or (iii) is correct, then
the self-complementarity implies that core(ΓΓ̄) is complete, which contradicts Proposi-
tion 3.2.
Remark 3.4. The claims in Proposition 3.2 and Theorem 3.3 are not true for some reg-
ular self-complementary graphs. In fact, there exists a regular self-complementary graph
Γ with a complete core such that core(ΓΓ̄) is complete (see [28, Example 3.5] or [31,
Example 5.5]).
Recall that a k-regular graph on n vertices is strongly regular with parameters (n, k, λ, µ)
if each pair of adjacent vertices has λ common neighbors and each pair of distinct non-
adjacent vertices has µ common neighbors. Theorem 3.5 was very recently proved in [28]
(see also the arXiv version [31, Theorem 5.7]). The proof relied on application of Lemma 2.4
together with several properties of the Lovász theta function and the graph spectrum. Here
we provide a sketch of an alternative proof that essentially copies the proofs in this section
and applies a remarkable result that Roberson recently proved [36].
Theorem 3.5 ([28]). If Γ is a strongly regular self-complementary graph, then ΓΓ̄ is a
core.
Sketch of a proof. To see that ΓΓ̄ does not have a complete core, we copy the proof of
Proposition 3.2, where we replace the application of Lemma 2.1 by its analog that can
be found in [14, Corolarry 3.8.6 and Theorem 3.8.4] and is valid for all strongly regular
graphs. From [36, Corollary 3.6] it follows that Γ is either a core or its core is complete.
Then we just copy the proof of Theorem 3.3, where we rely directly on Lemma 2.4 instead
on Corollary 2.6.
Recall from the introduction that many ‘nice’ graphs are either cores or their cores are
complete. Hence it is expected that many vertex-transitive self-complementary graphs fulfil
the assumptions in Theorem 3.3. Consequently, we state the following open problem.
Problem 3.6. Does there exist a vertex-transitive self-complementary graph Γ such that
ΓΓ̄ is not a core?
Note that edge-transitive self-complementary graphs are also arc-transitive (see [11]).
Since self-complementary graphs are always connected (cf. [11]), it follows that each edge-
transitive self-complementary graph is also vertex-transitive. However, such graphs are
always strongly regular (cf. [11]) and therefore their complementary prisms are cores by
Theorem 3.5. Despite the orders of vertex-transitive self-complementary graphs were fully
determined in [27], there is a major gap between the understanding of vertex-transitive
self-complementary graphs and the understanding of edge-transitive self-complementary
graphs. In fact, the later were completely characterized in [33]. Moreover, the first non-
Cayley vertex-transitive self-complementary graph was constructed only in 2001 [24], and
the construction is highly nontrivial. We believe that all these facts indicate that Prob-
lem 3.6 may be challenging.
In a distinct paper [32], we consider the only families of vertex-transitive self-comple-
mentary graphs the author is aware of, which are neither cores nor their cores are complete
graphs (i.e. they do not satisfy the assumption in Theorem 3.3). Unfortunately they do not
solve Problem 3.6.
M. Orel: The core of a vertex-transitive complementary prism 611
ORCID iDs
Marko Orel https://orcid.org/0000-0002-2544-2986
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P4.08 / 615–629
https://doi.org/10.26493/1855-3974.2688.2de
(Also available at http://amc-journal.eu)
Normal Cayley digraphs of dihedral groups with
the CI-property*
Jin-Hua Xie , Yan-Quan Feng † , Jin-Xin Zhou
School of Mathematics and Statistics, Beijing Jiaotong University,
Beijing, 100044, P. R. China
Received 2 September 2021, accepted 30 January 2023, published online 20 June 2023
Abstract
A Cayley (di)graph Cay(G,S) of a group G with respect to a set S ⊆ G is said to
be normal if the image of G under its right regular representation is normal in the au-
tomorphism group of Cay(G,S), and is called a CI-(di)graph if for every T ⊆ G with
Cay(G,S) ∼= Cay(G,T ), there is α ∈ Aut(G) such that Sα = T . A finite group G is
called a DCI-group or an NDCI-group if all Cayley digraphs or all normal Cayley digraphs
of G are CI-digraphs, respectively, and is called a CI-group or an NCI-group if all Cayley
graphs or all normal Cayley graphs of G are CI-graphs, respectively.
Motivated by a conjecture proposed by Ádám in 1967, CI-groups and DCI-groups have
been actively studied during the last fifty years by many researchers in algebraic graph
theory. It took about thirty years to obtain the classification of cyclic CI-groups and DCI-
groups, and recently, the first two authors, among others, classified cyclic NCI-groups and
NDCI-groups. Even though there are many partial results on dihedral CI-groups and DCI-
groups, their classification is still elusive. In this paper, we prove that a dihedral group of
order 2n is an NCI-group or an NDCI-group if and only if n = 2, 4 or n is odd. As a direct
consequence, we have that if a dihedral group D2n of order 2n is a DCI-group then n = 2
or n is odd-square-free, and that if D2n is a CI-group then n = 2, 9 or n is odd-square-
free, throwing some new light on classification of dihedral CI-groups and DCI-groups. As
a byproduct, we construct a non-CI Cayley graph of the dihedral group D8, but Holt and
Royle in 2020 claimed that D8 is a CI-group by an algorithm there.
Keywords: Dihedral group, CI-group, DCI-group, NCI-group, NDCI-group.
Math. Subj. Class. (2020): 05C25, 20B25
*The work was supported by the Fundamental Research Funds for the Central Universities (2022JBCG003),
the National Natural Science Foundation of China (12071023, 12161141005, 12271024) and the 111 Project of
China (B16002).
†Corresponding author.
E-mail addresses: jinhuaxie@bjtu.edu.cn (Jin-Hua Xie), yqfeng@bjtu.edu.cn (Yan-Quan Feng),
jxzhou@bjtu.edu.cn (Jin-Xin Zhou)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
616 Ars Math. Contemp. 23 (2023) #P4.08 / 615–629
1 Introduction
Graphs and digraphs considered in this paper are finite and simple, and groups are finite.
For a (di)graph Γ, we use V (Γ), E(Γ), Arc(Γ) and Aut(Γ) to denote the vertex set, edge
set, arc set, and automorphism group of Γ, respectively, where an arc means a directed edge
in a digraph and an ordered pair of adjacent vertices in a graph.
Let G be a group and S be a subset of G with 1 ̸∈ S. A digraph with vertex set G and
arc set {(g, sg) | g ∈ G, s ∈ S}, denoted by Cay(G,S), is called the Cayley digraph of G
with respect to S. If S is inverse-closed, that is, S = S−1 := {x−1 | x ∈ S}, then for two
adjacent vertices u and v in Cay(G,S), both (u, v) and (v, u) are arcs, and in this case, we
view Cay(G,S) as a graph by identifying the two arcs with one edge {u, v}.
Two Cayley (di)graphs Cay(G,S) and Cay(G,T ) are called Cayley isomorphic if there
is α ∈ Aut(G) such that Sα = T . Cayley isomorphic Cayley (di)graphs are isomorphic,
but the converse is not true. A subset S of G with 1 ̸∈ S is said to be a CI-subset if
Cay(G,S) ∼= Cay(G,T ), for some T ⊆ G with 1 ̸∈ T , implies that Cay(G,S) and
Cay(G,T ) are Cayley isomorphic. In this case, Cay(G,S) is called a CI-digraph, or a
CI-graph when S = S−1. A group G is called a DCI-group if all Cayley digraphs of G are
CI-digraphs, and a CI-group if all Cayley graphs of G are CI-graphs.
DCI-groups and CI-groups have been widely investigated over last fifty years, and they
have been reduced to some special restricted groups in [12, 28]. However, it is still very
difficult to determine whether a particular group is a DCI-group or a CI-group; see [2, 5,
11, 20, 26, 31, 32, 37, 38, 39, 40] for examples. In fact, it is even an open problem to
classify dihedral DCI-groups or CI-groups.
Ádám [1] conjectured that every finite cyclic group is a CI-group. Due to contributions
of many researchers like Elspas and Turner [16], Djoković [10], Turner [41], Babai [4],
Alspach and Parsons [3], Godsil [22] and Pálfy [33], the cyclic DCI-groups and CI-groups
were classified finally by Muzychuk [29, 30]. It follows that a cyclic group of order n is
a DCI-group if and only if n = mk where m = 1, 2, 4 and k is odd-square-free, and is
a CI-group if and only if either n = 8, 9, 18, or n = mk where m = 1, 2, 4 and k is
odd-square-free. The generalised dihedral group Dih(A) over an abelian group A is the
group ⟨A, b | b2 = 1, ab = a−1,∀a ∈ A⟩, and in particular, if A is a cyclic group of
order n, Dih(A) is the dihedral group D2n of order 2n. There are also some partial results
on dihedral DCI-groups or CI-groups. Babai [4] proved that the dihedral group D2p for a
prime p is a CI-group. Conder and Li [8] proved that D18 is a CI-group. In [12], it was
further proved that D6p (p a prime) is a DCI-group if and only if p ≥ 5, and is a CI-group
if and only if p ≥ 3. Recently, Dobson et al. [13] proved that if R is a generalised dihedral
CI-group, then for every odd prime p, the Sylow p-subgroup of R has order p, or 9, which
reduces dihedral DCI-groups to D2n with n = 2mk (m = 0, 1) and k odd-square-free and
dihedral CI-groups to D18 or D2n with n = 2mk (m = 0, 1) and k odd-square-free.
Let Cay(G,S) be a Cayley (di)graph of G with respect to a set S ⊆ G. For a given
g ∈ G, the right multiplication R(g) : x 7→ xg, x ∈ G, is an automorphism of Cay(G,S),
and R(G) := {R(g) | g ∈ G} is a regular group of automorphisms of Cay(G,S), that
is, the image of G under its right regular representation. A Cayley (di)graph Cay(G,S) is
said to be normal if R(G) is a normal subgroup of Aut(Cay(G,S)). Normality of Cayley
(di)graphs is very important because the automorphism groups of normal Cayley (di)graph
are actually known; see Godsil [21] or Proposition 2.2. Furthermore, the study of normality
of Cayley (di)graphs is currently a hot topic in algebraic graph theory, and we refer to [14,
15, 17, 18, 19, 34, 45, 46] for examples.
J.-H. Xie et al.: Normal Cayley digraphs of dihedral groups with the CI-property 617
A group G is called an NDCI-group or an NCI-group if all normal Cayley digraphs
or graphs of G are CI-digraphs or CI-graphs, respectively. Obviously, a DCI-group is an
NDCI-group and a CI-group is an NCI-group. Li [27] constructed some normal Cayley
digraphs of cyclic groups of order 2-powers that are not CI-graphs, and proposed the fol-
lowing problem: characterize normal Cayley digraphs which are not CI-digraphs. Similar
to DCI-groups and CI-groups, a natural problem is to classify finite NDCI-groups and
NCI-groups. Recently, Xie, Feng, Ryabov and Liu [43] classified cyclic NDCI-groups and
NCI-groups, and in this paper, we classify dihedral NDCI-groups and NCI-groups.
Theorem 1.1. Let n ≥ 2 be an integer and let D2n be the dihedral group of order 2n.
Then the following statements are equivalent:
(1) D2n is an NDCI-group;
(2) D2n is an NCI group;
(3) Either n = 2, 4 or n is odd.
Li [27, Problem 6.3(3)]) proposed the following problem: are all connected cubic Cay-
ley graphs CI-graphs? To prove Theorem 1.1, an infinite family of connected cubic normal
non-CI Cayley graphs are constructed in Lemma 4.1, which gives a negative answer to Li’s
problem. Classification of NDCI-groups and NCI-groups can be helpful for classification
of DCI-groups and CI-groups. In fact, by Theorem 1.1, together with [13], we have the
following corollary.
Corollary 1.2. If a dihedral group D2n of order 2n is a DCI-group then n = 2 or n is
odd-square-free, and if D2n is a CI-group then n = 2, 9 or n is odd-square-free.
We believe that the converse of Corollary 1.2 is true.
Conjecture 1.3. A dihedral group D2n of order 2n is a DCI-group if and only if n = 2 or
n is odd-square-free, and D2n is a CI-group if and only if n = 2, 9 or n is odd-square-free.
Note that Holt and Royle [24, 6.2] claimed that D8 is a CI-group, and this is not true by
Lemma 4.1, where a non-CI Cayley graph of D8 is constructed and its non-CI-property is
also checked by using MAGMA [7].
All the notation and terminologies used in this paper are standard, and for group and
graph concepts not defined here, we refer to [6, 9, 35, 36].
2 Preliminaries
In this section, we give some basic concepts and facts that will be needed later.
Let F be a free group of rank r and let G be a group generated by a set X of r elements,
say X = {x1, x2, . . . , xr}. Then there is a standard free presentation from F to G induced
by a bijective mapping from the free generators of F to X . Let G have a defining relation
set S, that is, G = ⟨X | S⟩. Let H be a group that can be generated by a set Y of r elements,
say Y = {y1, y2, . . . , yr}. The following is the well-known von Dyck’s Theorem [35,
2.2.1].
Proposition 2.1. Assume that all generators yi of H satisfy every relation in S by replacing
xi with yi. Then there is an epimorphism from G to H induced by xi 7→ yi for all 1 ≤ i ≤ r.
618 Ars Math. Contemp. 23 (2023) #P4.08 / 615–629
Let Cay(G,S) be a Cayley digraph of a group G with respect to S, and let Aut(G,S) =
{α ∈ Aut(G) | Sα = S}. Then Aut(G,S) is a subgroup of Aut(Cay(G,S))1, the stabilizer
of 1 in Aut(Cay(G,S)). By Godsil [21], the normalizer of R(G) in Aut(Cay(G,S)) is the
semiproduct R(G)⋊Aut(G,S), where R(g)α = R(gα) for all g ∈ G and α ∈ Aut(G,S),
and by [44, Propositions 1.3 and 1.5], we have the following.
Proposition 2.2. Let Cay(G,S) be a Cayley digraph of a group G with respect to S and let
A = Aut(Cay(G,S)). Then NA(R(G)) = R(G)⋊ Aut(G,S) and Cay(G,S) is normal if
and only if A1 = Aut(G,S).
Babai [4] gave a well-known criterion for a Cayley digraph to be a CI-digraph, that is,
a Cayley digraph Cay(G,S) is a CI-digraph if and only if every regular group of automor-
phisms of Cay(G,S) isomorphic to G, is conjugate to R(G) in Aut(Cay(G,S)). Based on
this, the following proposition is straightforward (also see [27, Corollary 6.9]).
Proposition 2.3. Let Cay(G,S) be a normal Cayley digraph of a group G with respect to
S. Then Cay(G,S) is a CI-digraph if and only if Aut(Cay(G,S)) has a unique regular
subgroup isomorphic to G, that is, R(G).
Let G be a finite group and let L ⊆ Aut(G). Write FG(L) = {g | gℓ = g for every ℓ ∈
L}, the fixed-points of L in G. Clearly, FG(L) ≤ G. The following proposition was given
in [42, Theorem 2.2], which is about non-normal Cayley digraphs.
Proposition 2.4. Let Cay(G,S) be a Cayley digraph. Let 1 ̸= L ≤ Aut(G,S) and H ⊴G
be such that for every right coset Hg in G, either L fixes Hg pointwise, or L fixes Hg
setwise and is transitive on Hg. Suppose that one of the following holds:
(1) |G : FG(L)| > 2;
(2) |G : FG(L)| = 2, and there is g ∈ G\FG(L) and h ∈ H such that hg ̸= h−1;
(3) |G : FG(L)| = 2, and there is 1 ̸= γ ∈ Aut(G,S) such that FG(⟨γ⟩) ̸= FG(L) and
γ fixes every coset of H in G setwise.
Then Cay(G,S) is non-normal.
3 Automorphisms and holomorphs of dihedral groups
In this section we collect some details about automorphism groups and holomorphs of
dihedral groups, which are crucial for the proof of Theorem 1.1.
Let G be a finite group and let g ∈ G. Denote by o(g) the order of g in G. Let p be
a prime and π a set of primes. Denote by Gp a Sylow p-subgroup of G. An element g of
G is called a π-element if all prime factors of o(g) belong to π, and a p′-element if o(g)
has no factor p. If G is soluble, denote by Gπ a Hall π-subgroup of G, and by Gp′ a Hall
p′-subgroup of G.
Let n be a positive integer. We first make the following convention:
n =
∏m
i=1
prii ≥ 2, Cn = Cpr11 ×· · ·×Cprmm = ⟨a1⟩×· · ·×⟨am⟩, a = a1 . . . am, (3.1)
where p1, . . . , pm are all distinct prime factors of n, Cprii = ⟨ai⟩
∼= Zprii for each 1 ≤ i ≤
m, and Cn = ⟨a⟩ ∼= Zn. By [36, Theorem 7.3], we have
Aut(Cn) = Aut(Cpr11 )× Aut(Cpr22 )× · · · × Aut(Cprmm ), (3.2)
J.-H. Xie et al.: Normal Cayley digraphs of dihedral groups with the CI-property 619
where Aut(Cprii ) is viewed as the subgroup of Aut(Cn) by identifying αi ∈ Aut(Cprii ) as
the automorphism of Cn induced by ai 7→ aαii and aj 7→ aj for all j ̸= i. Further we set
D2n = ⟨a, b | an = b2 = 1, bab = a−1⟩ = Cn ⋊ ⟨b⟩. (3.3)
By Proposition 2.1, for ai ∈ ⟨a⟩ with i ∈ Zn, there is an automorphism of D2n, denoted
by θai , which is induced by
θai : a 7→ a, b 7→ bai. (3.4)
Then o(θai) = o(ai) and ⟨θa⟩ ∼= Zn. Write ai = a1 . . . ai−1ai+1 . . . am. Then o(ai) =
n/prii , and hence ⟨θa⟩ = ⟨θai⟩ × ⟨θai⟩. In fact, ⟨θai⟩ and ⟨θai⟩ are the unique Sylow
pi-subgroup and the unique Hall p′i-subgroup of ⟨θa⟩, respectively. Clearly,
θaiθaj = θai+j and θkai = θaik for all i, j, k ∈ Zn. (3.5)
Again by Propostion 2.1, we also view Aut(Cn) as the subgroup of Aut(D2n) by iden-
tifying β ∈ Aut(Cn) as the automorphism of Aut(D2n) induced by a 7→ aβ and b 7→ b.
Then for each 1 ≤ i ≤ m, we have Aut(Cprii ) ≤ Aut(D2n).
Lemma 3.1. Let n ≥ 3. Then Aut(D2n) has the following properties.
(1) Aut(D2n) = ⟨θa⟩⋊Aut(Cn), and ⟨θa⟩ is the kernel of the natural action of Aut(D2n)
on Cn. Furthermore, θβx = θxβ for all x ∈ ⟨a⟩ and β ∈ Aut(Cn);
(2) For every 1 ≤ i ≤ m, ⟨θa⟩ ⋊ Aut(Cprii ) = (⟨θai⟩ ⋊ Aut(Cprii )) × ⟨θai⟩, where
ai = a1 . . . ai−1ai+1 . . . am;
(3) Assume p1 > p2 > · · · > pm. For every 1 ≤ i ≤ m, set πi = {p1, . . . , pi}. Then
Aut(D2n) has a normal Hall πi-subgroup and Aut(D2n)πi = ⟨θa1...ai⟩⋊Aut(Cn)πi ,
where Aut(Cn)πi = Aut(Cpr11 )πi × · · · × Aut(Cprii )πi .
Proof. It is easy to prove that Aut(D2n) = ⟨θa⟩ ⋊ Aut(Cn) (also see [25, Theorem 7.2]).
Let K be the kernel of Aut(D2n) acting on Cn. Then aγ = a for all γ ∈ K, which implies
⟨θa⟩ ≤ K. Furthermore, K is transitive on b⟨a⟩ as ⟨θa⟩ is transitive. Since D2n = ⟨a, b⟩
and K ≤ Aut(D2n), we have Kb = 1, implying that K is regular on b⟨a⟩. It follows that
|K| = |b⟨a⟩| = n, and hence K = ⟨θa⟩.
For β ∈ Aut(Cn), we have bβ = b, and for x ∈ ⟨a⟩, θx fixes ⟨a⟩ pointwise. Since β
fixes ⟨a⟩ setwise,
aθx
β
= (aβ
−1
)θxβ = aβ
−1β = a = aθxβ ,
and
bθx
β
= (bβ
−1
)θxβ = bθxβ = (bx)β = bxβ = bθxβ .
Since D2n = ⟨a, b⟩, we obtain θβx = θxβ . This completes the proof of part (1).
Recall that ⟨θa⟩ = ⟨θai⟩ × ⟨θai⟩. Let β ∈ Aut(Cprii ). Since ai = a1 . . . ai−1ai+1 . . .
am, we have a
β
i = ai, and θ
β
ai
= θ(ai)β = θai , that is, θaiβ = βθai . Thus,
⟨θa⟩⋊ Aut(Cprii ) = (⟨θai⟩⋊ Aut(Cprii ))× ⟨θai⟩.
This completes the proof of part (2).
620 Ars Math. Contemp. 23 (2023) #P4.08 / 615–629
Let 1 ≤ i ≤ m. Note that Aut(Cprii )
∼= Zpri−1i (pi−1) if pi is odd, and Aut(Cp
ri
i
) ∼=
Zpi × Zpri−2i if pi = 2 and ri ≥ 2. Since p1 > p2 > · · · > pm, we have Aut(Cp
rk
k
)πi = 1
for i < k ≤ m, and hence Aut(Cn)πi = Aut(Cpr11 )πi × · · · × Aut(Cprii )πi .
It is easy to see that θa1...ai = θa1 . . . θai has order p
r1
1 . . . p
ri
i . Then ⟨θa1...ai⟩ is
a Hall πi-subgroup of ⟨θa⟩, and hence ⟨θa1...ai⟩ is characteristic in ⟨θa⟩. Since ⟨θa⟩ ⊴
Aut(D2n), we have ⟨θa1...ai⟩⊴Aut(D2n), and ⟨θa1...ai⟩⋊Aut(Cn)πi is a Hall πi-subgroup
of Aut(D2n).
Recall that Aut(D2n) = ⟨θa⟩⋊Aut(Cn). To prove ⟨θa1...ai⟩⋊Aut(Cn)πi⊴Aut(D2n), it
suffices to show that Aut(Cn)θaπi ≤ ⟨θa1...ai⟩⋊Aut(Cn)πi , or alternatively, Aut(Cprjj )
θa
πi ≤
⟨θa1...ai⟩ ⋊ Aut(Cn)πi for every 1 ≤ j ≤ i. Now take α ∈ Aut(Cprjj )πi . Then a
α
k = ak
for every 1 ≤ k ≤ m with k ̸= j, and aαj ∈ ⟨aj⟩. It follows that
αθa = θ−1a αθa = θ
−1
a θ
α−1
a α = θa−1θaα−1α = θa−1aα−1α
= θ
a−1j a
α−1
j
α ∈ ⟨θaj ⟩Aut(Cprjj )πi ,
where
a−1aα
−1
= a−11 a
−1
2 . . . a
−1
m a1 . . . aj−1a
α−1
j aj+1 . . . am = a
−1
j a
α−1
j ∈ ⟨aj⟩.
Since 1 ≤ j ≤ i, we have
⟨θaj ⟩Aut(Cprjj )πi ≤ ⟨θa1...ai⟩⋊ Aut(Cn)πi ,
and hence Aut(C
p
rj
j
)θaπi ≤ ⟨θa1...ai⟩ ⋊ Aut(Cn)πi , as required. Thus, Aut(D2n) has a
normal Hall πi-subgroup, that is, Aut(D2n)πi = ⟨θa1...ai⟩⋊Aut(Cn)πi . This complete the
proof of part (3).
For a finite group G, the right regular representation R(G) and the automorphism
group Aut(G) are permutation groups on G. Furthermore, R(G) ∩ Aut(G) = 1, and
R(G)Aut(G) = R(G) ⋊ Aut(G), where R(g)α = R(gα) for all g ∈ G and α ∈ Aut(G).
The normalizer of R(G) in the symmetric group SG on G is called the holomorph of G,
denoted by Hol(G), and by [36, Lemma 7.16], Hol(G) = R(G)⋊ Aut(G). Now we have
Hol(D2n) = R(D2n)⋊ Aut(D2n) = R(D2n)⋊ (⟨θa⟩⋊ Aut(Cn)). (3.6)
Lemma 3.2. Let n be odd. Using the notations and formulae in Equations (3.1) – (3.6),
we have the following.
(1) For all d ∈ D2n and α ∈ Aut(D2n), ⟨R(a)⟩⟨θa⟩ = ⟨R(a)⟩ × ⟨θa⟩⊴ Hol(D2n);
(2) Assume p1 > p2 > · · · > pm. For each 1 ≤ i ≤ m, set πi = {p1, . . . , pi}. Then
Hol(D2n)πi ≤ (⟨R(a)⟩ × ⟨θa⟩) ⋊ Aut(Cn)πi ⊴ Hol(D2n), where Aut(Cn)πi =
Aut(Cpr11 )πi × · · · × Aut(Cprii )πi . Furthermore, ⟨R(b)⟩Aut(Cn)2 = ⟨R(b)⟩ ×
Aut(Cn)2 is a Sylow 2-subgroup of Hol(D2n), where Aut(Cn)2 = Aut(Cpr11 )2 ×
Aut(Cpr22 )2 × · · · × Aut(Cprmm )2;
(3) Let 1 ≤ i, j ≤ m with i ̸= j. Then Aut(Cprii ), under conjugacy, fixes each pj-
element in ⟨R(a)⟩ × ⟨θa⟩, and if α ∈ Aut(Cprii ) fixes an element of order p
ri
i in
⟨R(a)⟩ × ⟨θa⟩, then α = 1.
J.-H. Xie et al.: Normal Cayley digraphs of dihedral groups with the CI-property 621
Proof. Recall that R(d)α = R(dα) for all d ∈ D2n and α ∈ Aut(D2n). Then R(a)θa =
R(aθa) = R(a), that is, R(a) commutes with θa. Since ⟨R(a)⟩ ∩ ⟨θa⟩ = 1, we have
⟨R(a)⟩⟨θa⟩ = ⟨R(a)⟩ × ⟨θa⟩. Since n is odd, ⟨R(a)⟩ × ⟨θa⟩ is a Hall 2′-subgroup of
R(D2n) ⋊ ⟨θa⟩, and hence ⟨R(a)⟩ × ⟨θa⟩ is characteristic in R(D2n) ⋊ ⟨θa⟩. It follows
that ⟨R(a)⟩ × ⟨θa⟩ ⊴ Hol(D2n), as R(D2n) ⋊ ⟨θa⟩ ⊴ Hol(D2n) by Equation (3.6). This
completes the proof of part (1).
Now we prove part (2). By part (1), ⟨R(a)⟩× ⟨θa⟩⊴Hol(D2n), and by Equation (3.6),
Hol(D2n) = R(D2n)⋊ (⟨θa⟩⋊ Aut(Cn)) = (R(D2n)⋊ ⟨θa⟩)⋊ Aut(Cn).
Write A = ⟨R(a)⟩ × ⟨θa⟩. Then A ∩ Aut(Cn) = 1, and by Equation (3.6), (R(D2n) ⋊
⟨θa⟩)/A is a normal subgroup of order 2 in Hol(D2n)/A, and hence lies in the center of
Hol(D2n)/A. Thus,
Hol(D2n)/A = (R(D2n)⋊ ⟨θa⟩)/A× Aut(Cn)A/A,
where Aut(Cn)A/A ∼= Aut(Cn). Since Aut(Cn) is abelian, Hol(D2n)/A is abelian, and
therefore, (⟨R(a)⟩ × ⟨θa⟩)⋊ Aut(Cn)πi = A⋊ Aut(Cn)πi ⊴ Hol(D2n).
Since n is odd, all pi are odd and hence 2 ̸∈ πi for each 1 ≤ i ≤ m. Clearly,
Hol(D2n)/((R(D2n)⋊ ⟨θa⟩)⋊ Aut(Cn)πi) ∼= Aut(Cn)/Aut(Cn)πi .
Then for every Hall πi-subgroup Hol(D2n)πi , Hol(D2n)πi ≤ (R(D2n) ⋊ ⟨θa⟩) ⋊
Aut(Cn)πi , and it follows that Hol(D2n)πi ≤ (⟨R(a)⟩×⟨θa⟩)⋊Aut(Cn)πi , as |(R(D2n)⋊
⟨θa⟩) : (⟨R(a)⟩ × ⟨θa⟩)| = 2. By Lemma 3.1(3), Aut(Cn)πi = Aut(Cpr11 )πi × · · · ×
Aut(Cprii )πi .
Since Aut(Cn) fixes b, we have ⟨R(b)⟩Aut(Cn) = ⟨R(b)⟩ × Aut(Cn), and therefore,
⟨R(b)⟩Aut(Cn)2 = ⟨R(b)⟩ × Aut(Cn)2. By Equation (3.6), |Hol(D2n)2| = 2|Aut(Cn)2|,
so ⟨R(b)⟩ × Aut(Cn)2 is a Sylow 2-subgroup of Hol(D2n). Clearly, Aut(Cn)2 =
Aut(Cpr11 )2 × Aut(Cpr22 )2 × · · · × Aut(Cprmm )2. This completes the proof of part (2).
To prove part (3), let 1 ≤ i, j ≤ m with j ̸= i. Recall that pi is odd. By part (1),
⟨R(a)⟩ × ⟨θa⟩⊴ Hol(D2n).
Let α ∈ Aut(Cprii ) and let x be a pj-element in ⟨R(a)⟩ × ⟨θa⟩. Since ⟨R(aj)⟩ × ⟨θaj ⟩
is a normal Sylow pj-subgroup of ⟨R(a)⟩ × ⟨θa⟩, we may take x = R(aj)sθtaj for some
s, t ∈ Z
p
rj
j
. Since j ̸= i, aαj = aj and hence
xα = (R(aj)
sθtaj )
α = R(aαj )
sθtaαj = R(aj)
sθtaj = x.
Thus, Aut(Cprir ) fixes each pj-element in ⟨R(a)⟩ × ⟨θa⟩.
Now assume that α ∈ Aut(Cprii ) fixes an element of order p
ri
i in ⟨R(a)⟩ × ⟨θa⟩. Since
⟨R(ai)⟩ × ⟨θai⟩ is a normal Sylow pi-subgroup of ⟨R(a)⟩ × ⟨θa⟩, α fixes an element of
order prii in ⟨R(ai)⟩ × ⟨θai⟩, say R(ai)kθℓai for some k, ℓ ∈ Zprii . Since o(R(ai)) =
o(θai) = p
ri
i , we have (k, pi) = 1 or (ℓ, pi) = 1. Furthermore,
R(aki )θaℓi = R(ai)
kθℓai = (R(ai)
kθℓai)
α = R((aki )
α)θ(aℓi)α ,
and since ⟨R(a)⟩ ∩ ⟨θa⟩ = 1, we have (aki )α = aki and (aℓi)α = aℓi , which implies that
(ai)
α = ai as (k, pi) = 1 or (ℓ, pi) = 1. Thus, α = 1, completing the proof of part (3).
622 Ars Math. Contemp. 23 (2023) #P4.08 / 615–629
4 Proof of Theorem 1.1
First we construct normal Cayley graphs on dihedral groups which are not CI-graphs. Re-
call that D2n = ⟨a, b | an = b2 = 1, bab = a−1⟩, as given in Equation (3.3).
Lemma 4.1. Let Γ = Cay(D2n, {a, a−1, b}). Then we have the following.
(1) If n = 4 then Γ is non-normal, and if n ≥ 5 then Γ is normal;
(2) If n ≥ 4 is even, then Γ is not a CI-graph.
Proof. Let n ≥ 4. Write A = Aut(Γ) and S = {a, a−1, b}. It is easy to see that
Aut(D2n, S) = ⟨α⟩, where α is the automorphism of D2n induced by a 7→ a−1 and
b 7→ b.
Since S = S−1 and ⟨S⟩ = D2n, Γ is a connected cubic graph of order 2n. Clearly,
Γ ∼= Cn ×K2 is the ladder graph of order 2n, where Cn is the cycle of length n and K2 is
the complete graph of order 2 (the Cartesian product Γ1×Γ2 of two graphs Γ1 and Γ2 have
vertex set {(u1, u2) | ui ∈ V (Γi), i = 1, 2} and edge set {{(u1, u2), (v1, v2)} | either u1 =
v1 and (u2, v2) ∈ E(Γ2), or u2 = v2 and (u1, v1) ∈ E(Γ1)}). Note that C4 ∼= K2 ×K2,
and for n ≥ 5, Cn and K2 are relatively prime, that is, Cn cannot be a Cartesian product of
K2 and a graph because Cn contains no cycle of length 4 (see [23, Lemma 6.3]).
Assume n = 4. Then Γ ∼= C4 ×K2 ∼= K2 ×K2 ×K2 ∼= K4,4 − 4K2, the complete
bipartite graph K4,4 minus one factor. Then A ∼= S4 × C2, where S4 is the symmetric
group of degree 4. Since A ̸= R(D8)⋊ Aut(D8, S) = R(D8)⋊ ⟨α⟩, Γ is non-normal.
Assume that n ≥ 5. Since Cn and K2 are relatively prime, by [23, Corollary 6.12] we
have A ∼= Aut(Cn) × Aut(K2) ∼= D2n × Z2. It follows that |A : R(D2n)| = 2, and so
R(D2n)⊴A, that is, Γ is a normal Cayley graph. This completes the proof of part (1).
To prove that Γ is not a CI-graph for n ≥ 4 and n even, by the well-known Babai
criterion (see [4]), we only need to show that A has a regular dihedral subgroup, which is
not conjugate to R(D2n) in A.
Note that R(a)R(b) = R(a−1), R(b)α = R(b) and R(a)α = R(a−1). Thus, R(b)α is
an involution and R(a)(R(b)α) = R(a−1)α = R(a), that is, R(a) commutes with R(b)α.
Since R(a) has order n and n is even, R(ab)α = R(a)(R(b)α) has order n. Furthermore,
(R(ab)α)R(b) = R(ab)R(b)α = R(ba)α = αR(ba)α = αR(ba−1) = (R(ab)α)−1.
Thus, ⟨R(ab)α,R(b)⟩ is a dihedral group of order 2n.
Note that (R(ab)α)2 = R(ab)αR(ab)α = R(ab)R(ab)α = R(ab)R(a−1b) = R(a2).
Clearly, R(a2) has order n/2 and ⟨R(a2)⟩ ⊴ ⟨R(ab)α,R(b)⟩. Since ⟨R(a)⟩ is semireg-
ular on D2n with two orbits, ⟨R(a2)⟩ has four orbits on D2n, that is, ⟨a2⟩, a⟨a2⟩, b⟨a2⟩
and ba⟨a2⟩. The involution R(b) interchanges ⟨a2⟩ and b⟨a2⟩, and a⟨a2⟩ and ba⟨a2⟩. Fur-
thermore, R(ab)α interchanges ⟨a2⟩ and ba⟨a2⟩, and a⟨a2⟩ and b⟨a2⟩. It follows that
⟨R(ab)α,R(b)⟩ is transitive on D2n, and hence regular as |⟨R(ab)α,R(b)⟩| = 2n.
By MAGMA [7], ⟨R(ab)α,R(b)⟩ is not conjugate to R(D8) for n = 4, and so Γ is
not a CI-graph. Let n ≥ 5. Suppose Γ is a CI-graph. By part (1), Γ is normal, and by
Proposition 2.3, R(D2n) = ⟨R(ab)α,R(b)⟩, forcing α ∈ R(D2n), a contradiction. Thus,
Γ is not a CI-graph.
Now we are ready to prove Theorem 1.1.
J.-H. Xie et al.: Normal Cayley digraphs of dihedral groups with the CI-property 623
Proof of Theorem 1.1. Let n ≥ 2. First we prove (1) and (3) are equivalent, that is, D2n
is an NDCI-group if and only if either n = 2, 4 or n is odd. The necessity follows from
Lemma 4.1.
To prove the sufficiency, let n be odd, or n = 2, 4, and we only need to prove that
D2n is an NDCI-group. Let Γ = Cay(D2n, S) be a normal Cayley digraph. It suf-
fices to show that Γ is a CI-digraph. Note that we use the notations or formulae from
Equations (3.1) – (3.6). Let A = Aut(Γ). By Proposition 2.2, A = R(D2n)Aut(D2n, S) ≤
Hol(D2n) and A1 = Aut(D2n, S) ≤ Aut(D2n).
Assume n = 2. Then D4 ∼= C2 × C2, and D4 is an NDCI-group because C2 × C2
is a DCI-group. Assume n = 4. By Lemma 4.1, D8 is a non-DCI-group. However,
with the help of MAGMA [7], one may easily check that D8 is an NDCI-group; it also can
be proved by restricting the valency of Γ to be no more than 3 because the complement
of Γ is isomorphic to Cay(D8,D8\(S ∪ {1})) and by using the fact |A| is a divisor of
|Hol(D8)| = 64.
Assume that n is odd. Without loss of generality, we may further assume that p1 >
p2 > · · · > pm, where pi’s are all prime factors of n. For 1 ≤ i ≤ m, we may set
πi = {p1, p2, . . . , pi}. Recall that A = R(D2n)Aut(D2n, S) ≤ Hol(D2n) and A1 =
Aut(D2n, S) ≤ Aut(D2n). Since Cn is characteristic in D2n, A1 fixes Cn setwise. By
Lemma 3.1(1), ⟨θa⟩ is the kernel of Aut(D2n) on Cn, and therefore, the kernel of A1 on
Cn is ⟨θa⟩ ∩ A1. Then A1/(⟨θa⟩ ∩ A1) induces an subgroup of Aut(Cn), say B, and
|A1| = |⟨θa⟩ ∩ A1| · |B|. Note that Aut(Cn) is viewed as a subgroup of Aut(D2n), and so
is B, too. Then A1 ≤ ⟨⟨θa⟩ ∩ A1, B⟩. On the other hand, ⟨θa⟩ ∩ A1 is characteristic in
⟨θa⟩, and hence normal in Aut(D2n), forcing that (⟨θa⟩ ∩ A1)B = (⟨θa⟩ ∩ A1) ⋊ B. It
follows that ⟨⟨θa⟩ ∩ A1, B⟩ = (⟨θa⟩ ∩ A1)⋊ B, and since |A1| = |(⟨θa⟩ ∩ A1)| · |B|, we
have A1 = (⟨θa⟩ ∩A1)⋊B. Thus,
A = R(D2n)⋊A1, A1 = (⟨θa⟩ ∩A1)⋊B with B ≤ Aut(Cn).
Let G be a regular subgroup of A such that G ∼= D2n. To prove that Γ is a CI-digraph,
by Proposition 2.3 it suffices to show that G = R(D2n). We argue by contradiction, and
we suppose that G ̸= R(D2n).
By Lemma 3.2(2),
⟨R(b)⟩ × Aut(Cn)2 = ⟨R(b)⟩ × Aut(Cpr11 )2 × Aut(Cpr22 )2 × · · · × Aut(Cprmm )2
is a Sylow 2-subgroup of Hol(D2n), denote by HD2.
Let B2 be a Sylow 2-subgroup of B. Then we have B2 ≤ Aut(Cn)2, and hence
⟨R(b)⟩ ×B2 is a Sylow 2-subgroup of A. Since Sylow 2-subgroups of A are conjugate by
the second Sylow theorem and |G2| = 2, there is d ∈ A such that G∩ (⟨R(b)⟩×B2)d ̸= 1.
Thus, Gd
−1 ∩ (⟨R(b)⟩ × B2) ̸= 1. Write H = Gd
−1
. Then H ≤ A is regular on V (Γ),
and since R(D2n)⊴A and R(D2n) ̸= G, we have
H ̸= R(D2n) and |H ∩HD2| = 2.
By Equation (3.6) and Lemma 3.2(1),
Hol(D2n) = R(D2n)⋊ (⟨θa⟩⋊ Aut(Cn)) = ((⟨R(a)⟩ × ⟨θa⟩)⋊ ⟨R(b)⟩)⋊ Aut(Cn).
Since H ∼= D2n, we may assume that
H = ⟨v, w⟩ ∼= D2n, o(v) = n, o(w) = 2, vw = v−1, and w ∈ HD2.
624 Ars Math. Contemp. 23 (2023) #P4.08 / 615–629
Claim 1. v ∈ ⟨R(a)⟩ × ⟨θa⟩.
Proof of Claim 1. Since o(v) = n, we have o(vn/p
ri
i ) = prii . Write T0 = {1}, T1 =
{1, vn/p
r1
1 }, T2 = {1, vn/p
r1
1 , vn/p
r2
2 }, . . ., Tm = {1, vn/p
r1
1 , vn/p
r2
2 , . . . , vn/p
rm
m }.
Clearly, ⟨v⟩ = ⟨Tm⟩. To finish the proof of Claim 1, it suffices to show that Tm ⊆
⟨R(a)⟩ × ⟨θa⟩. To do this, we proceed by induction on k to show Tk ⊆ ⟨R(a)⟩ × ⟨θa⟩,
where 0 ≤ k ≤ m.
Clearly, T0 ⊆ ⟨R(a)⟩ × ⟨θa⟩, and we may let k > 0. By induction hypothesis, we may
assume that Tj ⊆ ⟨R(a)⟩ × ⟨θa⟩ for all 0 ≤ j < k and aim to show Tk ⊆ ⟨R(a)⟩ × ⟨θa⟩.
Since Tk = Tk−1 ∪ {vn/p
rk
k }, we only need to show that vn/p
rk
k ∈ ⟨R(a)⟩ × ⟨θa⟩.
Since o(vn/p
rk
k ) = prkk , v
n/p
rk
k is a πk-element. By Lemma 3.2(2), vn/p
rk
k ∈ (⟨R(a)⟩×
⟨θa⟩)⋊ (Aut(Cpr11 )πk × · · · × Aut(Cprkk )πk). Then we may write v
n/p
rk
k = xβ1β2 . . . βk,
where x ∈ ⟨R(a)⟩ × ⟨θa⟩ and βj ∈ Aut(Cprjj )πk for 1 ≤ j ≤ k.
Clearly, v commutes with every element in Tk−1, and so does vn/p
rk
k . Since ⟨R(a)⟩ ×
⟨θa⟩ is abelian, x commutes with every element in Tk−1 because Tk−1 ⊆ ⟨R(a)⟩ × ⟨θa⟩.
For every 1 ≤ ℓ ≤ k − 1, by Lemma 3.2(3) we have that if j ̸= ℓ then βj ∈ Aut(Cprjj )πk
commutes with every element of order prℓℓ in Tk−1, and then βℓ commutes with every
element of order prℓℓ in Tk−1 because βℓ = β
−1
ℓ−1 . . . β
−1
1 x
−1vn/p
rk
k β−1k . . . β
−1
ℓ+1, which
implies that βℓ commutes with an element of order prℓℓ in ⟨R(a)⟩×⟨θa⟩ as Tk−1 ⊆ ⟨R(a)⟩×
⟨θa⟩. Again by Lemma 3.2(3), we obtain βℓ = 1.
It follows vn/p
rk
k = xβk = R(y)θzβk for some y, z ∈ ⟨a⟩ and βk ∈ Aut(Cprkk )πk .
Since R(y) ∈ A, we have θzβk ∈ A, so θzβk ∈ A1 = (⟨θa⟩ ∩ A1) ⋊ B, where B ≤
Aut(Cn). Then there exist θz′ ∈ ⟨θa⟩ ∩ A1 and βk′ ∈ B such that θzβk = θz′βk′ . It
follows that θ−1z′ θz = βk′β
−1
k ∈ ⟨θa⟩ ∩ Aut(Cn) = 1, that is, θz = θz′ ∈ ⟨θa⟩ ∩ A1 ≤ A1
and βk = βk′ ∈ B ≤ A1.
Suppose βk ̸= 1. Recall that p1 > p2 > · · · > pk, πk = {p1, p2, . . . , pk} and
Aut(Cprkk )
∼= Z
p
rk−1
k (pk−1)
. Then every πk-element in Aut(Cprkk ) is a pk-element, and
since βk ∈ Aut(Cprkk )πk , o(βk) has order pk-power. Let β be the automorphism of D2n
induced by
β : ak 7→ a
p
rk−1
k +1
k , b 7→ b, ai 7→ ai for every i ̸= k.
Let L = ⟨β⟩. Then L is the unique subgroup of order pk of ⟨βk⟩. It follows that rk ≥ 2
and L ≤ ⟨βk⟩ ≤ A1 = Aut(D2n, S). Write
M = ⟨a1, . . . , ak−1, apkk , ak+1, . . . , am⟩, and K = ⟨a
p
rk−1
k
k ⟩.
Clearly, 1 < K ⊴ D2n. Note that every element of D2n has the form (a1a2 · · · am)t or
b(a1a2 · · · am)t for some t ∈ Zn. It is easy to see that the set of fixed-points of L in D2n
is M ∪ bM , that is, FD2n(L) = M ∪ bM . Furthermore, L is transitive on Kc for every
c ∈ D2n\(M ∪ bM), and |D2n : FD2n(L)| ≥ pk > 2 as pk is odd. By Proposition 2.4(1),
Γ is non-normal, a contradiction.
Then βk = 1. It follows that vn/p
rk
k = R(y)θzβk = R(y)θz ∈ ⟨R(a)⟩ × ⟨θa⟩ and
Tk ⊆ ⟨R(a)⟩ × ⟨θa⟩, as required. This completes the proof of Claim 1.
By Claim 1, we may assume that v = R(a)kθℓa for some k, ℓ ∈ Zn. Since θa fixes ⟨a⟩
pointwise, we have 1⟨v⟩ = ⟨ak⟩, the orbit of ⟨v⟩ containing 1 in D2n, and since ⟨v⟩ ≤ H is
J.-H. Xie et al.: Normal Cayley digraphs of dihedral groups with the CI-property 625
semiregular, |1⟨v⟩| = |⟨ak⟩| = o(v) = n, forcing o(ak) = n. Thus, ⟨ak⟩ = ⟨a⟩ and hence
(k, n) = 1. Since v ∈ H , if necessary we replace v by a power of v, and then one may let
v = R(a)θℓa for some ℓ ∈ Zn.
Recall that w ∈ H with o(w) = 2 and w ∈ HD2 = ⟨R(b)⟩ × Aut(Cn)2. If w ∈
Aut(Cn)2 = Aut(Cpr11 )2 × Aut(Cpr22 )2 × · · · × Aut(Cprmm )2, then w fixes 1, contradicting
the regularity of H . Thus, w ∈ R(b)Aut(Cn)2, and by Lemma 3.2(2), we have
w = R(b)ε1ε2 . . . εm, where εi ∈ Aut(Cprii )2 and ε
2
i = 1 for every 1 ≤ i ≤ m.
Claim 2. For every 1 ≤ k ≤ m, either prkk | ℓ and εk = 1, or (ℓ, pk) = 1 and εk ̸= 1.
Proof of Claim 2. Write ε = ε1 . . . εm ∈ Aut(Cn). Since H ∼= D2n, we have vR(b)ε =
vw = v−1 = R(a−1)θa−ℓ . By Lemma 3.2(1), we get
θR(b)a = θaR(b)
θaR(b) = θaR(b
θab) = θaR(a
−1) and (θℓa)
R(b) = θℓaR(a
−ℓ)
= R(a−ℓ)θaℓ .
Since vR(b) = (v−1)ε = R((aε)−1)θ(aε)−ℓ , we have
R(a−(ℓ+1))θaℓ = R(a
−1)R(a−ℓ)θaℓ = R(a)
R(b)(θℓa)
R(b) = vR(b) = R((aε)−1)θ(aε)−ℓ .
Since ⟨R(a)⟩ ∩ ⟨θa⟩ = 1, we deduce R(aℓ+1) = R(aε) and θaℓ = θ(aε)−ℓ . It follows that
aε = aℓ+1 and (aε)ℓ = a−ℓ. This yields aℓ(ℓ+2) = 1, so n | ℓ(ℓ + 2) and prkk | ℓ(ℓ + 2)
for every 1 ≤ k ≤ m. If pk | ℓ and pk | ℓ + 2, then pk | 2, which is impossible because n
is odd. It follows that either prkk | ℓ and (pk, ℓ+ 2) = 1, or p
rk
k | (ℓ+ 2) and (pk, ℓ) = 1.
Assume prkk | ℓ and (pk, ℓ+ 2) = 1. Since aε = aℓ+1, we have
aε = (a1a2 . . . am)
ε = aε11 a
ε2
2 . . . a
εm
m = a
ℓ+1 = aℓ+11 · a
ℓ+1
k−1aka
ℓ+1
k+1 . . . a
ℓ+1
m ,
implying aεkk = ak, that is, εk = 1.
Assume prkk | (ℓ + 2) and (pk, ℓ) = 1. Since (aε)ℓ = a−ℓ, we have (aℓk)εk = (aℓk)−1,
and so εk ̸= 1 because (pk, ℓ) = 1 implies o(aℓk) = p
rk
k . This completes the proof of
Claim 2.
If prkk | ℓ for all 1 ≤ k ≤ m, by Claim 2 we have εk = 1, and hence v = R(a)θℓa =
R(a) and w = R(b)ε1ε2 . . . εm = R(b), yielding H = R(D2n), a contradiction. Thus,
εk ̸= 1 for some k. To simplify notation, from now on we do not assume p1 > p2 > · · · >
pm any more, which has no confusion. Then we may assume that there exists 1 ≤ s ≤ m
such that (ℓ, p1 . . . ps) = 1, εj ̸= 1 for all 1 ≤ j ≤ s, prs+1s+1 . . . prmm | ℓ, and εj = 1 for
all s + 1 ≤ j ≤ m. It follows that o(aℓ) = pr11 . . . prss , forcing ⟨aℓ⟩ = ⟨a1a2 . . . as⟩ and
⟨θℓa⟩ = ⟨θa1a2...as⟩. Let L = ⟨θa1a2...as⟩.
Recall that v = R(a)θℓa ∈ H ≤ A. It follows from R(D2n) ≤ A that θℓa ∈ A, implying
θℓa ∈ A1 = Aut(D2n, S). Then L = ⟨θℓa⟩ ≤ Aut(D2n, S). Let K = ⟨a1a2 . . . as⟩.
Note that every coset of K has the form K(as+1 · · · am)t or Kb(as+1 · · · am)t with t ∈
Z
p
rs+1
s+1 ...p
rm
m
. Since θa1a2...as is the automorphism of D2n induced by a 7→ a and b 7→
ba1a2 . . . as, we have that FD2n(L) = ⟨a⟩, and for every coset of Kc in b⟨a⟩, L is transitive
on Kc. Clearly, K ⊴D2n as K is characteristic in ⟨a⟩, and |D2n : FD2n(L)| = 2.
626 Ars Math. Contemp. 23 (2023) #P4.08 / 615–629
Write γ = ε1 . . . εs. Then γ ̸= 1 and γ is the automorphism of D2n induced by
aγi = a
−1
i for 1 ≤ i ≤ s, a
γ
j = aj for s+ 1 ≤ j ≤ m, b
γ = b.
Then γ fixes each coset of K in D2n setwise, and FD2n(⟨γ⟩) ̸= ⟨a⟩ = FD2n(L). Noting
that w = R(b)ε1ε2 . . . εsεs+1 . . . εm = R(b)γ ∈ H ≤ A and R(b) ∈ R(D2n) ≤ A, we
have γ ∈ A, and hence γ ∈ A1 = Aut(D2n, S). By Proposition 2.4(3), Γ is non-normal, a
contradiction.
Thus, εk = 1 for all 1 ≤ k ≤ m. By Claim 2, ε = ε1 . . . εm = 1 and n =
pr11 p
r2
2 · · · prmm | ℓ. It follows that v = R(a)θℓa = R(a) and w = R(b)ε = R(b), im-
plying H = ⟨v, w⟩ = ⟨R(a), R(b)⟩ = R(D2n), a contradiction. This completes the proof
of the the equivalence of (1) and (3).
Now we are ready to finish the proof of Theorem 1.1 by proving that (2) and (3) are
equivalent, that is, D2n is an NCI-group if and only if either n = 2, 4 or n is odd. The
sufficiency follows from the fact that an NDCI-group is an NCI-group, and the necessity
follows from Lemma 4.1.
Proof of Corollary 1.2. Let the dihedral group D2n be a DCI-group. Then D2n is an
NDCI-group. By Theorem 1.1, n is 2, 4 or odd. By Lemma 4.1, D8 is a non-CI-group
and so a non-DCI-group. If n is odd, by [13, Theorem 1.2], n is square-free. Thus, n = 2
or n is odd-square-free. Now let D2n be a CI-group. Then D2n is an NCI-group. Again by
Theorem 1.1, either n is 2, 4, or n is odd. By [13, Theorem 1.2], if n is odd, then n = 9 or n
is square-free. Since D8 is a non-CI-group, we have n = 2, 9, or n is odd-square-free.
ORCID iDs
Jin-Hua Xie https://orcid.org/0000-0002-6547-0885
Yan-Quan Feng https://orcid.org/0000-0003-3214-0609
Jin-Xin Zhou https://orcid.org/0000-0002-8353-896X
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P4.09 / 631–648
https://doi.org/10.26493/1855-3974.2815.1e7
(Also available at http://amc-journal.eu)
On cubic bi-Cayley graphs of p-groups*
Na Li
Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China
Young Soo Kwon †
Department of Mathematics, Yeungnam University, 280 Daehak-ro, Gyeongsan,
Gyeongbuk 38541, Korea
Jin-Xin Zhou ‡
Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China
Received 26 January 2022, accepted 27 April 2023, published online 21 June 2023
Abstract
A graph is called a Cayley graph (or bi-Cayley graph, respectively) of a group G if
it has a group G of automorphisms acting semiregularly on the vertices with exactly one
orbit (or two orbits, respectively). It is known every Cayley graph is vertex-transitive. In
this paper, we first present a classification of connected cubic non-Cayley vertex-transitive
bi-Cayley graphs of a finite p-group H , where p > 3 is a prime and the derived subgroup
of H is either cyclic or isomorphic to Zp × Zp. This is then used to give a classification of
connected cubic non-Cayley vertex-transitive graphs of order 2p4 for each prime p.
Keywords: Bi-Cayley graph, vertex-transitive, non-Cayley.
Math. Subj. Class. (2020): 05C25, 20B25
*The authors are grateful to the referees for helpful remarks and suggestions. This work was partially supported
by the Fundamental Research Funds for the Central Universities (2022JBCG003), the National Natural Science
Foundation of China (12071023, 12161141005).
†The second author was supported by the Basic Science Research Program through the National Research
Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B05048450).
‡Corresponding author.
E-mail addresses: 20118007@bjtu.edu.cn (Na Li), ysookwon@ynu.ac.kr (Young Soo Kwon),
jxzhou@bjtu.edu.cn (Jin-Xin Zhou)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
632 Ars Math. Contemp. 23 (2023) #P4.09 / 631–648
1 Introduction
In this paper, we shall describe an investigation of non-Cayley vertex-transitive bi-Cayley
graphs of finite p-groups. To explain this, we first introduce some terminology.
Let G be a permutation group on a set Ω and α ∈ Ω. The stabilizer of α in G is the
subgroup Gα = {g ∈ G | αg = α}. If Gα = 1 for every α ∈ Ω, then G is said to be
semiregular on Ω, and if G is transitive and semiregular on Ω, then we say that G is regular
on Ω.
Let Γ be a finite simple connected graph with vertex set V (Γ) and E(Γ). We use Aut(Γ)
to denote the full automorphism group of Γ. Note that Aut(Γ) is a permutation group on
V (Γ). A graph Γ is vertex-transitive if Aut(Γ) is transitive on V (Γ), and a vertex-transitive
graph Γ is called non-Cayley vertex-transitive if Aut(Γ) has no regular subgroups.
Let R,L and S be subsets of a group H such that R = R−1, L = L−1 and R ∪ L
does not contain the identity element of H . The bi-Cayley graph BiCay(H,R,L, S) of H
relative to R,L, S is a graph having vertex set the union of the following two copies of H:
H0 = {h0 | h ∈ H} and H1 = {h1 | h ∈ H},
and edge set
{{h0, g0} | gh−1 ∈ R} ∪ {{h1, g1} | gh−1 ∈ L} ∪ {{h0, g1} | gh−1 ∈ S}.
In the study of bi-Cayley graphs, much work has been devoted to construct and clas-
sify non-Cayley vertex-transitive graphs, see, for example, [19, 20, 23, 25]. In [25], the
following problem was proposed:
Problem A ([25, Problem 1]). Characterize cubic non-Cayley vertex-transitive bi-Cayley
graphs of a p-group for an odd prime p.
In [25], Problem A was partially solved for cubic bi-Cayley graphs of regular p-groups.
In this paper, we make a progress towards Problem A by classifying non-Cayley vertex-
transitive bi-Cayley graphs of a p-group H such that the derived group of H is cyclic or
isomorphic to Zp × Zp, where p > 3 is prime. The following is our first main result.
Theorem 1.1. Let p > 3 be a prime and let H be a finite p-group such that the derived
subgroup of H is either cyclic or isomorphic to Zp × Zp. Let Γ be a connected cubic bi-
Cayley graph of H . Then Γ is non-Cayley vertex-transitive if and only if Γ is isomorphic
to one of the graphs given in Constructions I – V (see Section 4).
Note that every cubic non-Cayley vertex-transitive graph of order 2p or 2p2 (p a prime)
is a generalized Petersen graph, see [13, 21]. In [25], all connected cubic non-Cayley
vertex-transitive graphs of order 2p3 were classified for each prime p. Applying the above
theorem, our next main theorem gives a classification of connected cubic non-Cayley
vertex-transitive graphs of order 2p4 for each prime p.
Theorem 1.2. Let p be a prime. Then a connected cubic graph Γ of order 2p4 is a non-
Cayley vertex-transitive graph if and only if Γ is isomorphic to Γ(p,m, n, s; t)(m+n+3 =
4,m > n ≥ 0, s ≥ 0), Γ(p, 1; k, i) or ∆(p, 1; k) (see Section 4 for the construction of
these graphs).
N. Li et al.: On cubic bi-Cayley graphs of p-groups 633
2 Preliminaries
We first introduce some notation about groups. For a positive integer, let Zn be the cyclic
group of order n and Z∗n be the multiplicative group of Zn consisting of numbers coprime
to n. A semidirect product of a group N by a group M is denoted by N ⋊ M . If H is a
subgroup of a group G, the centralizer and normalizer of H in G are denoted by CG(H)
and NG(H), respectively. The automorphism group, the center, the derived subgroup and
the Frattini subgroup of a group G will be represented by Aut(G), Z(G), G′ and Φ(G),
respectively.
2.1 Cayley graphs
Given a finite group G and a self-inverse subset S ⊆ G \ {1}, the Cayley graph Γ =
Cay(G,S) on G relative to S is a graph with vertex set G and edge set {{g, sg} | g ∈
G, s ∈ S}. Let Aut(G,S) := {α ∈ Aut(G)|Sα = S}. Then Aut(G,S) ≤ Aut(Γ)1.
For any g ∈ G, R(g) is the permutation of G defined by R(g) : x 7→ xg for x ∈ G.
Set R(G) := {R(g) | g ∈ G}. It is well-known that R(G) is a regular subgroup of
Aut(Cay(G,S)). By [9, Lemma 2.1], we have NAut(Γ) = R(G) ⋊ Aut(G,S). Clearly,
every Cayley graph is vertex-transitive. In general, we have the following proposition.
Proposition 2.1 ([1, Lemma 16.3]). A graph Γ is isomorphic to a Cayley graph of a group
G if and only if Aut(Γ) has a subgroup isomorphic to G, acting regularly on the vertex set
of Γ.
The next two propositions about Cayley graphs will be used in the sequel.
Proposition 2.2 ([22, Theorem 1.1]). Let G be a finite p-group with an odd prime p, and
let Γ = Cay(G,S) be connected and of valency 4. Assume that Γ is X-edge-transitive,
where G ≤ X ≤ Aut(Γ). Then either G is normal in X , or X has a normal subgroup R
such that R ≤ G, and Γ is a normal cover of ΓR, and moreover, the quotient graph ΓR
and the quotient group X̄ satisfy one of the following conditions (1) and (2) (Here, for any
R ≤ Y ≤ X and x ∈ X , let Ȳ = Y/R and x̄ = xR.):
(1) ΓR ∼= K5(the complete graph of order 5) and A5 ≤ X̄ ≤ S5.
(2) X̄ has a unique minimal normal subgroup N̄ ∼= Znp with n = pm such that
(i) Ḡ = N̄ ⋊ M̄ , where M̄ ∼= Zpm with m ⩾ 1,
(ii) X̄ = N̄ ⋊ ((H̄⋊M̄).Ō) and X̄1Ḡ = H̄.Ō, where H̄ ∼= Z
l
2 with 2 ≤ l ≤ n and
Ō ∼= Zt with t = 1 or 2, such that there exist b̄1, ..., b̄n ∈ N̄ and h̄1, ..., h̄n ∈
H̄ such that N̄ = ⟨b̄1, ..., b̄n⟩, ⟨b̄i, h̄i⟩ ∼= D2p and H̄ = ⟨h̄i⟩ × CH̄(b̄i) for
1 ≤ i ≤ n.
A graph Γ is called symmetric if Aut(Γ) is transitive on the arcs of Γ. By [8, Corol-
lary 3.4], we have the following result.
Proposition 2.3. Every connected cubic symmetric graph of order 2pn is a Cayley graph
whenever n ≥ 1 and p ≥ 7 is a prime.
634 Ars Math. Contemp. 23 (2023) #P4.09 / 631–648
2.2 Basic properties of bi-Cayley graphs
In this subsection, we let Γ be a connected bi-Cayley graph BiCay(H,R,L, S) of a group
H . The following lemma provides some basic properties of Γ (see [24, Lemma 3.1]).
Lemma 2.4. The following hold.
(1) H is generated by R ∪ L ∪ S.
(2) Up to graph isomorphism, S can be chosen to contain the identity of H .
For each g ∈ H , let
R(g) : hi 7→ (hg)i, ∀i ∈ Z2, h ∈ H. (2.1)
Set R(H) = {R(g) | g ∈ H}. Then R(H) is a semiregular subgroup of Aut(Γ) with H0
and H1 as its two orbits. In general, a graph Γ is isomorphic to a bi-Cayley graph of a
group H if and only if Γ admits a group of automorphisms which is isomorphic to H and
acts semiregularly on the vertices with two orbits.
For α ∈ Aut(H) and x, y, g ∈ H , let
δα,x,y : h0 7→ (xhα)1, h1 7→ (yhα)0, ∀h ∈ H,
σα,g : h0 7→ (hα)0, h1 7→ (ghα)1, ∀h ∈ H.
(2.2)
Set
I = {δα,x,y | α ∈ Aut(H) s.t. Rα = x−1Lx, Lα = y−1Ry, Sα = y−1S−1x},
F = {σα,g | α ∈ Aut(H) s.t. Rα = R, Lα = g−1Lg, Sα = g−1S}.
(2.3)
The following theorem determines the normalizer of R(H) in Aut(Γ).
Theorem 2.5 ([24, Theorem 1.1]). Let Γ = BiCay(H,R,L, S) be a connected bi-Cayley
graph of a group H . Then NAut(Γ)(R(H)) = R(H)⋊F if I = ∅ and NAut(Γ)(R(H)) =
R(H)⟨F, δα,x,y⟩ if I ̸= ∅ and δα,x,y ∈ I . Furthermore, if I ̸= ∅, then for any δα,x,y ∈ I ,
we have
(1) ⟨R(H), δα,x,y⟩ acts transitively on V (Γ);
(2) if α has order 2 and x = y = 1, then Γ is isomorphic to the Cayley graph Cay(H̄,
R ∪ αS), where H̄ = H ⋊ ⟨α⟩.
2.3 Normal covers
Let Γ be a graph. Assume that G ≤ Aut(Γ) is vertex-transitive on Γ. Let N be a normal
subgroup of G such that N is intransitive on V (Γ). The normal quotient graph ΓN of Γ
relative to N is defined as the graph with vertices the orbits of N on V (Γ) and with two
different orbits adjacent if there exists an edge in Γ between the vertices lying in those two
orbits. If ΓN and Γ have the same valency, then we say that Γ is a normal N -cover of ΓN .
By [12, Lemma 2.5], we have the following lemma.
Lemma 2.6. Let Γ be a connected cubic graph such that G ≤ Aut(Γ) is arc-transitive on
Γ. Suppose that there exists N G such that N has at least three orbits on V (Γ). Then
N. Li et al.: On cubic bi-Cayley graphs of p-groups 635
(1) Γ is a normal N -cover of the quotient graph ΓN of Γ relative to N .
(2) N acts semiregularly on V (Γ), N is the kernel of G acting on V (ΓN ) and G/N ≤
Aut(ΓN ).
(3) G/N is arc-transitive on ΓN .
3 Cubic bi-Cayley graphs of groups of odd order
In this section, we shall give a description of cubic bi-Cayley graphs of groups of odd order,
and prove a sufficient condition for such graphs being non-Cayley vertex-transitive.
Lemma 3.1. Let Γ be a connected cubic non-Cayley vertex-transitive graph. Let A ≤
Aut(Γ) be vertex- but not arc-transitive on Γ. Take any vertex v of Γ. Then there exists a
unique neighbor v∗ such that Av = Av∗ . In particular, {v, v∗} is a block of imprimitivity
of A on V (Γ).
Proof. Since Γ is not a Cayley graph, one has Av > 1. Assume that |Av| is divisible by
p, where p > 3 is an odd prime. Then Av contains an element α of order p. Note that
each orbit of ⟨α⟩ has length either 1 or p. Since ⟨α⟩ fixes v and Γ has valency 3, the con-
nectedness of Γ implies that each orbit of ⟨α⟩ has length 1, and so α = 1, a contradiction.
Since Γ is not arc-transitive, it follows that 3 ∤ |Av|. Thus Av is a nontrivial 2-group for
every v ∈ V (Γ). Let A∗v be the kernel of Av acting on the neighborhood of v in Γ. Then
Av/A
∗
v ≤ Z2. If Av/A∗v = 1, then Av = A∗v fixes all neighbors of v, and then since Γ is
connected and A is vertex-transitive on Γ, Av would fix all vertices of Γ, forcing Av = 1,
a contradiction. Thus, Av/A∗v ∼= Z2. So there exists a unique neighbor, say v∗, of v such
that Av = Av∗ .
For any g ∈ A, it is easily verified that Avg = Agv = A
g
v∗ = A(v∗)g . It follows
that either {v, v∗} = {vg, (v∗)g} or {v, v∗} ∩ {vg, (v∗)g} = ∅. So {v, v∗} is a block of
imprimitivity of A on V (Γ).
Definition 3.2. Let Γ be a connected cubic non-Cayley vertex-transitive graph. Let A ≤
Aut(Γ) be vertex- but not arc-transitive on Γ. Set
M(Γ) = {{v, v∗} | v, v∗ ∈ V (Γ), v ∼ v∗, Av = Av∗}.
The matching quotient graph ΓM of Γ is the graph with vertex set M(Γ) and with two
elements in M(Γ) adjacent in ΓM if there exists an edge of Γ between them.
In the above definition, it is easy to see that M(Γ) is A-invariant and A induces a
group of automorphisms of ΓM. By [14, Lemma 9 & Theorem 10], we have the following
lemma.
Lemma 3.3. Let Γ be a connected cubic non-Cayley vertex-transitive graph, and let A ≤
Aut(Γ) be vertex- but not arc-transitive on Γ. Then ΓM is a connected tetravalent A-arc-
transitive graph, that is, A acts faithfully on M(Γ) and is arc-transitive ΓM.
The following proposition gives a description of connected cubic bi-Cayley graphs of
groups of odd order.
636 Ars Math. Contemp. 23 (2023) #P4.09 / 631–648
Proposition 3.4. Let Γ = BiCay(H,R,L, S) be a connected cubic bi-Cayley graph of
a group H of odd order. Let A ≤ Aut(Γ) be such that R(H) ≤ A. If Γ is a non-
Cayley vertex-transitive graph and A is vertex- but not arc-transitive on Γ, then one of the
following holds.
(1) Γ ∼= BiCay(H, ∅, ∅, {1, x, y}), and ΓM ∼= Cay(H, {x, y, x−1, y−1}), where H =
⟨x, y⟩. Furthermore, H has no automorphism α such that xα = x−1 and yα = y−1.
In particular, R(H) is not normal in A.
(2) Γ ∼= BiCay(H, {a, a−1}, {b, b−1}, {1}), and ΓM ∼= Cay(H, {a, b, a−1, b−1}), where
H = ⟨a, b⟩. Furthermore, H has no automorphism α such that aα = a−1 and
bα = b−1.
Proof. Assume that Γ is a non-Cayley vertex-transitive graph. Clearly, Γ has 2|H| vertices
as Γ is a bi-Cayley of the group H . Since A ≤ Aut(Γ) is vertex- but not arc-transitive on
Γ, by Lemma 3.3, A acts faithfully on M(Γ) (see Definition 3.2). As H has odd order, for
any {u, u∗} ∈ M(Γ), we have |{u, u∗} ∩Hi| = 1 with i = 1, 2, where Hi(i = 0, 1) are
the two orbits of R(H) on V (Γ). It follows that R(H) acts regularly on M(Γ), and so
ΓM is a Cayley graph of H . By Lemma 2.4, we may assume that S contains the identity 1
of H . Then {10, 11} is an edge of Γ. Since H has odd order, we have |R| = |L| = 0 or 2.
If |R| = |L| = 0, then we may let S = {1, x, y}. Since Γ is connected, by Lemma 2.4
we have H = ⟨S⟩ = ⟨x, y⟩. We may assume that {10, s1} ∈ M(Γ) with s ∈ S. Then
A10 = As1 . Define a map on V (Γ) as follows:
L(s) : h0 7→ h0, h1 7→ (s−1h)1,∀h ∈ H.
It is easy to verify that L(s) is a permutation on V (Γ). Let
Γ′ = BiCay(H, ∅, ∅, s−1S).
For each edge {h0, g1} of Γ, we have gh−1 ∈ S and so s−1gh−1 ∈ s−1S. It fol-
lows that {h0, (s−1g)1} is an edge of Γ′. Clearly, both Γ and Γ′ have valency 3, so
L(s) is an isomorphism between Γ and Γ′. For any edge e of Γ′ and for any α ∈ A,
we have eL(s)
−1αL(s) ∈ E(Γ′). This implies that L(s)−1AL(s) ≤ Aut(Γ′). Let B =
L(s)−1AL(s). Since L(s) fixes 10, one has B10 = L(s)
−1A10L(s). It is easy to see
that L(s)−1As1L(s) = B11 . Since A10 = As1 , one has B10 = B11 . Then {11, 10} ∈
M(Γ′). Therefore, without loss of generality, we may assume that {11, 10} ∈ M(Γ).
Then we have M(Γ) = {{h0, h1} | h ∈ H}. By Lemma 3.3, ΓM has valency 4.
So {x0, x1}, {y0, y1}, {(x−1)0, (x−1)1}, {(y−1)0, (y−1)1} are just the four neighbors
of {11, 10} in ΓM. Clearly, {11, 10}R(h) = {h0, h1} for each h ∈ H , so we have
ΓM ∼= Cay(H, {x, y, x−1, y−1}).
If H has an automorphism, say α, such that xα = x−1 and yα = y−1, then α has order
2 and Sα = S−1, and then by Theorem 2.5, R(H)⋊ ⟨δα,1,1⟩ acts regularly on V (Γ). This
is contrary to the fact that Γ is a non-Cayley vertex-transitive graph.
By Lemma 3.3, ΓM is A-arc-transitive. If R(H) is normal in A, then Aut(H, {x, y,
x−1, y−1}) is transitive on {x, y, x−1, y−1}. So Aut(H) must have an involution inter-
changing the two pairs (x, y) and (x−1, y−1), which is impossible by the argument in the
above paragraph. Thus, we obtain part (1).
Now assume that |R| = |L| = 2. In this case, we must have M(Γ) = {{h0, h1} |
h ∈ H}. Assume that R = {a, a−1} and L = {b, b−1}. Again, by Lemma 3.3, ΓM
N. Li et al.: On cubic bi-Cayley graphs of p-groups 637
has valency 4. So {a0, a1}, {b0, b1}, {(a−1)0, (a−1)1}, {(b−1)0, (b−1)1} are just the four
neighbors of {11, 10} in ΓM. As {11, 10}R(h) = {h0, h1} for each h ∈ H , we have
ΓM ∼= Cay(H, {a, b, a−1, b−1}). If H has an automorphism, say α, such that aα = a−1
and bα = b−1, then α has order 2 and Rα = L, and then by Theorem 2.5, R(H)⋊ ⟨δα,1,1⟩
acts regularly on V (Γ). This is contrary to the fact that Γ is a non-Cayley vertex-transitive
graph. Then part (2) happens.
We now give a sufficient condition for a cubic bi-Cayley graph of a group of odd order
in (2) of Proposition 3.4 being non-Cayley vertex-transitive.
Theorem 3.5. Let Γ = BiCay(H, {a, a−1}, {b, b−1}, {1}) be a connected cubic bi-Cayley
of a group H of odd order, where a, b ∈ H . If Aut(H, {a, b, a−1, b−1}) ∼= Z4, then Γ is a
non-Cayley vertex-transitive graph.
Proof. Assume that Aut(H, {a, b, a−1, b−1}) ∼= Z4. Then H has an automorphism α
such that aα = b and bα = a−1. So α swaps {a, a−1} and {b, b−1}. By Theorem 2.5,
⟨R(H), δα,1,1⟩ acts transitively on V (Γ).
Suppose on the contrary that Γ is a Cayley graph. First, assume that Γ is symmetric.
Let N be the largest normal subgroup of Aut(Γ) contained in R(H). Note that each orbit
Hi(i = 1 or 2) of R(H) on V (Γ) induces a subgraph which is a union of cycles of
odd length. If N = R(H), then R(H) ⊴ Aut(Γ) and then each Hi will be a block of
imprimitivity of Aut(Γ) on V (Γ). Since Γ is symmetric and connected, it follows that H1 =
H2 = V (Γ), a contradiction. This implies that N < R(H) and Γ is non-bipartite. By
Lemma 2.6, the quotient graph ΓN of Γ relative to N is a cubic graph with Aut(Γ)/N as an
arc-transitive group of automorphisms. Clearly, R(H)/N ≤ Aut(Γ)/N acts semiregularly
on V (ΓN ) with two orbits. So ΓN is a bi-Cayley graph of R(H)/N . If R(H)/N has a
subgroup, say M/N , which is normal in Aut(ΓN ), then M/NAut(Γ)/N . Since N is the
largest normal subgroup of Aut(Γ) contained in R(H), it follows that M/N is trivial. Thus,
the only proper subgroup of R(H)/N which is normal in Aut(ΓN ) is the identity subgroup.
As ΓN has order 2|H|/|N | and |H| is odd, by [7, Theorem 7.1], either ΓN is 3-arc-regular
and has order 6, 10, 110 or 506, or ΓN is 4-arc-regular and has order 14, 506 or 2162. (One
may the definition of s-arc-regular graphs in [7, page 145].) Furthermore, since Γ is not
bipartite, by inspecting the list of cubic symmetric graphs of order at most 10000 [6], we
get that either ΓN ∼= F10 or F506A, or ΓN ∼= F2162A. However, by Magma [3], each of
F10,F506A,F2162A is a non-Cayley graph, a contradiction. (For an integer n, if there is
a unique cubic symmetric graph of order n up to graph isomorphism, then we use Fn to
denote this graph, and if there are more than one cubic symmetric graph of order n up to
graph isomorphism, then we use FnA, FnB, etc. to denote the corresponding graphs (see
[4]).)
Now assume that Γ is nonsymmetric. Then Aut(Γ)10 is a 2-group. Since |Aut(Γ)| =
2|H||Aut(Γ)10 |, R(H) is 2′-Hall subgroup of Aut(Γ). Since Γ is supposed to be a Cayley
graph, Aut(Γ) has a subgroup, say G, acting regularly on V (Γ). Then |G| = 2|H|, and
since |H| is odd, G is solvable, and so G has a 2′-Hall subgroup, say B. By [10, Theorem
A], R(H) and B are conjugate in Aut(Γ). Without loss of generality, we may assume
R(H) ≤ G. Then R(H)  G. Then {H0, H1} is G-invariant. As 11 is the unique
neighbor of 10 contained in H1, for each b ∈ G, either {10, 11}b = {10, 11} or {10, 11}b ∩
{10, 11} = ∅. Since G is regular on V (Γ), there exists b ∈ G mapping 10 to 11, and then
{10, 11}b = {10, 11}. It follows that b interchanges 10 and 11, and hence b is an involution.
638 Ars Math. Contemp. 23 (2023) #P4.09 / 631–648
By Theorem 2.5, there exists an automorphism, say α, of H and x, y ∈ H such that δα,x,y
swaps 10 and 11. By the definition of δα,x,y (see Eqs. 2.2–2.3), we have x = y = 1
and α swaps {a, a−1} and {b, b−1}. However, since Aut(H, {a, b, a−1, b−1}) ∼= Z4, the
only involution in Aut(H, {a, b, a−1, b−1}) must fix both {a, a−1} and {b, b−1} setwise, a
contradiction.
4 Five families of cubic VNC-graphs
In this section, we shall apply Theorem 3.5 to construct five families of cubic VNC graphs.
Construction I Let p be an odd prime, and m > n ≥ 0 and let t be an integer such that
t2 ≡ −1(mod pm−n). Let
H(p,m, n, s; t) = ⟨x, y, z | xp
m
= yp
m
= zp
s
= 1, z = [x, y], [x, z] =[y, z] = 1,
xp
n
= yp
n
⟩,
and Γ(p,m, n, s; t) = BiCay(H(p,m, n, s; t), {x, x−1}, {yt, y−t}, {1}).
Lemma 4.1. The graph Γ(p,m, n, s; t) is a connected cubic non-Cayley vertex-transitive
graph with 2pm+n+s vertices.
Proof. Let G = H(p,m, n, s; t) and Σ = Γ(p,m, n, s; t). Clearly, G′ = ⟨z⟩ ∼= Zps and
G/G′ = ⟨xG′, yG′⟩. By the definition of H(p,m, n, s; t), we have ⟨x⟩∩⟨z⟩ = ⟨y⟩∩⟨z⟩ =
1. Since xp
n
= yp
n
, one has G/G′ = ⟨xG′⟩ × ⟨xy−1G′⟩ ∼= Zpm × Zpn . So G has order
pm+n+s, and so Σ has 2pm+n+s vertices.
Let u = yt, v = x−r and w = [u, v], where r ∈ Z∗pm is such that rt ≡ 1(mod pm).
Note that t2 ≡ −1(mod pm−n). Clearly, upm = vpm = 1. Since z = [x, y] ∈ Z(G), one
has w = [u, v] = [yt, x−r] = [y, x]−tr = ztr, and so wp
s
= 1. Also, [u,w] = [v, w] = 1.
Since rt ≡ 1(mod pm) and t2 ≡ −1(mod pm−n), one has r = −t(mod pm−n). So upn =
ytp
n
= xtp
n
= x−rp
n
= vp
n
. Thus, u, v, w have the same relations as do x, y, z. So G has
an automorphism, say α, such that xα = yt and yα = x−r. Then (yt)α = x−tr = x−1,
and then α ∈ Aut(G, {x, x−1, yt, y−t}). To complete the proof, by Proposition 3.4, it
suffices to prove that Aut(G, {x, x−1, yt, y−t}) = ⟨α⟩.
By way of contradiction, assume that Aut(G, {x, x−1, yt, y−t}) > ⟨α⟩. Then there
would exist β ∈ Aut(G, {x, x−1, yt, y−t}) such that xβ = yt and (yt)β = x. Then
(xH ′)β = ytH ′ and (ytH ′)β = xH ′. Since yH ′ = x · (x−1y)H ′, one has (xH ′)β =
ytH ′ = xt · (x−1y)tH ′. It follows that
xH ′ = (ytH ′)β = (xtH ′)β · ((x−1y)tH ′)β = xt
2
H ′ · (x−1y)t
2
H ′ · ((x−1y)t)βH ′.
As xp
n
= yp
n
, one has (x−1y)p
n
H ′ = H ′, and hence xp
n
H ′ = xt
2pnH ′. It follows
that x(t
2−1)pn ∈ H ′, and since ⟨x⟩ ∩ H ′ = 1, one has x(t2−1)pn = 1. This implies that
t2 ≡ 1(mod pm−n). However, it is assumed that t2 ≡ −1(mod pm−n) and m > n. This
forces that pm−n | 2, a contradiction.
Construction II Let p be an odd prime, and m be an integer. Take i, k ∈ Z∗p and assume
that k2 ≡ −1(mod p). Let
H(p,m; k, i) = ⟨a, b, c | ap
m+1
= cp = 1, ap
m
= bkp
m
, c = [a, b], [a, c] =aip
m
,
[b, c] = bip
m
⟩
N. Li et al.: On cubic bi-Cayley graphs of p-groups 639
and let Γ(p,m; k, i) = BiCay(H(p,m; k, i), {a, a−1}, {b, b−1}, {1}).
Lemma 4.2. The graph Γ(p,m; k, i) is a connected cubic non-Cayley vertex-transitive
graph with 2p2m+2 vertices.
Proof. Let G = H(p,m; k, i) and let Σ = G(p,m; k, i). Clearly, G′ = ⟨c⟩ × ⟨apm⟩ ∼=
Zp × Zp and ap
m ∈ Z(G). Letting N = ⟨apm⟩, we have
G/N = ⟨aN, bN, cN | ap
m
N = bp
m
N = cpN = N, cN = [a, b]N,[cN, aN ] =
[cN, bN ] = N⟩.
So |G/N | = p2m+1 and so |G| = p2m+2. Thus, Σ has 2p2m+2 vertices.
Let u = b, v = a−1 and w = [u, v]. Since ap
m
= bkp
m
and k2 ≡ −1(mod p), one
has bp
m
= a−kp
m
. So we have up
m
= vkp
m
and up
m+1
= 1. As w = [u, v] = [b, a−1] =
[a, b]a
−1
= ca
−1
, one has wp = 1. As ap
m
= bkp
m
, one has [a, c] = aip
m ∈ Z(G) and
[b, c] = bip
m ∈ Z(G). It follows that [u,w] = [b, ca−1 ] = [b, [a, c]c] = [b, c] = bipm =
uip
m
, and [v, w] = [a−1, ca
−1
] = [a−1, [a, c]c] = [a−1, c] = [a, c]−1 = a−ip
m
= vip
m
.
Thus u, v, w have the same relations as do a, b, c. So H has an automorphism, say α, such
that aα = b and bα = a−1. This implies α ∈ Aut(G, {a, a−1, b, b−1}). To complete the
proof, by Proposition 3.4, it suffices to prove that Aut(G, {a, a−1, b, b−1}) = ⟨α⟩.
Suppose on the contrary that Aut(G, {a, a−1, b, b−1}) > ⟨α⟩. Then there would exist
β ∈ Aut(G, {a, a−1, b, b−1}) such that aβ = b and bβ = a. Then bpm = (apm)β =
(bkp
m
)β = akp
m
. It follows that ap
m
= ak
2pm and hence k2 ≡ 1(mod p). However, since
it is assumed that k2 ≡ −1(mod p), one has p | 2, a contradiction.
Construction III Let p be an odd prime, and m > n > 0 be integers. Take i ∈ Z∗p
and s ∈ Z∗pn such that s2 ≡ −1(mod pn). Let H(p,m, n; s, i) be a group with the follow
presentation:
⟨a, b, c | ap
m+1
= cp = 1, ap
m−n
= bsp
m−n
, c = [a, b], [a, c] = aip
m
, [b, c] = bip
m
⟩
and Γ(p,m, n; s, i) = BiCay(H(p,m, n; s, i), {a, a−1}, {b, b−1}, {1}).
Lemma 4.3. The graph Γ(p,m, n; s, i) is a connected cubic non-Cayley vertex-transitive
graph with 2p2m−n+2 vertices.
Proof. Let G = H(p,m, n; s, i) and let Σ = Γ(p,m, n; s, i). Clearly, G′ = ⟨c⟩×⟨apm⟩ ∼=
Zp ×Zp, ap
m ∈ Z(G) and G/G′ = ⟨aG′, bG′⟩. Since apm , bpm ∈ G′, one has G′ ∩ ⟨a⟩ =
G′ ∩ ⟨b⟩ = ⟨apm⟩. Since apm−n = bspm−n , one has ⟨aG′⟩ ∩ ⟨bG′⟩ = ⟨apm−nG′⟩ ∼= Zpn ,
and so |G/G′| = p2m−n. It follows that G/G′ = ⟨aG′⟩× ⟨ab−sG′⟩ ∼= Zpm ×Zpm−n , and
hence |G| = p2m−n+2. Thus, Σ has 2p2m−n+2 vertices.
Let u = b, v = a−1 and w = [u, v]. Since ap
m−n
= bsp
m−n
and s2 ≡ −1(mod pn),
one has bp
m−n
= a−sp
m−n
. So we have up
m−n
= vsp
m−n
and up
m+1
= 1. As w =
[u, v] = [b, a−1] = [a, b]a
−1
= ca
−1
, one has wp = 1. As ap
m−n
= bsp
m−n
, one has
[a, c] = aip
m ∈ Z(G) and [b, c] = bipm ∈ Z(G). It follows that [u,w] = [b, ca−1 ] =
[b, [a, c]c] = [b, c] = bip
m
= uip
m
, and [v, w] = [a−1, ca
−1
] = [a−1, [a, c]c] = [a−1, c] =
[a, c]−1 = a−ip
m
= vip
m
. Thus u, v, w have the same relations as do a, b, c. So G has an
automorphism, say α, such that aα = b and bα = a−1, and hence α ∈ Aut(G, {a, a−1, b,
640 Ars Math. Contemp. 23 (2023) #P4.09 / 631–648
b−1}). To complete the proof, by Proposition 3.4, it suffices to prove that Aut(G, {a, a−1, b,
b−1}) = ⟨α⟩.
Suppose on the contrary that Aut(G, {a, a−1, b, b−1}) > ⟨α⟩. Then there would exist
β ∈ Aut(G, {a, a−1, b, b−1}) such that aβ = b and bβ = a. Then bpm−n = (apm−n)β =
(bsp
m−n
)β = asp
m−n
. It follows that ap
m−n
= as
2pm−n and hence s2 ≡ 1(mod pn).
However, since it is assumed that s2 ≡ −1(mod pn), one has p | 2, a contradiction.
Construction IV Let p be an odd prime, and m be an integer. Take k ∈ Z∗p such that
k2 ≡ −1(mod p). Let H(p,m; k) be a group with the following representation:
⟨a, b, c, d | ap
m
= bp
m
= cp = dp = 1, c = [a, b], [a, c] = d, [b, c] = dk, [a, d] = [b, d] = 1⟩
and let ∆(p,m; k) = BiCay(H(p,m; k), {a, a−1}, {b, b−1}, {1}).
Lemma 4.4. The graph ∆(p,m; k) is a connected cubic non-Cayley vertex-transitive
graph with 2p2m+2.
Proof. Let G = H(p,m; k) and let Σ = ∆(p,m; k). Clearly, G′ = ⟨c⟩ × ⟨d⟩ ∼= Zp × Zp,
d ∈ Z(G) and G/G′ = ⟨aG′, bG′⟩. Letting N = ⟨d⟩, we have
G/N = ⟨aN, bN, cN | ap
m
N = bp
m
N = cpN = N, cN = [a, b]N,[cN, aN ] =
[cN, bN ] = N⟩.
So |G/N | = p2m+1 and so |G| = p2m+2. Thus, Σ has 2p2m+2 vertices.
Let u = b, v = a−1, x = [u, v] and y = [u, x]. Clearly, up
m
= vp
m
= 1. Note that
x = [u, v] = [b, a−1] = [a, b]a
−1
= aca−1c−1ac(ac)−1c = d(ac)
−1
c = dc.
Then y = [u, x] = [b, dc] = [b, c] = dk. It follows that xp = yp = 1. Furthermore,
[v, x] = [a−1, c] = [c, a]a
−1
= d−1 = dk
2
= yk.
Clearly, [u, y] = [v, y] = 1 due to [a, d] = [b, d] = 1. Thus u, v, x, y have the same
relations as do a, b, c, d. So G has an automorphism, say α, such that aα = b and bα = a−1.
This implies α ∈ Aut(G, {a, a−1, b, b−1}). To complete the proof, by Proposition 3.4, it
suffices to prove that Aut(G, {a, a−1, b, b−1}) = ⟨α⟩.
Suppose on the contrary that Aut(G, {a, a−1, b, b−1}) > ⟨α⟩. Then there would exist
β ∈ Aut(G, {a, a−1, b, b−1}) such that aβ = b and bβ = a. Then cβ = [b, a] = c−1 and
dβ = [b, c−1] = [c, b]c
−1
= d−k. So (dk)β = d−k
2
= d since k2 ≡ −1(mod p). On
the other hand, (dk)β = [bβ , cβ ] = [a, c−1] = d−1. It follows that d = d−1 which is
impossible since d has order p > 2.
Construction V Let p be an odd prime, and let m > n > 0 be integers. Take k ∈
Z∗p and t ∈ Z∗pn such that t2 ≡ −1(mod pn) and t ≡ k(mod p). Take j ∈ Zp. Let
G(p,m, n; t, k, j) be a group with the following representation:
⟨a, b, c, d | ap
m
= cp = dp = 1, ap
m−n
= btp
m−n
dj , c = [a, b], [a, c] = d, [b, c] = dk,
[a, d] = [b, d] = 1⟩
and let Θ(p,m, n; t, k, j) = BiCay(G(p,m, n; t, k, j), {a, a−1}, {b, b−1}, {1}).
N. Li et al.: On cubic bi-Cayley graphs of p-groups 641
Lemma 4.5. The graph Θ(p,m, n; t, k, j) is a connected cubic non-Cayley vertex-transitive
graph with 2p2m−n+2 vertices.
Proof. Let G = G(p,m, n; t, k) and let Σ = Θ(p,m, n; t, k). Clearly, G′ = ⟨c⟩ × ⟨d⟩ ∼=
Zp×Zp, d ∈ Z(G) and G/G′ = ⟨aG′, bG′⟩. Furthermore, G′∩⟨a⟩ = G′∩⟨b⟩ = 1. Since
ap
m−n
= btp
m−n
dj , one has ⟨aG′⟩∩⟨bG′⟩ = ⟨apm−nG′⟩ ∼= Zpn , and so |G/G′| = p2m−n.
It follows that G/G′ = ⟨aG′⟩ × ⟨ab−tG′⟩ ∼= Zpm × Zpm−n , and hence |G| = p2m−n+2.
Thus, Σ has 2p2m−n+2 vertices.
Let u = b, v = a−1, x = [u, v] and y = [u, x]. Note that
x = [u, v] = [b, a−1] = [a, b]a
−1
= aca−1c−1ac(ac)−1c = d(ac)
−1
c = dc.
Then y = [u, x] = [b, dc] = [b, c] = dk. It follows that xp = yp = 1.
As ap
m−n
= btp
m−n
dj and t2 ≡ −1(mod pn), one has upm−n = bpm−n = a−tpm−ndjt
= vtp
m−n
yj (due to t ≡ k(mod p)), and hence upm = 1. Furthermore,
[v, x] = [a−1, c] = [c, a]a
−1
= d−1 = dk
2
= yk.
Clearly, [u, y] = [v, y] = 1 due to [a, d] = [b, d] = 1. Thus u, v, x, y have the same
relations as do a, b, c, d. So G has an automorphism, say α, such that aα = b and bα = a−1,
and hence α ∈ Aut(G, {a, a−1, b, b−1}). To complete the proof, by Proposition 3.4, it
suffices to prove that Aut(G, {a, a−1, b, b−1}) = ⟨α⟩.
Suppose on the contrary that Aut(G, {a, a−1, b, b−1}) > ⟨α⟩. Then there would exist
β ∈ Aut(G, {a, a−1, b, b−1}) such that aβ = b and bβ = a. Then cβ = [b, a] = c−1 and
dβ = [b, c−1] = [c, b]c
−1
= d−k. So (dk)β = d−k
2
= d since k2 ≡ −1(mod p). On
the other hand, (dk)β = [bβ , cβ ] = [a, c−1] = d−1. It follows that d = d−1 which is
impossible since d has order p > 2.
5 Proof of Theorem 1.1
The goal of this section is to prove Theorem 1.1. Let G be a finite p-group with p a
prime. By [18, 5.3.2], if |G : Φ(G)| = pr then every generating set of G has a subset of
r elements which also generates G. In particular, G/Φ(G) ∼= Zrp. This implies that all
minimal generating sets of a finite p-group G have the same cardinality, called the rank of
G and denoted by d(G).
Lemma 5.1. Let p be an odd prime and let H be a finite p-group such that d(H ′) ≤ 2. Let
Γ be a connected cubic bi-Cayley of H . Let A ≤ Aut(Γ) be such that R(H) ≤ A. If Γ is a
non-Cayley vertex-transitive graph and A is vertex- but not arc-transitive on Γ, then either
p = 3 and d(H ′) = 2, or R(H)A.
Proof. Assume that Γ is a non-Cayley vertex-transitive graph and A is vertex- but not
arc-transitive on Γ. Assume further that R(H) is not normal in A. We shall prove that
p = 3 and d(H ′) = 2. By Proposition 3.4, ΓM is a connected tetravalent Cayley graph
of H , and by Lemma 3.3, ΓM is A-arc-transitive and R(H) ≤ A. For convenience, we
may identify R(H) with H and let Σ = ΓM. Applying Proposition 2.2 to the A-arc-
transitive Cayley graph Σ of odd order group H , we see that A has a normal subgroup
R such that R ≤ H , Σ is a normal cover of ΣR, and moreover, ΣR and A/R satisfy
642 Ars Math. Contemp. 23 (2023) #P4.09 / 631–648
(1) or (2) of Proposition 2.2. Since Γ is non-symmetric, for each v ∈ V (Γ), Av is a 2-
group, and so A is a {2, p}-group. It follows that A is solvable. If Proposition 2.2(1)
happens, then A5 ≤ A/R ≤ S5, contradicting that A is solvable. If Proposition 2.2(2)
happens, then H/R has two subgroups N/R and M/R such that H/R = N/R ⋊ M/R
with M/R ∼= Zpm and N/R ∼= Znp , where n = pm. Then H ′ ≤ N . Since M ∩ N = R,
one has M ∩H ′R = R, and so MH ′R/H ′R ∼= M/R ∼= Zpm . It follows that H/H ′R =
N/H ′R×MH ′/H ′R. If N = H ′R, then H ′/(H ′∩R) ∼= N/R ∼= Znp . As d(H ′) ≤ 2, one
has n = pm ≤ 2, contrary to that p is an odd prime. Thus, N > H ′R. By Proposition 3.4,
we have d(H) ≤ 2, and so N/H ′R ∼= Zp. Then H ′/(H ′∩R) ∼= H ′R/R ∼= Zn−1p . Again,
as d(H ′) ≤ 2, one has n − 1 = pm − 1 ≤ 2, and hence pm ≤ 3. Thus, d(H ′) = 2 and
pm = 3. This completes the proof.
We shall fulfill our task of the proof of Theorem 1.1 by the following three lemmas.
Lemma 5.2. Let p > 3 be a prime and let H be a finite p-group such that either H ′ ∼=
Zp × Zp or H ′ is cyclic. Let Γ be a connected cubic bi-Cayley of H . If Γ is a symmetric
non-Cayley vertex-transitive graph, then Γ ∼= Γ(5, 1, 0, 0; 2) or Γ(5, 2; 2, 4) (see Construc-
tions I & II).
Proof. Assume that Γ is symmetric and non-Cayley. Since p > 3, by Proposition 2.3, we
have p = 5. Let |H| = 5n. Then R(H) is a Sylow 5-subgroup of Aut(Γ). Let P be the
maximal normal 5-subgroup of Aut(Γ). By [16, Lemma 18], we have |P | = 5n or 5n−1. If
|P | = 5n, then P = R(H). Let Q be a Sylow 2-subgroup of Aut(Γ). Then A = R(H)⋊Q
is vertex- but not arc-transitive on Γ. Since R(H)  A, by Proposition 3.4, we have Γ ∼=
BiCay(H, {a, a−1}, {b, b−1}, {1}) with ⟨a, b⟩ = H . However, as R(H) = P  Aut(Γ)
and since Γ is symmetric, the two orbits H0 and H1 of R(H) do not contain edges of Γ, a
contradiction.
If |P | = 5n−1, then by Proposition 2.6, Γ is a normal cover of the quotient graph
ΓP of Γ relative to P , and Aut(Γ)/P is arc-transitive on ΓP . Clearly, ΓP has order 10.
By inspecting the list of cubic symmetric graphs of order at most 10000 (see [6]), ΓP
is isomorphic to the Petersen graph. It follows that Γ is not bipartite. So we have Γ =
BiCay(H,R,L, S) with |R| = |L| > 0. By Proposition 2.4, we may assume that 1 ∈ S.
Since |H| is odd and Γ has valency 3, one has R = {a, a−1} and L = {b, b−1}. Since Γ
is connected, we have H = ⟨a, b⟩ by Proposition 2.4. Thus, d(H) ≤ 2. For convenience
of the statement, in the following proof of this lemma, we shall identify R(H) with H .
Clearly, P is maximal in H , so H ′ ≤ P . It follows that P/H ′ ≤ H/H ′ and P ′ ≤ H ′. If
H ′ = 1, then by [11, Theorem 1.1] or [23, Proposition 5.1], we have H ∼= Z5 and Γ is just
the Petersen graph. By Magma [3], Γ(5, 1, 0, 0; 2) is a symmetric cubic graph of order 10.
So Γ ∼= Γ(5, 1, 0, 0; 2).
Now assume that H ′ > 1. Then P > 1. Since P/H ′ ≤ H/H ′, one has d(P/H ′) ≤ 2.
As P/H ′ ∼= (P/P ′)/(H ′/P ′) and since either H ′ ∼= Z5 × Z5 or H ′ is cyclic, one has
d(P/P ′) ≤ 4. Since P ′ is characteristic in P and P  Aut(Γ), one has P ′  Aut(Γ).
Consider the quotient graph ΓP ′ of Γ relative to P ′. By Proposition 2.6, Γ is a normal
cover of ΓP ′ , and Aut(Γ)/P ′ is arc-transitive on ΓP ′ . As P/P ′ Aut(Γ)/P ′, the quotient
graph, denote by ∆, of ΓP ′ relative to P/P ′ is isomorphic to ΓP and so isomorphic to the
Petersen graph. So ΓP ′ is a normal P/P ′-cover of the Petersen graph. Note that P/P ′ is
abelian. By [5, Theorem 7.1], we have P/P ′ ∼= Z5 × Z5 × Z5 as d(P/P ′) ≤ 4. Since
d(P/H ′) ≤ 2, one has P ′ < H ′. Since either H ′ ∼= Z5 × Z5 or H ′ is cyclic, it follows
N. Li et al.: On cubic bi-Cayley graphs of p-groups 643
that either H ′/P ′ ∼= Z5 or P ′ = 1 and H ′ ∼= Z5 × Z5. In the former case, H/P ′ is an
inner abelian 5-group. Clearly, ΓP ′ is a bi-Cayley graph of H/P ′. By [16, Lemma 19] and
[17, Lemma 4.7], we have H/P ′  Aut(ΓP ′), and hence H/P ′  Aut(Γ)/P ′. It follows
that H  Aut(Γ), contrary to the maximality of P . For the latter case, we have P ′ = 1
and P ∼= Z5 × Z5 × Z5. So Γ is a cubic symmetric non-Cayley vertex-transitive graph
of order 1250. By inspecting the list of cubic symmetric graphs of order at most 10000
(see [6]), up to isomorphism, there exists a unique cubic symmetric non-Cayley graph of
order 1250, and by Magam [3], we see that Γ(5, 2; 2, 4) (see Construction II) is a cubic
symmetric non-Cayley graph of order 1250. Thus, Γ ∼= Γ(5, 2; 2, 4).
The next lemma will classify cubic non-Cayley vertex-transitive bi-Cayley graphs of a
p-group H in case that H ′ is cyclic, where p > 3 is a prime.
Lemma 5.3. Let p > 3 be a prime and let H be a finite p-group such that H ′ is cyclic.
Let Γ = BiCay(H,R,L, S) be a connected cubic bi-Cayley of H . Then Γ is a non-Cayley
vertex-transitive graph if and only if Γ is isomorphic to the graphs given in Construction I.
Proof. From Lemma 4.1 we obtain the sufficiency.
Next we prove the necessity. If Γ is symmetric, then by Lemma 5.2, we have Γ ∼=
Γ(5, 1, 0, 0; 2). Now assume that Γ is not symmetric. Let A = Aut(Γ). By Lemma 5.1, we
have R(H)A, and then by Proposition 3.4, we may let Γ = BiCay(H, {a, a−1}, {b, b−1},
{1}), and ΓM ∼= Cay(H, {a, b, a−1, b−1}), where H = ⟨a, b⟩. Furthermore, H has no au-
tomorphisms interchanging the two pairs (a, a−1) and (b, b−1). As ΓM is A-arc-transitive
by Lemma 3.3 and since R(H)  A, it follows that Aut(H, {a, b, a−1, b−1}) is transi-
tive on {a, b, a−1, b−1}. Consequently, we have Aut(H, {a, b, a−1, b−1}) ∼= Z4. This
implies that H has an automorphism, say α, such that aα = b and bα = a−1. Then
[a, b]α = [b, a−1] = [a, b]a
−1
and [a, b]α
2
= [a−1, b−1] = [a, b](ab)
−1
.
Since H = ⟨a, b⟩, one has H ′ = ⟨[a, b]h | h ∈ H⟩, and since d(H ′) = 1, one has
H ′ = ⟨[a, b]⟩. It follows that [a, b]α = [a, b]k and [a, b]α2 = [a, b]k2 , where k is an integer
coprime with p. Since α has order 4, one has k4 ≡ 1(mod |H ′|). On the other hand, as
we already have [a, b]α = [a, b]a
−1
. It follows that kp
r ≡ 1(mod |H ′|), where pr is the
order of a. Thus, k ≡ 1(mod |H ′|). This implies that [a, b] = [a, b]α = [a, b]a−1 and then
[a, b] = [a, b]α
2
= [a−1, b−1] = [a, b](ab)
−1
. Consequently, we obtain that [a, b] commutes
with both a and b, and so [a, b] is contained in the center Z(H) of H . This implies that
H ′ = ⟨[a, b]⟩.
Assume that H/H ′ ∼= Zpm × Zpn with m ≥ n. Then ap
m
, bp
m ∈ H ′. Let c = [a, b].
Assume that c has order ps. Since c ∈ Z(H), one has cpm = [apm , b] = 1 and hence
m ≥ s. Suppose that apm = cpt with t ≤ s. Then bpm = (apm)α = (cpt)α = cpt = apm ,
implying that α fixes ap
m
. Since aα
2
= a−1, one has a−p
m
= ap
m
and so ap
m
= 1.
Thus, ap
m
= bp
m
= 1. Then ⟨a⟩ ∩ ⟨c⟩ = ⟨b⟩ ∩ ⟨c⟩ = 1. As H/H ′ ∼= Zpm × Zpn ,
one has |⟨aH ′⟩⟨bH ′⟩| = pm+n. It follows that |⟨aH ′⟩ ∩ ⟨bH ′⟩| = pm−n, and hence
⟨apnH ′⟩ = ⟨bpnH ′⟩. Then apn = bℓpncλ, with ℓ ∈ Z∗pm−n and λ ∈ Zps . Now we
have bp
n
= (ap
n
)α = (bℓp
n
cλ)α = a−ℓp
n
cλ. Then ap
n
= bℓp
n
cλ = a−ℓ
2pnc2λ. Since
⟨a⟩ ∩ ⟨c⟩ = 1, one has c2λ = 1 and hence cλ = 1. So apn = a−ℓ2pn . Hence, either m = n
or ℓ2 ≡ −1(mod pm−n). If m = n, then we have
H = ⟨a, b, c | ap
m
= bp
m
= cp
s
= 1, c = [a, b], [a, c] = [b, c] = 1⟩.
644 Ars Math. Contemp. 23 (2023) #P4.09 / 631–648
It is easy to see that β : a 7→ b, b 7→ a induces an automorphism of H . Clearly, β ∈
Aut(H, {a, b, a−1, b−1}). This, however, is impossible since Aut(H, {a, b, a−1, b−1} =
⟨α⟩ ∼= Z4.
Now assume that m > n and let x = a, y = bℓ and z = [x, y]. Then z = cp
ℓ
, and
H = H(p,m, n, s; t) = ⟨x, y, z | xp
m
= zp
s
= 1, z = [x, y], [x, z] =[y, z] = 1,
xp
n
= yp
n
⟩.
Let t ∈ Z∗pm−n be such that tℓ ≡ 1(mod p
m−n). Then b = yt and hence we obtain that
Γ = BiCay(H, {x, x−1}, {yt, y−t}, {1}) = Γ(p,m, n, s; t).
Finally, we shall classify cubic non-Cayley vertex-transitive bi-Cayley graphs of a p-
group H in case H ′ ∼= Zp × Zp and p > 3 is prime.
Lemma 5.4. Let p > 3 be a prime and let H be a finite p-group such that H ′ ∼= Zp ×
Zp. Let Γ = BiCay(H,R,L, S) be a connected cubic bi-Cayley of H . Then Γ is a non-
Cayley vertex-transitive graph if and only if Γ is isomorphic to one of the graphs given
Constructions II – V.
Proof. The sufficiency can be obtained from Lemmas 4.2 – 4.5.
In the following, we prove the necessity. Let A = Aut(Γ). If Γ is symmetric, then
by Lemma 5.2, we have Γ ∼= Γ(5, 2; 2, 4) (see Construction II). Now assume that Γ is
not symmetric. By Lemma 5.1, we have R(H)  A, and then by Proposition 3.4, we
may let Γ = BiCay(H, {a, a−1}, {b, b−1}, {1}), and ΓM ∼= Cay(H, {a, b, a−1, b−1}),
where H = ⟨a, b⟩. Furthermore, H has no automorphisms interchanging the two pairs
(a, a−1) and (b, b−1). As ΓM is A-arc-transitive by Lemma 3.3 and since R(H)  A, it
follows that Aut(H, {a, b, a−1, b−1}) is transitive on {a, b, a−1, b−1}. Consequently, we
have Aut(H, {a, b, a−1, b−1}) ∼= Z4. This implies that H has an automorphism, say α,
such that aα = b and bα = a−1.
Since H ′ ∼= Zp × Zp and H ′ = ⟨[a, b]h | h ∈ H⟩, [a, b] has order p and [a, b] is not in
the center of H . Let
H3 = ⟨[[g, h], k]r | g, h, k ∈ {a, b}, r ∈ H⟩.
Then H3 H . Since H is a p-group, one has H3 < H ′. So H3 ∼= Zp as H ′ ∼= Zp × Zp.
Since H is a p-group, one has H3 ∩ Z(H) > 1, implying that H3 ≤ Z(H). As [a, b] is
not in the center of H , one has H ′ ∩ Z(H) = H3. Let c = [a, b]. We shall first prove the
following claim:
Claim 1 There exists k ∈ Z∗p such that k2 ≡ −1(mod p) and [b, c] = [a, c]k.
A direct computation shows that cα = ca
−1
and cα
2
= c(ab)
−1
. Using the fact that
H3 ≤ Z(H), we obtain the following:
[a, c]α = [b, ca
−1
] = [b, [a−1, c−1]c] = [b, c],
[a, c]α
2
= [b, c]α = [a−1, ca
−1
] = [a−1, [a−1, c−1]c] = [a, c]−1,
[a, c]α
3
= [b, c]−1.
Then [a, c], [b, c] ̸= 1, and since H3 ∼= Zp, one has H3 = ⟨[a, c]⟩ = ⟨[b, c]⟩. As α is an
automorphism of H of order 4, one has Hα3 = H3. We may let [a, c]
α = [a, c]k with
N. Li et al.: On cubic bi-Cayley graphs of p-groups 645
k ∈ Z∗p. Since [a, c]α = [b, c], we have [b, c] = [a, c]k, and since [a, c]α
2
= [a, c]−1, one
has k2 ≡ −1(mod p), as claimed.
Assume that ap
m ∈ H ′ for some integer m > 0. Then bpm = (apm)α ∈ H ′. Then
(H/H3)
′ = H ′/H3 = ⟨cH3⟩ ∼= Zp. Clearly, Φ(H ′) = 1. If H/H3 is metacyclic, then
by [2, Theorem 2.3], H is metacyclic. This, however is impossible since H ′ ∼= Zp × Zp.
Thus, H/H3 is not metacyclic. Since H ′/H3 ∼= Zp, it follows that H/H3 is inner abelian
(see, for example, [2, Lemma 2.5]). Clearly, H/H3 = ⟨aH3⟩(⟨bH3, cH3⟩). Assume that
⟨aH3⟩ ∩ ⟨bH3, cH3⟩ ∼= Zpn with n < m. We shall consider the following two cases:
Case 1: n = 0.
In this case, H/H3 is generated by aH3, bH3, cH3 with the following defining rela-
tions:
ap
m
H3 = b
pmH3 = c
pH3 = H3, cH3 = [aH3, bH3], [cH3, aH3] = [cH3, bH3] = H3.
If ⟨a⟩ ∩H3 ̸= 1, then a has order pm+1 and H3 = ⟨ap
m⟩. Then [a, c] = aipm for some
i ∈ Z∗p. As [a, c]α = [b, c] and aα = b, one has [b, c] = bip
m
. Again, since aα = b, b also
has order pm+1 and H3 = ⟨bp
m⟩. By Claim 1, we have apm = bkpm with k ∈ Z∗p and
k2 ≡ −1(mod p). This implies that H is isomorphic to the following group:
H(p,m; k, i) = ⟨a, b, c | ap
m+1
= cp = 1, ap
m
= bkp
m
, c = [a, b], [a, c] = aip
m
,
[b, c] = bip
m
⟩.
This is just the group given in Construction II. So Γ ∼= Γ(p,m; k, i).
If ⟨a⟩∩H3 = 1, then a has order pm. Since aα = b, b also has order pm. Let d = [a, c].
By Claim 1, we have dα = [b, c] = dk with k ∈ Z∗p and k2 ≡ −1(mod p). This implies
that H is isomorphic to the following group as given in Construction IV:
⟨a, b, c, d | ap
m
= bp
m
= cp = dp = 1, c = [a, b], [a, c] = d, [b, c] = dk, [a, d] = [b, d] = 1⟩.
It follows that Γ ∼= ∆(p,m; k).
Case 2: n > 0.
In this case, we have ap
m−n
H3 = b
tpm−ndH3 with t ∈ Z∗pn and d ∈ ⟨c⟩. Then
dH3 = a
pm−nb−tp
m−n
H3 ∈ ⟨ab−tH3⟩ as H/H3 is inner abelian. If dH3 ̸= H3, then we
have H ′/H3 = ⟨dH3⟩ ≤ ⟨ab−tH3⟩ and then ⟨ab−tH3⟩H/H3. This implies that H/H3
is metacyclic since H/H3 = ⟨ab−tH3, bH3⟩. This is a contradiction. Thus, dH3 = H3
and so d = 1 as d ∈ ⟨c⟩. It follows that apm−nH3 = btp
m−n
H3 with t ∈ Z∗pn . In particular,
⟨apm−n , H3⟩ = ⟨bp
m−n
, H3⟩.
Since aα = b and bα = a−1, it follows that bp
m−n
H3 = (a
pm−nH3)
α = (btp
m−n
H3)
α
= a−tp
m−n
H3 = b
−t2pm−nH3. Consequently, we obtain that t2 ≡ −1(mod pn).
Now we see that H/H3 is generated by aH3, bH3, cH3 with the following defining
relations:
bp
m
H3 = c
pH3 = H3, a
pm−nH3 = b
tpm−nH3, cH3 = [aH3, bH3],[cH3, aH3] =
[cH3, bH3] = H3.
If ⟨a⟩ ∩H3 ̸= 1, then a has order pm+1 and H3 = ⟨ap
m⟩. Since 0 < n < m, one has
H3 ≤ ⟨ap
m−n⟩ ∩ ⟨btpm−n⟩. As ⟨apm−n , H3⟩ = ⟨bp
m−n
, H3⟩, one has ⟨ap
m−n⟩ = ⟨bpm−n⟩.
646 Ars Math. Contemp. 23 (2023) #P4.09 / 631–648
Then ap
m−n
= bsp
m−n
for some s ∈ Z∗pn . As (ap
m−n
)α = bp
m−n
and (bp
m−n
)α =
a−p
m−n
, we have s2 ≡ −1(mod pn). Since H3 = ⟨[a, c]⟩, [a, c] = aip
m
for some i ∈ Z∗p.
As [a, c]α = [b, c] and aα = b, one has [b, c] = bip
m
.This implies that H is isomorphic to
the following group:
H(p,m, n; s, i) = ⟨a, b, c | ap
m+1
= cp = 1, ap
m−n
= bsp
m−n
, c = [a, b],[a, c] = aip
m
,
[b, c] = bip
m
⟩.
This is just the group given in Construction III, and so Γ ∼= Γ(p,m, n; s, i).
If ⟨a⟩ ∩ H3 = 1, then a has order pm. Since aα = b, b also has order pm. Let
d = [a, c]. By Claim 1, we have dα = [b, c] = dk with k ∈ Z∗p and k2 ≡ −1(mod p). As
ap
m−n
H3 = b
tpm−nH3, one has ap
m−n
= btp
m−n
dj for some j ∈ Zp. Considering the
image of ap
m−n
= btp
m−n
dj under α, we have t ≡ k(mod p).
This implies that H is isomorphic to the following group which is just the group given
in Construction V:
G(p,m, n; t, k, j) = ⟨a, b, c, d | apm = cp = dp = 1, apm−n = btpm−ndj ,
c = [a, b], [a, c] = d, [b, c] = dk, [a, d] = [b, d] = 1⟩.
Consequently, we obtain that Γ ∼= Θ(p,m, n; t, k, j). This completes the proof.
6 Proof of Theorem 1.2
In this final section, we shall prove Theorem 1.2 which gives a classification of cubic VNC-
graphs of order p4 for each prime p.
Proof of Theorem 1.2. By Lemmas 4.1, 4.2 and 4.4, we see that the graphs Γ(p,m, n, s; t)
(m+ n+ 3 = 4,m > n ≥ 0, s ≥ 0), Γ(p, 1; k, i) and ∆(p, 1; k) are connected cubic non-
Cayley vertex-transitive graphs of order 2p4. So we only need to prove the sufficiency.
Let Γ be a connected cubic non-Cayley vertex-transitive graph of order 2p4. Assume
first that Γ is symmetric. By [8, Corollary 3.4], every connected cubic symmetric graph of
order 2pn is a Cayley graph whenever p ≥ 7 and n ≥ 1, and by [24, Theorem 1.2], every
connected cubic symmetric graph order a 2-power is a Cayley graph. So we have p = 3
or 5. Then by inspecting the list of cubic symmetric graphs of order at most 10000 (see
[6]), we obtain that p = 5, and up to isomorphism, there exists a unique cubic symmetric
non-Cayley graph of order 1250. Thus, Γ ∼= Γ(5, 2; 2, 4) by Lemma 5.2.
Assume now that Γ is not symmetric. If p ≤ 3, then by inspecting the list of cubic
vertex-transitive graphs of order up to 1280 (see [15]), we see that every cubic vertex-
transitive graph of order 32 or 162 is a Cayley graph, a contradiction. Thus, we may assume
that p > 3. Since Γ is not symmetric, the stabilizer Aut(Γ)v of any vertex v ∈ V (Γ) in
Aut(Γ) is a 2-group. Let P be a Sylow p-subgroup. Then P acts semiregularly on V (Γ)
with two orbits. So Γ is a bi-Cayley graph of P . By Lemma 5.1, we have P  Aut(Γ),
and then by Proposition 3.4, we may let Γ ∼= BiCay(P, {a, a−1}, {b, b−1}, {1}), where
P = ⟨a, b⟩. Then |H ′| ≤ p2, and so either H ′ is cyclic or H ′ ∼= Z2p. By Lemmas 5.3
and 5.4, we have Γ is isomorphic to Γ(p,m, n, s; t)(m + n + 3 = 4,m > n ≥ 0, s ≥ 0),
Γ(p, 1; k, i) or ∆(p, 1; k) since Γ has order 2p4.
N. Li et al.: On cubic bi-Cayley graphs of p-groups 647
ORCID iDs
Na Li https://orcid.org/0009-0001-4308-7543
Young Soo Kwon https://orcid.org/0000-0002-1765-0806
Jin-Xin Zhou https://orcid.org/0000-0002-8353-896X
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P4.10 / 649–682
https://doi.org/10.26493/1855-3974.2948.f25
(Also available at http://amc-journal.eu)
Using a q-shuffle algebra to describe the
basic module V (Λ0) for the quantized
enveloping algebra Uq(ŝl2)*
Paul Terwilliger †
Department of Mathematics, University of Wisconsin 480 Lincoln Drive,
Madison, WI 53706-1388 USA
Received 25 August 2022, accepted 25 April 2023, published online 26 June 2023
Abstract
We consider the quantized enveloping algebra Uq(ŝl2) and its basic module V (Λ0).
This module is infinite-dimensional, irreducible, integrable, and highest-weight. We de-
scribe V (Λ0) using a q-shuffle algebra in the following way. Start with the free associative
algebra V on two generators x, y. The standard basis for V consists of the words in x, y. In
1995 M. Rosso introduced an associative algebra structure on V, called a q-shuffle algebra.
For u, v ∈ {x, y} their q-shuffle product is u ⋆ v = uv + q(u,v)vu, where (u, v) = 2 (resp.
(u, v) = −2) if u = v (resp. u ̸= v). Let U denote the subalgebra of the q-shuffle algebra
V that is generated by x, y. Rosso showed that the algebra U is isomorphic to the positive
part of Uq(ŝl2). In our first main result, we turn U into a Uq(ŝl2)-module. Let U denote
the Uq(ŝl2)-submodule of U generated by the empty word. In our second main result, we
show that the Uq(ŝl2)-modules U and V (Λ0) are isomorphic. Let V denote the subspace
of V spanned by the words that do not begin with y or xx. In our third main result, we
show that U = U ∩V.
Keywords: Quantized enveloping algebra, q-Serre relations, basic module, q-shuffle algebra.
Math. Subj. Class. (2020): 17B37, 05E14, 81R50
*No funding was received for conducting this study. All data generated or analysed during this study are
included in this published article.
†The author thanks Pascal Baseilhac for many conversations about Uq(ŝl2) and U+q . The author has no
relevant financial or non-financial interests to disclose.
E-mail address: terwilli@math.wisc.edu (Paul Terwilliger)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
650 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
1 Introduction
The quantized enveloping algebra Uq(ŝl2) is associative, noncommutative, and infinite di-
mensional. A presentation by generators and relations is given in Appendix B below. The
algebra Uq(ŝl2) has a subalgebra U+q , called the positive part. The algebra U
+
q has a pre-
sentation involving two generators A,B and two relations, called the q-Serre relations:
[A, [A, [A,B]q]q−1 ] = 0, [B, [B, [B,A]q]q−1 ] = 0.
Both U+q and Uq(ŝl2) are well known in algebraic combinatorics [4,21,26], representation
theory [11,12,15,47], and mathematical physics [6,7,17,27]. In the following paragraphs,
we describe a few situations in which U+q and Uq(ŝl2) play a role.
There is an object in algebraic combinatorics called a tridiagonal pair [21]. Roughly speak-
ing, this is a pair of diagonalizable linear maps on a finite-dimensional vector space, that
each act on the eigenspaces of the other one in a block-tridiagonal fashion. According
to [21, Example 1.7], for a finite-dimensional irreducible U+q -module V on which the gen-
erators A,B are not nilpotent, the pair A,B act on V as a tridiagonal pair. The resulting
tridiagonal pair is said to have q-geometric type or q-Serre type. This type of tridiagonal
pair is described in [1–3, 22–26, 36, 48].
Another object in algebraic combinatorics is a partially ordered set Y called the Young
lattice [35, page 288]. The elements of Y are the Young diagrams (partitions), and the
partial order is given by diagram inclusion. Define a vector space V consisting of the
formal linear combinations of Y . In the Hayashi realization [4, Theorem 10.6] the vector
space V becomes an integrable Uq(ŝl2)-module with the following features. Each Young
diagram λ is a weight vector. The Uq(ŝl2)-generators E0, E1, F0, F1 act on λ as follows.
Color the boxes of λ alternating blue and red, with the top left box colored blue. The
generator E0 (resp. E1) sends λ to a linear combination of the Young diagrams µ obtained
from λ by removing a blue box (resp. red box). In this linear combination the µ-coefficient
is a power of q that depends on the location of the box λ/µ. Similarly, F0 (resp. F1) sends
λ to a linear combination of the Young diagrams µ obtained from λ by adding a blue box
(resp. red box). In this linear combination the µ-coefficient is a power of q that depends on
the location of the box µ/λ. The Uq(ŝl2)-submodule of V generated by the empty Young
diagram is denoted V (Λ0) and called the basic representation. The Uq(ŝl2)-module V (Λ0)
is infinite-dimensional, irreducible, integrable, and highest-weight. For more detail about
V (Λ0) see [4, Chapter 10], [20, Section 9], [27, Chapter 5].
Next we recall an embedding, due to M. Rosso [33, 34] of U+q into a q-shuffle algebra.
Start with a free associative algebra V on two generators x, y. These generators are called
letters. For n ≥ 0, a word of length n in V is a product of letters ℓ1ℓ2 · · · ℓn. We interpret
the word of length 0 to be the multiplicative identity in V; this word is called trivial and
denoted by 1. The words in V form a basis for the vector space V; this basis is called
standard. In [33, 34] M. Rosso introduced an associative algebra structure on V, called a
q-shuffle algebra. For letters u, v their q-shuffle product is u ⋆ v = uv + q(u,v)vu, where
(u, v) = 2 (resp. (u, v) = −2) if u = v (resp. u ̸= v). In [34, Theorem 15] Rosso gave an
injective algebra homomorphism ♮ from U+q into the q-shuffle algebra V, that sends A 7→ x
and B 7→ y.
We mention some applications of the map ♮ : U+q → V. In [16] I. Damiani obtained a
Poincaré-Birkhoff-Witt (or PBW) basis for U+q whose elements are defined recursively.
In [11, Proposition 6.1] J. Beck obtained another PBW basis for U+q by adjusting some of
P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 651
the elements in the Damiani PBW basis. In [40] (resp. [45]) we applied the map ♮ to the
Damiani (resp. Beck) PBW basis, and expressed the images in the standard basis for V. We
gave the images in closed form [40, Theorem 1.7], [45, Theorem 7.1]. The map ♮ is used
in [39] to define the alternating elements of U+q . In [39, Theorem 10.1] a set of alternating
elements is shown to form a PBW basis for U+q . This PBW basis is said to be alternating
[39, Definition 10.3]. In [38] we used the alternating elements to obtain a central extension
U+q of U+q . The algebra U+q is defined by generators and relations. These generators, said
to be alternating, are in bijection with the alternating elements of U+q . By [38, Lemma 3.3]
there exists a surjective algebra homomorphism U+q → U+q that sends each alternating
generator of U+q to the corresponding alternating element in U+q . In [38, Lemma 3.6] this
homomorphism is adjusted to obtain an algebra isomorphism U+q → U+q ⊗ F[z1, z2, . . .]
where F is the ground field and {zn}∞n=1 are mutually commuting indeterminates. By [38,
Theorem 10.2] the alternating generators form a PBW basis for U+q . The algebra U+q is
called the alternating central extension of U+q [38, 46]. We remark that U+q is related to
the work of Baseilhac, Koizumi, Shigechi concerning the q-Onsager algebra and integrable
lattice models [8, 10]. See [5–7, 9, 37, 41–44, 46] for related work.
Turning to the present paper, our goal is to describe the basic Uq(ŝl2)-module V (Λ0) using
the q-shuffle algebra V. We have three main results, which are summarized below. Let
End(V) denote the algebra consisting of the linear maps from V to V. We now define
some maps X,Y,K in End(V). The map X (resp. Y ) is the automorphism of the free
algebra V that sends x 7→ qx and y 7→ y (resp. x 7→ x and y 7→ qy). Define K = X2Y −2.
Define the maps A∗L, B
∗
L, A
∗
R, B
∗
R in End(V) that send 1 7→ 0 and for a nontrivial word
w = ℓ1ℓ2 · · · ℓn in V,
A∗Lw = ℓ2 · · · ℓnδℓ1,x, B∗Lw = ℓ2 · · · ℓnδℓ1,y,
A∗Rw = ℓ1 · · · ℓn−1δℓn,x, B∗Rw = ℓ1 · · · ℓn−1δℓn,y.
Here δr,s is the Kronecker delta. Define the maps Aℓ, Bℓ, Ar, Br in End(V) such that for
v ∈ V,
Aℓv = x ⋆ v, Bℓv = y ⋆ v, Arv = v ⋆ x, Brv = v ⋆ y.
Let U denote the subalgebra of the q-shuffle algebra V that is generated by x, y. By con-
struction the map ♮ : U+q → U is an algebra isomorphism. Our first main result is that U
becomes a Uq(ŝl2)-module on which the Uq(ŝl2)-generators act as follows:
generator E0 F0 K±10 E1 F1 K
±1
1 D
±1
action on U A∗R
qArK
−1−q−1Aℓ
q−q−1 q
±1K∓1 B∗R
BrK−Bℓ
q−q−1 K
±1 X∓1
Let U denote the submodule of the Uq(ŝl2)-module U that is generated by the vector 1.
Our second main result is that the Uq(ŝl2)-modules U and V (Λ0) are isomorphic. Let V
denote the intersection of the kernel of B∗L and the kernel of (A
∗
L)
2. The vector space V
has a basis consisting of the words in V that do not begin with y or xx. Note that the sum
V = F1+ Fx+ xyV is direct. Our third main result is that U = U ∩V.
The paper is organized as follows. Section 2 contains some preliminaries. In Section 3
we recall the algebra U+q and discuss its basic properties. In Section 4 we describe the
free algebra V. In Section 5 we describe the maps X,Y,K in End(V). In Section 6 we
652 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
describe the maps A∗L, B
∗
L, A
∗
R, B
∗
R in End(V). In Section 7 we describe the q-shuffle
algebra V. In Section 8 we describe the maps Aℓ, Bℓ, Ar, Br in End(V). In Section 9 we
describe the subalgebra U of the q-shuffle algebra V. In Sections 10, 11 we give our main
results, which are Theorems 10.1, 10.7, 11.11. In Section 12 we describe some variations
on Theorem 10.1. In Appendix A we display some relations that are satisfied by the maps
from the main body of the paper. In Appendix B we give a presentation of Uq(ŝl2). In
Appendix C we give a basis for some of the weight spaces of the Uq(ŝl2)-module U.
In Appendix D we show how the Uq(ŝl2)-generators act on the bases in Appendix C. In
Appendix E we discuss a linear algebraic situation that comes up in Section 11.
2 Preliminaries
We now begin our formal argument. Recall the natural numbers N = {0, 1, 2, . . .} and
integers Z = {0,±1,±2, . . .}. Let F denote a field with characteristic zero. Throughout
this paper, every vector space we discuss is understood to be over F. Every algebra we
discuss is understood to be associative, over F, and have a multiplicative identity. A subal-
gebra has the same multiplicative identity as the parent algebra. Let A denote an algebra.
An automorphism of A is an algebra isomorphism A → A. The opposite algebra Aopp
consists of the vector space A and the multiplication map A × A → A, (a, b) 7→ ba. An
antiautomorphism of A is an algebra isomorphism A → Aopp.
We recall a few concepts from linear algebra. Let V denote a vector space, and consider
an F-linear map T : V → V . The map T is said to be nilpotent whenever there exists a
positive integer n such that Tn = 0. The map T is said to be locally nilpotent whenever
for all v ∈ V there exists a positive integer n such that Tnv = 0. If T is nilpotent then
T is locally nilpotent. If T is locally nilpotent and the dimension of V is finite, then T is
nilpotent.
Throughout the paper, fix a nonzero q ∈ F that is not a root of unity. Recall the notation
[n]q =
qn − q−n
q − q−1
n ∈ N.
3 The positive part of Uq(ŝl2)
Later in the paper, we will discuss the quantized enveloping algebra Uq(ŝl2). For now,
we consider a subalgebra U+q of Uq(ŝl2), called the positive part. Shortly we will give a
presentation of U+q by generators and relations.
For elements X ,Y in any algebra, define their commutator and q-commutator by
[X ,Y] = XY − YX , [X ,Y]q = qXY − q−1YX .
Note that
[X , [X , [X ,Y]q]q−1 ] = X 3Y − [3]qX 2YX + [3]qXYX 2 − YX 3. (3.1)
Definition 3.1 (See [30, Corollary 3.2.6]). Define the algebra U+q by generators A, B and
relations
[A, [A, [A,B]q]q−1 ] = 0, [B, [B, [B,A]q]q−1 ] = 0. (3.2)
P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 653
We call U+q the positive part of Uq(ŝl2). The relations (3.2) are called the q-Serre relations.
We mention some symmetries of U+q .
Lemma 3.2. There exists an automorphism σ of U+q that sends A ↔ B. Moreover σ2 =
id, where id denotes the identity map.
Lemma 3.3 (See [40, Lemma 2.2]). There exists an antiautomorphism † of U+q that fixes
each of A, B. Moreover †2 = id.
Lemma 3.4 (See [41, Lemma 3.4]). The maps σ, † commute.
Definition 3.5. Let τ denote the composition of σ and †. Note that τ is an antiautomor-
phism of U+q that sends A ↔ B. We have τ2 = id.
Next we describe a grading for the algebra U+q . The q-Serre relations are homogeneous
in both A and B. Therefore, the algebra U+q has a N2-grading for which A and B are
homogeneous, with degrees (1, 0) and (0, 1) respectively. For (r, s) ∈ N2 let U+q (r, s)
denote the (r, s)-homogeneous component of the grading. The dimension of U+q (r, s) is
described by a generating function, as we now discuss. Let t and u denote commuting
indeterminates.
Definition 3.6. Define the generating function
Φ(t, u) =
∞∏
n=1
1
1− tnun−1
1
1− tnun
1
1− tn−1un
.
Using (1− z)−1 = 1 + z + z2 + · · · we expand the above generating function as a power
series:
Φ(t, u) =
∑
(r,s)∈N2
dr,st
rus, dr,s ∈ N.
For notational convenience, define dr,−1 = 0 and d−1,s = 0 for r, s ∈ N.
Example 3.7 (See [39, Example 3.4]). For 0 ≤ r, s ≤ 6 we display dr,s in the (r, s)-entry
of the matrix below: 
1 1 1 1 1 1 1
1 2 3 3 3 3 3
1 3 6 8 9 9 9
1 3 8 14 19 21 22
1 3 9 19 32 42 48
1 3 9 21 42 66 87
1 3 9 22 48 87 134

We have Φ(t, u) = Φ(u, t). Moreover dr,s = ds,r for (r, s) ∈ N2.
Lemma 3.8 (See [39, Definition 3.2, Corollary 3.7]). For (r, s) ∈ N2 we have
dr,s = dimU
+
q (r, s).
654 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
Our next goal is to show that dr,s−1 ≤ dr,s and dr−1,s ≤ dr,s for (r, s) ∈ N2. To reach the
goal, we modify the generating function Φ(t, u) in the following way.
Definition 3.9. Define the generating function
∆(t, u) =
∞∏
n=1
1
1− tnun−1
1
1− tnun
1
1− tnun+1
. (3.3)
Lemma 3.10. We have ∆(t, u) = Φ(t, u)(1− u) and ∆(u, t) = Φ(t, u)(1− t). Moreover
∆(t, u) =
∑
(r,s)∈N2
(
dr,s − dr,s−1
)
trus, ∆(u, t) =
∑
(r,s)∈N2
(
dr,s − dr−1,s
)
trus.
Proof. Use Definitions 3.6, 3.9.
Lemma 3.11. For (r, s) ∈ N2 we have dr,s−1 ≤ dr,s and dr−1,s ≤ dr,s.
Proof. Expand the right-hand side of (3.3) as a power series. In this power series, the coeffi-
cient of trus is nonnegative for (r, s) ∈ N2. The result follows in view of Lemma 3.10.
Our next general goal is to compute max{dr,s|s ∈ N} for r ∈ N, and max{dr,s|r ∈ N}
for s ∈ N. To reach the goal, we will use the concept of a partition.
For n ∈ N, a partition of n is a sequence λ = {λi}∞i=1 of natural numbers such that
λi ≥ λi+1 for i ≥ 1 and n =
∑∞
i=1 λi. Let pn denote the number of partitions of n. For
example,
n 0 1 2 3 4 5 6
pn 1 1 2 3 5 7 11
Define the generating function for partitions:
p(t) =
∑
n∈N
pnt
n. (3.4)
The following result is well known; see for example [13, Theorem 8.3.4].
p(t) =
∞∏
n=1
1
1− tn
. (3.5)
We expand the generating function
(
p(t)
)3
as a power series:(
p(t)
)3
=
∑
n∈N
µnt
n, µn ∈ N. (3.6)
Consider the coefficients {µn}n∈N. For example,
n 0 1 2 3 4 5 6
µn 1 3 9 22 51 108 221
P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 655
Proposition 3.12. For r ∈ N we have
µr = max{dr,s|s ∈ N}. (3.7)
For s ∈ N we have
µs = max{dr,s|r ∈ N}. (3.8)
Proof. First, for r ∈ N we verify (3.7). Let µ′r denote right-hand side of (3.7). We show
that µr = µ′r. By Lemma 3.11, we may view
µ′r =
∑
s∈N
(dr,s − dr,s−1).
By this and Lemma 3.10,
∆(t, 1) =
∑
(r,s)∈N2
(
dr,s − dr,s−1
)
tr =
∑
r∈N
µ′rt
r. (3.9)
Set u = 1 in (3.3), and evaluate the result using (3.5), (3.6). This yields
∆(t, 1) =
∞∏
n=1
1
(1− tn)3
=
(
p(t)
)3
=
∑
r∈N
µrt
r. (3.10)
Comparing (3.9), (3.10) we obtain µr = µ′r for r ∈ N. We have verified (3.7). The second
assertion in the proposition statement follows from the first assertion in the proposition
statement and the comment above Lemma 3.8.
Our next general goal is to embed U+q into a q-shuffle algebra. For this q-shuffle algebra the
underlying vector space is a free algebra on two generators. This free algebra is described
in the next section.
4 The free algebra V
Let x, y denote noncommuting indeterminates. Let V denote the free algebra with genera-
tors x, y. By a letter in V we mean x or y. For n ∈ N, a word of length n in V is a product
of letters ℓ1ℓ2 · · · ℓn. We interpret the word of length 0 to be the multiplicative identity in
V; this word is called trivial and denoted by 1. The vector space V has a basis consisting
of its words; this basis is called standard.
We mention some symmetries of the free algebra V. For the next four lemmas, the proofs
are routine and omitted.
Lemma 4.1. There exists an automorphism σ of the free algebra V that sends x ↔ y.
Moreover σ2 = id.
Lemma 4.2. There exists an antiautomorphism † of the free algebra V that fixes each of x,
y. Moreover †2 = id.
Lemma 4.3. The map σ from Lemma 4.1 commutes with the map † from Lemma 4.2.
656 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
Lemma 4.4. There exists an antiautomorphism τ of the free algebra V that sends x ↔ y.
The map τ is the composition the map σ from Lemma 4.1 and the map † from Lemma 4.2.
We have τ2 = id.
Example 4.5. The automorphism σ sends
xxx ↔ yyy, xxyy ↔ yyxx, xyxxyy ↔ yxyyxx.
The antiautomorphism † sends
xxx ↔ xxx, xxyy ↔ yyxx, xyxxyy ↔ yyxxyx.
The antiautomorphism τ sends
xxx ↔ yyy, xxyy ↔ xxyy, xyxxyy ↔ xxyyxy.
The free algebra V has a N2-grading for which x and y are homogeneous, with degrees
(1, 0) and (0, 1) respectively. For (r, s) ∈ N2 let V(r, s) denote the (r, s)-homogeneous
component of the grading. These homogeneous components are described as follows. Let
w = ℓ1ℓ2 · · · ℓn denote a word in V. The x-degree of w is the cardinality of the set {i|1 ≤
i ≤ n, ℓi = x}. The y-degree of w is the cardinality of the set {i|1 ≤ i ≤ n, ℓi = y}. For
(r, s) ∈ N2 the subspace V(r, s) has a basis consisting of the words in V that have x-degree
r and y-degree s. The dimension of V(r, s) is equal to the binomial coefficient
(
r+s
r
)
. By
construction V(0, 0) = F1. By construction, the sum V =
∑
(r,s)∈N2 V(r, s) is direct.
Example 4.6. The following is a basis for the vector space V(2, 3):
xxyyy, xyxyy, xyyxy, xyyyx, yxxyy,
yxyxy, yxyyx, yyxxy, yyxyx, yyyxx.
Let End(V) denote the algebra consisting of the F-linear maps from V to V. Let I denote
the identity in End(V).
5 The maps X , Y , K
In this section we describe some maps X , Y , K in End(V) that will be used in our main
results.
Definition 5.1. Let X denote the automorphism of the free algebra V that sends x 7→ qx
and y 7→ y. Let Y denote the automorphism of the free algebra V that sends x 7→ x and
y 7→ qy.
Example 5.2. The map X sends
xxx 7→ q3xxx, xxyy 7→ q2xxyy, xyxxyy 7→ q3xyxxyy.
The map Y sends
xxx 7→ xxx, xxyy 7→ q2xxyy, xyxxyy 7→ q3xyxxyy.
Lemma 5.3. For (r, s) ∈ N2 the maps X and Y act on V(r, s) as qrI and qsI , respectively.
P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 657
Proof. By the description of V(r, s) above Example 4.6.
By construction the maps X , Y are invertible, and they commute.
Definition 5.4. Define K = X2Y −2. Thus K is the automorphism of the free algebra V
that sends x 7→ q2x and y 7→ q−2y.
Example 5.5. The map K sends
xxx 7→ q6xxx, xxyy 7→ xxyy, xyxxyy 7→ xyxxyy.
Lemma 5.6. For (r, s) ∈ N2 the map K acts on V(r, s) as q2r−2sI .
Proof. By Lemma 5.3 and Definition 5.4.
Lemma 5.7. The following diagrams commute:
V X
±1
−−−−→ V
σ
y yσ
V −−−−→
Y ±1
V
V Y
±1
−−−−→ V
σ
y yσ
V −−−−→
X±1
V
V K
±1
−−−−→ V
σ
y yσ
V −−−−→
K∓1
V
V X
±1
−−−−→ V
†
y y†
V −−−−→
X±1
V
V Y
±1
−−−−→ V
†
y y†
V −−−−→
Y ±1
V
V K
±1
−−−−→ V
†
y y†
V −−−−→
K±1
V
V X
±1
−−−−→ V
τ
y yτ
V −−−−→
Y ±1
V
V Y
±1
−−−−→ V
τ
y yτ
V −−−−→
X±1
V
V K
±1
−−−−→ V
τ
y yτ
V −−−−→
K∓1
V
Proof. Routine.
6 The maps A∗L, B
∗
L, A
∗
R, B
∗
R
In this section we recall from [32] some maps A∗L, B
∗
L, A
∗
R, B
∗
R in End(V) that will be
used in our main results. First we mention some notation. The Kronecker delta δr,s is equal
to 1 if r = s, and 0 if r ̸= s.
Definition 6.1 (See [32, Lemma 4.3]). Define the maps A∗L, B∗L, A∗R, B∗R in End(V) as
follows. For a nontrivial word w = ℓ1ℓ2 · · · ℓn in V,
A∗Lw = ℓ2 · · · ℓnδℓ1,x, B∗Lw = ℓ2 · · · ℓnδℓ1,y,
A∗Rw = ℓ1 · · · ℓn−1δℓn,x, B∗Rw = ℓ1 · · · ℓn−1δℓn,y.
Moreover
A∗L1 = 0, B
∗
L1 = 0, A
∗
R1 = 0, B
∗
R1 = 0. (6.1)
658 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
Example 6.2. The maps A∗L, B∗L, A∗R, B∗R are illustrated in the table below.
w x y xx xy yx yy
A∗Lw 1 0 x y 0 0
B∗Lw 0 1 0 0 x y
A∗Rw 1 0 x 0 y 0
B∗Rw 0 1 0 x 0 y
Lemma 6.3. For v ∈ V,
A∗L(xv) = v, A
∗
L(yv) = 0, B
∗
L(xv) = 0, B
∗
L(yv) = v,
A∗R(vx) = v, A
∗
R(vy) = 0, B
∗
R(vx) = 0, B
∗
R(vy) = v.
Proof. Use Definition 6.1.
For notational convenience, define V(r,−1) = 0 and V(−1, s) = 0 for r, s ∈ N.
Lemma 6.4. For (r, s) ∈ N2 we have
A∗LV(r, s) ⊆ V(r − 1, s), B∗LV(r, s) ⊆ V(r, s− 1),
A∗RV(r, s) ⊆ V(r − 1, s), B∗RV(r, s) ⊆ V(r, s− 1).
Proof. By Definition 6.1 or Lemma 6.3.
Lemma 6.5. The maps A∗L, B∗L, A∗R, B∗R are locally nilpotent on the vector space V.
Proof. We mentioned above Example 4.6 that the sum V =
∑
(r,s)∈N2 V(r, s) is direct.
The result follows from this and Lemma 6.4.
Next we describe how the maps X , Y are related to the maps A∗L, B
∗
L, A
∗
R, B
∗
R.
Lemma 6.6. We have
XA∗L = q
−1A∗LX, XB
∗
L = B
∗
LX, XA
∗
R = q
−1A∗RX, XB
∗
R = B
∗
RX,
Y A∗L = A
∗
LY, Y B
∗
L = q
−1B∗LY, Y A
∗
R = A
∗
RY, Y B
∗
R = q
−1B∗RY.
Proof. By Lemmas 5.3, 6.4.
The next result is about A∗L and B
∗
L; a similar result holds for A
∗
R and B
∗
R. Observe that
the sum V = F1+ xV+ yV is direct.
Lemma 6.7. The following (i) – (v) hold:
(i) kerA∗L has a basis consisting of the words in V that do not begin with x;
(ii) kerA∗L = F1+ yV;
(iii) kerB∗L has a basis consisting of the words in V that do not begin with y;
(iv) kerB∗L = F1+ xV;
(v) kerA∗L ∩ kerB∗L = F1.
P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 659
Proof. Use Definition 6.1 and the observation above the lemma statement.
The following result appears in [32]; we give a short proof for the sake of completeness.
Lemma 6.8 (See [32, Lemma 4.6]). Let W denote a nonzero subspace of V that is closed
under A∗L and B
∗
L. Then 1 ∈ W .
Proof. For n ∈ N define Vn =
∑
r+s≤n V(r, s). Note that V0 = F1. We have Vn−1 ⊆
Vn for n ≥ 1, and V = ∪n∈NVn. For n ≥ 1 we have A∗LVn ⊆ Vn−1 and B∗LVn ⊆ Vn−1,
in view of Lemma 6.4. Since W ̸= 0, there exists n ∈ N such that W ∩ Vn ̸= 0. Assume
for the moment that n = 0. Then 1 ∈ W and we are done. Next assume that n ≥ 1.
Without loss, we may assume that W ∩ Vn−1 = 0. Pick 0 ̸= v ∈ W ∩ Vn. We have
A∗Lv ∈ W ∩ Vn−1 = 0 and B∗Lv ∈ W ∩ Vn−1 = 0, so v ∈ F1 in view of Lemma 6.7(v).
By construction 0 ̸= v ∈ W , so 1 ∈ W .
Lemma 6.9. Let W denote a nonzero subspace of V that is closed under A∗R and B∗R.
Then 1 ∈ W .
Proof. The †-image W † is a subspace of V that is invariant under A∗L and B∗L. We have
1 ∈ W † by Lemma 6.8, and 1† = 1 by construction, so 1 ∈ W .
Lemma 6.10. The following diagrams commute:
V
A∗L−−−−→ V
σ
y yσ
V −−−−→
B∗L
V
V
B∗L−−−−→ V
σ
y yσ
V −−−−→
A∗L
V
V
A∗R−−−−→ V
σ
y yσ
V −−−−→
B∗R
V
V
B∗R−−−−→ V
σ
y yσ
V −−−−→
A∗R
V
V
A∗L−−−−→ V
†
y y†
V −−−−→
A∗R
V
V
B∗L−−−−→ V
†
y y†
V −−−−→
B∗R
V
V
A∗R−−−−→ V
†
y y†
V −−−−→
A∗L
V
V
B∗R−−−−→ V
†
y y†
V −−−−→
B∗L
V
V
A∗L−−−−→ V
τ
y yτ
V −−−−→
B∗R
V
V
B∗L−−−−→ V
τ
y yτ
V −−−−→
A∗R
V
V
A∗R−−−−→ V
τ
y yτ
V −−−−→
B∗L
V
V
B∗R−−−−→ V
τ
y yτ
V −−−−→
A∗L
V
Proof. Routine.
7 The q-shuffle algebra V
In the previous sections we discussed the free algebra V. There is another algebra structure
on V, called the q-shuffle algebra. This algebra was introduced by Rosso [33, 34] and
described further by Green [18]. We will adopt the approach of [18], which is suited to our
purpose. The q-shuffle product is denoted by ⋆. To describe this product, we first consider
some special cases. We have 1 ⋆ v = v ⋆ 1 = v for v ∈ V. For letters u, v we have
u ⋆ v = uv + vuq(u,v), where
660 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
( , ) x y
x 2 −2
y −2 2
Thus
x ⋆ x = (1 + q2)xx, x ⋆ y = xy + q−2yx,
y ⋆ x = yx+ q−2xy, y ⋆ y = (1 + q2)yy.
For a letter u and a nontrivial word v = v1v2 · · · vn in V,
u ⋆ v =
n∑
i=0
v1 · · · viuvi+1 · · · vnq(v1,u)+(v2,u)+···+(vi,u), (7.1)
v ⋆ u =
n∑
i=0
v1 · · · viuvi+1 · · · vnq(vn,u)+(vn−1,u)+···+(vi+1,u). (7.2)
For example
y ⋆ (xxx) = yxxx+ q−2xyxx+ q−4xxyx+ q−6xxxy,
(xxx) ⋆ y = q−6yxxx+ q−4xyxx+ q−2xxyx+ xxxy.
For nontrivial words u = u1u2 · · ·ur and v = v1v2 · · · vs in V,
u ⋆ v = u1
(
(u2 · · ·ur) ⋆ v
)
+ v1
(
u ⋆ (v2 · · · vs)
)
q(u1,v1)+(u2,v1)+···+(ur,v1),
u ⋆ v =
(
u ⋆ (v1 · · · vs−1)
)
vs +
(
(u1 · · ·ur−1) ⋆ v
)
urq
(ur,v1)+(ur,v2)+···+(ur,vs).
For example, assume r = 2 and s = 2. Then
u ⋆ v = u1u2v1v2 + u1v1u2v2q
(u2,v1) + u1v1v2u2q
(u2,v1)+(u2,v2) + v1u1u2v2q
(u1,v1)+(u2,v1)
+ v1u1v2u2q
(u1,v1)+(u2,v1)+(u2,v2) + v1v2u1u2q
(u1,v1)+(u1,v2)+(u2,v1)+(u2,v2).
The map σ from Lemma 4.1 is an automorphism of the q-shuffle algebra V. The map † from
Lemma 4.2 is an antiautomorphism of the q-shuffle algebra V. The map τ from Lemma 4.4
is an antiautomorphism of the q-shuffle algebra V. Above Example 4.6 we mentioned an
N2-grading of the free algebra V. This is also an N2-grading for the q-shuffle algebra V.
See [19, 29, 31, 32, 38–40] for more information about the q-shuffle algebra V.
8 The maps Aℓ, Bℓ, Ar, Br
In this section we recall from [32] some maps Aℓ, Bℓ, Ar, Br in End(V) that will be used
in our main results.
Definition 8.1 (See [32, Definition 7.1]). Define the maps Aℓ, Bℓ, Ar, Br in End(V) as
follows. For v ∈ V,
Aℓv = x ⋆ v, Bℓv = y ⋆ v, Arv = v ⋆ x, Brv = v ⋆ y.
P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 661
Example 8.2. The maps Aℓ, Bℓ, Ar, Br are illustrated in the table below.
v 1 x y xy
Aℓv x q[2]qxx xy + q
−2yx q[2]qxxy + xyx
Bℓv y q
−2xy + yx q[2]qyy q
−1[2]qxyy + yxy
Arv x q[2]qxx q
−2xy + yx q−1[2]qxxy + xyx
Brv y xy + q
−2yx q[2]qyy q[2]qxyy + yxy
Lemma 8.3. For (r, s) ∈ N2 we have
AℓV(r, s) ⊆ V(r + 1, s), BℓV(r, s) ⊆ V(r, s+ 1),
ArV(r, s) ⊆ V(r + 1, s), BrV(r, s) ⊆ V(r, s+ 1).
Proof. By Definition 8.1 and the description of V(r, s) above Example 4.6.
Next we describe how the maps X , Y are related to the maps Aℓ, Bℓ, Ar, Br.
Lemma 8.4. We have
XAℓ = qAℓX, XBℓ = BℓX, XAr = qArX, XBr = BrX,
Y Aℓ = AℓY, Y Bℓ = qBℓY, Y Ar = ArY, Y Br = qBrY.
Proof. By Lemmas 5.3, 8.3.
Lemma 8.5. The following diagrams commute:
V Aℓ−−−−→ V
σ
y yσ
V −−−−→
Bℓ
V
V Bℓ−−−−→ V
σ
y yσ
V −−−−→
Aℓ
V
V Ar−−−−→ V
σ
y yσ
V −−−−→
Br
V
V Br−−−−→ V
σ
y yσ
V −−−−→
Ar
V
V Aℓ−−−−→ V
†
y y†
V −−−−→
Ar
V
V Bℓ−−−−→ V
†
y y†
V −−−−→
Br
V
V Ar−−−−→ V
†
y y†
V −−−−→
Aℓ
V
V Br−−−−→ V
†
y y†
V −−−−→
Bℓ
V
V Aℓ−−−−→ V
τ
y yτ
V −−−−→
Br
V
V Bℓ−−−−→ V
τ
y yτ
V −−−−→
Ar
V
V Ar−−−−→ V
τ
y yτ
V −−−−→
Bℓ
V
V Br−−−−→ V
τ
y yτ
V −−−−→
Aℓ
V
Proof. Routine.
662 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
9 The subspace U
In this section we discuss a subspace U ⊆ V that will be used in our main results.
Definition 9.1. Let U denote the subalgebra of the q-shuffle algebra V that is generated by
x, y.
The algebra U is described as follows. By [33, Theorem 13] or [18, page 10],
x ⋆ x ⋆ x ⋆ y − [3]qx ⋆ x ⋆ y ⋆ x+ [3]qx ⋆ y ⋆ x ⋆ x− y ⋆ x ⋆ x ⋆ x = 0,
y ⋆ y ⋆ y ⋆ x− [3]qy ⋆ y ⋆ x ⋆ y + [3]qy ⋆ x ⋆ y ⋆ y − x ⋆ y ⋆ y ⋆ y = 0.
So in the q-shuffle algebra V the elements x, y satisfy the q-Serre relations. Therefore,
there exists an algebra homomorphism ♮ from U+q to the q-shuffle algebra V, that sends
A 7→ x and B 7→ y. The map ♮ has image U by Definition 9.1, and is injective by [34,
Theorem 15]. Consequently ♮ : U+q → U is an algebra isomorphism. By construction the
following diagrams commute:
U+q
♮−−−−→ V
σ
y yσ
U+q −−−−→
♮
V
U+q
♮−−−−→ V
†
y y†
U+q −−−−→
♮
V
U+q
♮−−−−→ V
τ
y yτ
U+q −−−−→
♮
V
(9.1)
Consequently U is invariant under each of σ, †, τ . Earlier we mentioned an N2-grading for
both the algebra U+q and the q-shuffle algebra V. These gradings are related as follows.
The algebra U has an N2-grading inherited from U+q via ♮. With respect to this grad-
ing, for (r, s) ∈ N2 the (r, s)-homogeneous component of U is the ♮-image of the (r, s)-
homogeneous component of U+q . We denote this homogeneous component by U(r, s). By
construction,
U(r, s) = V(r, s) ∩ U, (r, s) ∈ N2. (9.2)
By construction U(0, 0) = F1. By construction, the sum U =
∑
(r,s)∈N2 U(r, s) is direct.
By Lemma 3.8 and the construction,
dr,s = dimU(r, s), (r, s) ∈ N2. (9.3)
Lemma 9.2. For (r, s) ∈ N2 the following hold on U(r, s):
X = qrI, Y = qsI, K = q2r−2sI.
Proof. By Lemmas 5.3, 5.6 and since U(r, s) ⊆ V(r, s).
Lemma 9.3. The vector space U is invariant under each of
X±1, Y ±1, K±1.
Proof. By Lemma 9.2 and since U =
∑
(r,s)∈N2 U(r, s).
P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 663
Lemma 9.4 (See [32, Proposition 9.1]). The vector space U is invariant under each of
A∗L, B
∗
L, A
∗
R, B
∗
R.
For notational convenience, define U(r,−1) = 0 and U(−1, s) = 0 for r, s ∈ N.
Lemma 9.5. For (r, s) ∈ N2 we have
A∗LU(r, s) ⊆ U(r − 1, s), B∗LU(r, s) ⊆ U(r, s− 1),
A∗RU(r, s) ⊆ U(r − 1, s), B∗RU(r, s) ⊆ U(r, s− 1).
Proof. By (9.2) and Lemmas 6.4, 9.4.
Lemma 9.6. The subspace U is invariant under each of
Aℓ, Bℓ, Ar, Br.
Proof. By Definitions 9.1, 8.1.
Lemma 9.7. For (r, s) ∈ N2 we have
AℓU(r, s) ⊆ U(r + 1, s), BℓU(r, s) ⊆ U(r, s+ 1),
ArU(r, s) ⊆ U(r + 1, s), BrU(r, s) ⊆ U(r, s+ 1).
Proof. By (9.2) and Lemmas 8.3, 9.6.
In [32, Propositions 9.1, 9.3] there are many relations satisfied by the maps
K, K−1, A∗L, B
∗
L, A
∗
R, B
∗
R, Aℓ, Bℓ, Ar, Br.
For convenience we reproduce these relations in Appendix A. These relations will be used
in our main results.
10 The Uq(ŝl2)-module U and its submodule U
We now bring in the algebra Uq(ŝl2). The definition of this algebra can be found in Ap-
pendix B. In the present section, we turn the vector space U into a Uq(ŝl2)-module, and
describe the submodule U generated by the vector 1.
The following is our first main result.
Theorem 10.1. The vector space U becomes a Uq(ŝl2)-module on which the Uq(ŝl2)-
generators act as follows:
generator E0 F0 K±10 E1 F1 K
±1
1 D
±1
action on U A∗R
qArK
−1−q−1Aℓ
q−q−1 q
±1K∓1 B∗R
BrK−Bℓ
q−q−1 K
±1 X∓1
Proof. This is routinely checked using Lemmas 9.3, 9.4, 9.6 along with the relations in
Lemmas 6.6, 8.4 and Appendix A. Among the things to check, is that qArK−1−q−1Aℓ and
BrK−Bℓ satisfy the q-Serre relations. This can be checked easily using [32, Lemma 10.3,
Corollary 10.4].
664 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
Consider the Uq(ŝl2)-module U from Theorem 10.1. Recall the N2-grading of U from
around (9.2). Next we describe how the Uq(ŝl2)-generators act on the homogeneous com-
ponents of this grading.
Lemma 10.2. For (r, s) ∈ N2 the following hold on U(r, s):
K0 = q
2s−2r+1I, K1 = q
2r−2sI, D = q−rI. (10.1)
Moreover
E0U(r, s) ⊆ U(r − 1, s), F0U(r, s) ⊆ U(r + 1, s), (10.2)
E1U(r, s) ⊆ U(r, s− 1), F1U(r, s) ⊆ U(r, s+ 1). (10.3)
Proof. By Lemmas 9.2, 9.5, 9.7 and the data in Theorem 10.1.
In this paragraph we recall a few concepts about Uq(ŝl2)-modules; see for example [20,
Section 3.2]. Let W denote a Uq(ŝl2)-module. A weight space for W is a common
eigenspace for the action of K0,K1, D on W . The sum of these weight spaces is direct.
We call W a weight module whenever W is equal to the sum of its weight spaces. If W is
a weight module, then every submodule of W is a weight module [20, Proposition 3.2.1].
We return our attention to the Uq(ŝl2)-module U from Theorem 10.1. We mentioned above
Lemma 9.2 that the sum U =
∑
(r,s)∈N2 U(r, s) is direct. By this and (10.1), the Uq(ŝl2)-
module U is a weight module, and its weight spaces are the nonzero subspaces among
U(r, s) (r, s ∈ N). Note that these weight spaces have finite dimension.
It turns out that the Uq(ŝl2)-module U is not irreducible. Next we consider its submodules.
Definition 10.3. Let U denote the submodule of the Uq(ŝl2)-module U that is generated
by the vector 1.
Lemma 10.4. For the Uq(ŝl2)-module U,
(i) U is contained in every nonzero submodule of the Uq(ŝl2)-module U;
(ii) U is the unique irreducible submodule of the Uq(ŝl2)-module U.
Proof. (i) Let W denote a nonzero submodule of the Uq(ŝl2)-module U. The vector space
W is invariant under A∗R, B
∗
R by Theorem 10.1, so 1 ∈ W by Lemma 6.9. The Uq(ŝl2)-
module U is generated by 1, so U ⊆ W .
(ii) By (i) above.
Next we consider how the Uq(ŝl2)-generators act on the vector 1.
Lemma 10.5. For the Uq(ŝl2)-module U,
K01 = q1, K11 = 1, D1 = 1,
E01 = 0, F
2
0 1 = 0, E11 = 0, F11 = 0.
Proof. This is routinely checked using the data in Theorem 10.1 and Lemma 10.2.
P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 665
There is a well known Uq(ŝl2)-module V (Λ0) that is said to be basic; see [20, page 221].
The module V (Λ0) is highest weight, integrable, and level one; see [4, Chapter 10] and [27,
Chapter 5]. The module V (Λ0) is characterized as follows.
Lemma 10.6 (See [27, pages 63, 64]). There exists a Uq(ŝl2)-module V (Λ0) with the
following property: V (Λ0) is generated by a nonzero vector v such that
K0v = qv, K1v = v, Dv = v,
E0v = 0, F
2
0 v = 0, E1v = 0, F1v = 0.
Moreover V (Λ0) is irreducible, infinite-dimensional, and unique up to isomorphism of
Uq(ŝl2)-modules.
The following is our second main result.
Theorem 10.7. The Uq(ŝl2)-modules U and V (Λ0) are isomorphic.
Proof. By Definition 10.3 and Lemmas 10.5, 10.6.
Descriptions of V (Λ0) can be found in [4, Chapter 10] and [20, Section 9] and [27, Chap-
ter 5]; see also [14, Section 20.4] and [28, Chapter 14]. Our next general goal is to describe
V (Λ0) from the point of view of U.
Definition 10.8. For (r, s) ∈ N2 define U(r, s) = U ∩ U(r, s).
The Uq(ŝl2)-module U is a weight module, and its weight spaces are the nonzero subspaces
among U(r, s) (r, s ∈ N). More detail is given in the next result.
Lemma 10.9. The following (i) – (iv) hold for the Uq(ŝl2)-module U.
(i) U(0, 0) = F1.
(ii) The sum U =
∑
(r,s)∈N2 U(r, s) is direct.
(iii) For (r, s) ∈ N2 the following hold on U(r, s):
K0 = q
2s−2r+1I, K1 = q
2r−2sI, D = q−rI.
(iv) For (r, s) ∈ N2,
E0U(r, s) ⊆ U(r − 1, s), F0U(r, s) ⊆ U(r + 1, s),
E1U(r, s) ⊆ U(r, s− 1), F1U(r, s) ⊆ U(r, s+ 1),
where U(r,−1) = 0 and U(−1, s) = 0.
Proof. (i) By Definition 10.8 and since U(0, 0) = F1.
(ii) Since U is a weight module.
(iii), (iv) By Lemma 10.2 and Definition 10.8.
Our next goal is to describe the weight space dimensions for the Uq(ŝl2)-module U. Recall
the partition numbers {pn}n∈N from Section 3.
666 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
Proposition 10.10. For (r, s) ∈ N2 the vector space U(r, s) ̸= 0 if and only if r ≥ (r−s)2.
In this case dimU(r, s) = pn, where n = r − (r − s)2.
Proof. For the Uq(ŝl2)-module V (Λ0) the weight space dimensions are described in [20,
pages 221, 222]. The result follows from that description and Theorem 10.7 above.
Example 10.11. For 0 ≤ r, s ≤ 6 the dimension of U(r, s) is given in the (r, s)-entry of
the matrix below: 
1 0 0 0 0 0 0
1 1 1 0 0 0 0
0 1 2 1 0 0 0
0 0 2 3 2 0 0
0 0 1 3 5 3 1
0 0 0 1 5 7 5
0 0 0 0 2 7 11

Compare the above matrix with the one in Example 3.7.
Next we describe the generating function
∑
(r,s)∈N2 dimU(r, s)t
rus.
Define the generating function
ϕ(t, u) =
∑
n∈Z
tn
2
un
2−n. (10.4)
Note that
ϕ(t, u) = 1 + t+ tu2 + t4u2 + t4u6 + · · · .
Proposition 10.12. We have∑
(r,s)∈N2
dimU(r, s)trus = p(tu)ϕ(t, u),
where p(t) is from (3.4) and ϕ(t, u) is from (10.4).
Proof. This is a reformulation of Proposition 10.10.
In Appendix C, we give a basis for each nonzero U(r, s) such that r + s ≤ 10.
11 A characterization of the Uq(ŝl2)-module U
In order to motivate this section, we glance at the basis vectors displayed in Appendix C.
Each displayed vector is a linear combination of some words in V that do not begin with y
or xx. Consequently, each displayed vector is contained in the kernel of B∗L and the kernel
of (A∗L)
2. Using this observation, we will characterize the Uq(ŝl2)-module U.
Definition 11.1. Let V denote the intersection of the kernel of B∗L and the kernel of (A∗L)2.
Note that V is a subspace of the vector space V.
We have several comments about V.
P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 667
Lemma 11.2. The vector space V has a basis consisting of the words in V that do not
begin with y or xx.
Proof. By Definitions 6.1, 11.1.
Lemma 11.3. The sum V = F1+ Fx+ xyV is direct.
Proof. By Lemma 11.2.
We are going to show that U = U ∩V. We will do this in several steps. In the first step,
we show that U ∩V is a submodule of the Uq(ŝl2)-module U.
Lemma 11.4. The vector space V is invariant under each of
X±1, Y ±1, K±1, A∗R, B
∗
R.
Proof. Use Definitions 5.1, 5.4 and Lemma 11.3.
Lemma 11.5. The vector space V is invariant under each of
qArK
−1 − q−1Aℓ, BrK −Bℓ.
Proof. We will use Lemma 11.3. The map qArK−1 − q−1Aℓ sends 1 7→ (q − q−1)x and
x 7→ 0. The map BrK −Bℓ sends 1 7→ 0 and
x 7→ q2x ⋆ y − y ⋆ x = q2(xy + q−2yx)− (yx+ q−2xy) = (q2 − q−2)xy.
Pick (r, s) ∈ N2 and a word w ∈ V(r, s). The map qArK−1 − q−1Aℓ sends
xyw 7→ q1+2s−2r(xyw) ⋆ x− q−1x ⋆ (xyw). (11.1)
Using (7.2) we obtain
(xyw) ⋆ x = xy(w ⋆ x) + [2]qq
2r−2s−1xxyw. (11.2)
Using (7.1) we obtain
x ⋆ (xyw) = q[2]qxxyw + xy(x ⋆ w). (11.3)
By (11.1) – (11.3) the map qArK−1 − q−1Aℓ sends
xyw 7→ xy
(
q1+2s−2rw ⋆ x− q−1x ⋆ w
)
.
The map BrK −Bℓ sends
xyw 7→ q2r−2s(xyw) ⋆ y − y ⋆ (xyw). (11.4)
Using (7.2) we obtain
(xyw) ⋆ y = xy(w ⋆ y) + q2s−2r+2xyyw + q2s−2ryxyw. (11.5)
Using (7.1) we obtain
y ⋆ (xyw) = yxyw + q−2xyyw + xy(y ⋆ w). (11.6)
By (11.4) – (11.6) the map BrK −Bℓ sends
xyw 7→ xy
(
(q2 − q−2)yw + q2r−2sw ⋆ y − y ⋆ w
)
.
The result follows from the above comments.
668 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
Lemma 11.6. The vector space U ∩V is a submodule of the Uq(ŝl2)-module U.
Proof. By Theorem 10.1 and Lemmas 11.4, 11.5.
Lemma 11.7. The vector space U is a submodule of the Uq(ŝl2)-module U ∩V.
Proof. We have 1 ∈ U by the comment below (9.2). We have 1 ∈ V by Lemma 11.3. So
1 ∈ U ∩V. The result follows in view of Definition 10.3 and Lemma 11.6.
The Uq(ŝl2)-module U ∩ V is a weight module, and its weight spaces are the nonzero
subspaces among U(r, s) ∩ V (r, s ∈ N). We will return to the Uq(ŝl2)-module U ∩ V
after some comments about U.
Lemma 11.8. The Uq(ŝl2)-generators E0, E1 are locally nilpotent on the Uq(ŝl2)-module
U.
Proof. By Lemma 6.5 and Theorem 10.1.
Lemma 11.9. The Uq(ŝl2)-generators F0, F1 are not locally nilpotent on the Uq(ŝl2)-
module U.
Proof. The words xx and y are contained in U, but Fn0 (xx) ̸= 0 and Fn1 y ̸= 0 for all
n ∈ N.
Lemma 11.10. The Uq(ŝl2)-generators F0, F1 are locally nilpotent on the Uq(ŝl2)-module
U ∩V.
Proof. First consider F0. Assume that there exists v ∈ U ∩V such that Fn+10 v ̸= 0 for all
n ∈ N. We will get a contradiction. By our comments below Lemma 11.7, we may assume
without loss of generality that v ∈ U(r, s) for some (r, s) ∈ N2. We have r ≥ 1 and s ≥ 1,
by Lemma 11.3 and F 20 1 = 0 and F0x = 0. Let n ∈ N. By (10.2) and Lemma 11.6, we
have Fn+10 v ∈ U(r + n + 1, s) ∩ V. In particular F
n+1
0 v ∈ V, so (A∗L)2F
n+1
0 v = 0
and B∗LF
n+1
0 v = 0 in view of Definition 11.1. We have A
∗
LF
n+1
0 v ̸= 0, by Lemma 6.7(v)
and since B∗LF
n+1
0 v = 0. We have A
∗
LF
n+1
0 v ∈ U(r + n, s) by Lemma 9.5. By these
comments 0 ̸= A∗LF
n+1
0 v ∈ ker(A∗L)∩U(r+n, s). Therefore 0 ̸= ker(A∗L)∩U(r+n, s).
The map A∗L is locally nilpotent by Lemma 6.5. In Appendix A we find A
∗
LAℓ−q2AℓA∗L =
I . The vector space U is invariant under A∗L and Aℓ. By these comments, we may apply
Appendix E with S = A∗L and T = Aℓ and V = U. By Lemma 17.8 the map A∗L is
surjective on U. By this and Lemma 9.5, A∗LU(r + n, s) = U(r + n − 1, s). By this and
0 ̸= ker(A∗L) ∩ U(r + n, s), we obtain dimU(r + n − 1, s) < dimU(r + n, s). Since
n ∈ N is arbitrary,
dimU(r − 1, s) < dimU(r, s) < dimU(r + 1, s) < dimU(r + 2, s) < · · ·
This contradicts (3.8) and (9.3), so F0 is locally nilpotent.
Next we consider F1. Assume that there exists v ∈ U ∩ V such that Fm+11 v ̸= 0 for all
m ∈ N. We will get a contradiction. By our comments below Lemma 11.7, we may assume
without loss of generality that v ∈ U(r, s) for some (r, s) ∈ N2. We have r ≥ 1 and s ≥ 1,
by Lemma 11.3 and F11 = 0 and F 31 x = 0. Let m ∈ N. By (10.3) and Lemma 11.6, we
have Fm+11 v ∈ U(r, s + m + 1) ∩ V. In particular F
m+1
1 v ∈ V, so B∗LF
m+1
1 v = 0 in
view of Definition 11.1. By these comments 0 ̸= Fm+11 v ∈ ker(B∗L) ∩ U(r, s +m + 1).
P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 669
Therefore 0 ̸= ker(B∗L)∩U(r, s+m+1). The map B∗L is locally nilpotent by Lemma 6.5.
In Appendix A we find B∗LBℓ − q2BℓB∗L = I . The vector space U is invariant under
B∗L and Bℓ. By these comments, we may apply Appendix E with S = B
∗
L and T = Bℓ
and V = U. By Lemma 17.8 the map B∗L is surjective on U. By this and Lemma 9.5,
B∗LU(r, s+m+1) = U(r, s+m). By this and 0 ̸= ker(B∗L)∩U(r, s+m+1), we obtain
dimU(r, s+m) < dimU(r, s+m+ 1). Since m ∈ N is arbitrary,
dimU(r, s) < dimU(r, s+ 1) < dimU(r, s+ 2) < dimU(r, s+ 3) < · · ·
This contradicts (3.7) and (9.3), so F1 is locally nilpotent.
The following is our third main result.
Theorem 11.11. We have U = U ∩V.
Proof. By Lemmas 11.8, 11.10 the Uq(ŝl2)-generators E0, E1, F0, F1 are locally nilpotent
on the Uq(ŝl2)-module U ∩V. Therefore, the Uq(ŝl2)-module U ∩V is integrable in the
sense of [4, Definition 4.2]. By this and [20, Theorem 3.5.4], the Uq(ŝl2)-module U∩V is
completely reducible. By Lemma 11.7, U is a submodule of the Uq(ŝl2)-module U ∩ V.
By these comments, there exists a submodule W of the Uq(ŝl2)-module U ∩ V such that
the sum U ∩V = U +W is direct. Assume for the moment that W ̸= 0. Then U ⊆ W
by Lemma 10.4(i), for a contradiction. Consequently W = 0, so U = U ∩V.
12 Variations on the theme
In Theorem 10.1 we turned the vector space U into a Uq(ŝl2)-module. In this section
we describe three more ways to do this. Each way yields a Uq(ŝl2)-module U that is
isomorphic to the one in Theorem 10.1.
Proposition 12.1. For each row in the table below, the vector space U becomes a Uq(ŝl2)-
module on which the Uq(ŝl2)-generators act as indicated:
generator E0 F0 K±10 E1 F1 K
±1
1 D
±1
action on U B∗R
qBrK−q−1Bℓ
q−q−1 q
±1K±1 A∗R
ArK
−1−Aℓ
q−q−1 K
∓1 Y ∓1
action on U A∗L
qAℓK
−1−q−1Ar
q−q−1 q
±1K∓1 B∗L
BℓK−Br
q−q−1 K
±1 X∓1
action on U B∗L
qBℓK−q−1Br
q−q−1 q
±1K±1 A∗L
AℓK
−1−Ar
q−q−1 K
∓1 Y ∓1
The above three Uq(ŝl2)-modules U are isomorphic to the Uq(ŝl2)-module U in Theo-
rem 10.1. For row 1 (resp. row 2) (resp. row 3), a Uq(ŝl2)-module isomorphism is given by
the restriction of σ (resp. †) (resp. τ ) to U.
Proof. Below (9.1) we mentioned that U is invariant under each of σ, †, τ . The result
follows from this along with Lemmas 9.3, 9.4, 9.6 and Lemmas 5.7, 6.10, 8.5.
13 Appendix A: Some relations
In this appendix we list some relations satisfied by the maps
K, K−1, A∗L, B
∗
L, A
∗
R, B
∗
R, Aℓ, Bℓ, Ar, Br.
670 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
Proposition 13.1 (See [32, Proposition 9.1]). We have
KA∗L = q
−2A∗LK, KB
∗
L = q
2B∗LK,
KA∗R = q
−2A∗RK, KB
∗
R = q
2B∗RK,
KAℓ = q
2AℓK, KBℓ = q
−2BℓK,
KAr = q
2ArK, KBr = q
−2BrK,
A∗LA
∗
R = A
∗
RA
∗
L, B
∗
LB
∗
R = B
∗
RB
∗
L,
A∗LB
∗
R = B
∗
RA
∗
L, B
∗
LA
∗
R = A
∗
RB
∗
L,
AℓAr = ArAℓ, BℓBr = BrBℓ,
AℓBr = BrAℓ, BℓAr = ArBℓ,
A∗LBr = BrA
∗
L, B
∗
LAr = ArB
∗
L,
A∗RBℓ = BℓA
∗
R, B
∗
RAℓ = AℓB
∗
R,
A∗LBℓ = q
−2BℓA
∗
L, B
∗
LAℓ = q
−2AℓB
∗
L,
A∗RBr = q
−2BrA
∗
R, B
∗
RAr = q
−2ArB
∗
R,
A∗LAℓ − q2AℓA∗L = I, A∗RAr − q2ArA∗R = I,
B∗LBℓ − q2BℓB∗L = I, B∗RBr − q2BrB∗R = I,
A∗LAr −ArA∗L = K, B∗LBr −BrB∗L = K−1,
A∗RAℓ −AℓA∗R = K, B∗RBℓ −BℓB∗R = K−1,
A3ℓBℓ − [3]qA2ℓBℓAℓ + [3]qAℓBℓA2ℓ −BℓA3ℓ = 0,
B3ℓAℓ − [3]qB2ℓAℓBℓ + [3]qBℓAℓB2ℓ −AℓB3ℓ = 0,
A3rBr − [3]qA2rBrAr + [3]qArBrA2r −BrA3r = 0,
B3rAr − [3]qB2rArBr + [3]qBrArB2r −ArB3r = 0.
Proposition 13.2 (See [32, Proposition 9.3]). The following relations hold on U:
(A∗L)
3B∗L − [3]q(A∗L)2B∗LA∗L + [3]qA∗LB∗L(A∗L)2 −B∗L(A∗L)3 = 0,
(B∗L)
3A∗L − [3]q(B∗L)2A∗LB∗L + [3]qB∗LA∗L(B∗L)2 −A∗L(B∗L)3 = 0,
(A∗R)
3B∗R − [3]q(A∗R)2B∗RA∗R + [3]qA∗RB∗R(A∗R)2 −B∗R(A∗R)3 = 0,
(B∗R)
3A∗R − [3]q(B∗R)2A∗RB∗R + [3]qB∗RA∗R(B∗R)2 −A∗R(B∗R)3 = 0.
P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 671
14 Appendix B: The algebra Uq(ŝl2)
In this appendix we recall the quantized enveloping algebra Uq(ŝl2). We will generally
follow the approach of Ariki [4, Section 3.3]. We will refer to the matrix
A =
(
2 −2
−2 2
)
.
We index the rows and columns of A by 0, 1.
Definition 14.1 (See [4, Definition 3.16]). Define the algebra Uq(ŝl2) by generators
K±1i , D
±1, Ei, Fi, i ∈ {0, 1}
and the following relations. For i, j ∈ {0, 1},
KiK
−1
i = K
−1
i Ki = 1, DD
−1 = D−1D = 1,
[Ki,Kj ] = 0, [D,Ki] = 0,
KiEjK
−1
i = q
Ai,jEj , KiFjK
−1
i = q
−Ai,jFj ,
DE0D
−1 = qE0, DF0D
−1 = q−1F0,
[D,E1] = 0, [D,F1] = 0,
[Ei, Fj ] = δi,j
Ki −K−1i
q − q−1
,
[Ei, [Ei, [Ei, Ej ]q]q−1 ] = 0, [Fi, [Fi, [Fi, Fj ]q]q−1 ] = 0, i ̸= j.
Note 14.2. The Ariki notation is related to our notation as follows.
Ariki notation our notation
v q
ti Ki
vd D
αj(hi) Ai,j
Note 14.3 (See [4, Section 3.3]). The algebra Uq(ŝl2) is sometimes called the quantum
algebra of type A(1)1 .
15 Appendix C: The subspaces U(r, s)
In this appendix, we give a basis for each nonzero U(r, s) such that r + s ≤ 10.
r s basis for U(r, s)
0 0 1
r s basis for U(r, s)
1 0 x
r s basis for U(r, s)
1 1 xy
672 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
r s basis for U(r, s)
2 1 xyx
1 2 xyy
r s basis for U(r, s)
2 2 xyxy, xyyx
r s basis for U(r, s)
3 2 xyxyx, xyyxx
2 3 xyxyy + xyyxy
r s basis for U(r, s)
4 2 xyxyxx+ [3]qxyyxxx
3 3 xyxyxy, xyyxxy, xyyxyx+ xyxyyx
r s basis for U(r, s)
4 3 xyxyxyx, xyyxyxx+ xyxyyxx+ xyyxxyx,
xyxyxxy + [3]qxyyxxxy + xyyxxyx
3 4 xyyxxyy, xyyxyxy + xyxyyxy + xyxyxyy
r s basis for U(r, s)
5 3 [3]qxyyxyxxx+ [3]qxyxyyxxx+ [2]
2
qxyyxxyxx
+2xyxyxyxx+ xyxyxxyx+ [3]qxyyxxxyx
4 4 xyxyxyxy, xyyxxyyx,
xyyxyxxy + xyxyyxxy + xyyxxyxy,
xyxyxxyy + [3]qxyyxxxyy + xyyxxyxy,
xyyxyxyx+ xyxyyxyx+ xyxyxyyx
r s basis for U(r, s)
5 4 [3]qxyyxyxxxy + [3]qxyxyyxxxy + [2]
2
qxyyxxyxxy
+2xyxyxyxxy + xyxyxxyxy + [3]qxyyxxxyxy
+xyyxyxxyx+ xyxyyxxyx+ xyyxxyxyx,
xyxyxyxyx, xyyxxyyxx,
xyxyxxyyx+ [3]qxyyxxxyyx+ xyyxxyxyx,
xyyxyxyxx+ xyxyyxyxx+ xyxyxyyxx
+xyyxyxxyx+ xyxyyxxyx+ xyyxxyxyx
4 5 xyyxyxxyy + xyxyyxxyy + xyyxxyxyy + xyyxxyyxy,
xyxyxyyxy + xyxyyxyxy + xyyxyxyxy + xyxyxyxyy,
[3]qxyxyxxyyy + [3]
2
qxyyxxxyyy + [3]qxyyxxyxyy + xyxyxyxyy
P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 673
r s basis for U(r, s)
6 4 [3]qxyyxyxxxyx+ [3]qxyxyyxxxyx+ [2]
2
qxyyxxyxxyx
+2xyxyxyxxyx+ xyxyxxyxyx+ [3]qxyyxxxyxyx
+xyyxyxxyxx+ xyxyyxxyxx+ xyyxxyxyxx
+3xyxyxyxyxx+ [3]qxyyxyxyxxx+ [3]qxyxyyxyxxx
+[3]qxyxyxyyxxx+ [3]qxyyxyxxyxx+ [3]qxyxyyxxyxx+ [3]qxyyxxyxyxx,
[3]qxyyxxyyxxx+ xyxyxxyyxx+ [3]qxyyxxxyyxx+ xyyxxyxyxx
5 5 xyxyxyxyxy,
xyxyxyxyyx+ xyxyxyyxyx+ xyxyyxyxyx+ xyyxyxyxyx,
xyxyxyyxxy + xyxyyxyxxy + xyyxyxyxxy + xyxyyxxyxy
+xyyxxyxyxy + xyyxyxxyxy,
xyyxxyyxyx+ xyyxxyxyyx+ xyyxyxxyyx+ xyxyyxxyyx,
xyyxxyyxxy,
[3]qxyyxxyxyyx+ xyxyxxyyxy + [3]qxyyxxxyyxy + [3]
2
qxyyxxxyyyx
+[3]qxyxyxxyyyx+ xyxyxyxyyx+ xyyxxyxyxy,
xyxyyxxyxy + [3]qxyxyyxxxyy + 2xyxyxyxxyy + xyyxyxxyxy
+[3]qxyyxyxxxyy + [2]
2
qxyyxxyxxyy + 2xyyxxyxyxy + [3]qxyyxxxyxyy
+[3]qxyyxxxyyxy + xyxyxxyxyy + xyxyxxyyxy
4 6 [2]q[3]qxyyxyxxyyy + [2]q[3]qxyxyyxxyyy + [2]q[3]qxyyxxyxyyy
+[2]q[3]qxyyxxyyxyy + [2]qxyxyxyyxyy + [2]qxyxyyxyxyy
+[2]qxyyxyxyxyy + [2]q[3]qxyxyxyxyyy + [3]q[4]qxyxyxxyyyy
+[3]2q[4]qxyyxxxyyyy + [3]q[4]qxyyxxyxyyy
16 Appendix D: Some matrix representations
In this appendix we consider the Uq(ŝl2)-module U from Definition 10.3. We display
the matrices that represent the actions of E0, F0,K0, E1, F1,K1, D on the bases in Ap-
pendix C.
On U(0, 0):
K0 :
(
q
)
, K1 :
(
1
)
, D :
(
1
)
.
From U(1, 0) to U(0, 0):
E0 :
(
1
)
, E1 :
(
0
)
From U(0, 0) to U(1, 0):
F0 :
(
1
)
, F1 :
(
0
)
On U(1, 0):
K0 :
(
q−1
)
, K1 :
(
q2
)
, D :
(
q−1
)
.
From U(1, 1) to U(1, 0):
E0 :
(
0
)
, E1 :
(
1
)
From U(1, 0) to U(1, 1):
F0 :
(
0
)
, F1 :
(
[2]q
)
674 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
On U(1, 1):
K0 :
(
q
)
, K1 :
(
1
)
, D :
(
q−1
)
.
From U(2, 1) +U(1, 2) to U(1, 1):
E0 :
(
1 0
)
, E1 :
(
0 1
)
From U(1, 1) to U(2, 1) +U(1, 2):
F0 :
(
1
0
)
, F1 :
(
0
[2]q
)
On U(2, 1) +U(1, 2):
K0 : diag(q
−1, q3), K1 : diag(q
2, q−2), D : diag(q−2, q−1).
From U(2, 2) to U(2, 1) +U(1, 2):
E0 :
(
0 0
0 1
)
, E1 :
(
1 0
0 0
)
From U(2, 1) +U(1, 2) to U(2, 2):
F0 :
(
0 1
0 [3]q
)
, F1 :
(
[2]q 0
[2]q 0
)
On U(2, 2):
K0 : diag(q, q), K1 : diag(1, 1), D : diag(q
−2, q−2).
From U(3, 2) +U(2, 3) to U(2, 2):
E0 :
(
1 0 0
0 1 0
)
, E1 :
(
0 0 1
0 0 1
)
From U(2, 2) to U(3, 2) +U(2, 3):
F0 :
1 10 [2]2q
0 0
 , F1 :
 0 00 0
[2]q 0

On U(3, 2) +U(2, 3):
K0 : diag(q
−1, q−1, q3), K1 : diag(q
2, q2, q−2), D : diag(q−3, q−3, q−2).
From U(4, 2) +U(3, 3) to U(3, 2) +U(2, 3):
E0 :
 1 0 0 0[3]q 0 0 0
0 0 0 1
 , E1 :
0 1 0 00 0 1 0
0 0 0 0

P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 675
From U(3, 2) +U(2, 3) to U(4, 2) +U(3, 3):
F0 :

0 1 0
0 0 2
0 0 [2]2q
0 0 [3]q
 , F1 :

0 0 0
[2]q 0 0
0 [2]q 0
[2]q 0 0

On U(4, 2) +U(3, 3):
K0 : diag(q
−3, q, q, q), K1 : diag(q
4, 1, 1, 1), D : diag(q−4, q−3, q−3, q−3).
From U(4, 3) +U(3, 4) to U(4, 2) +U(3, 3):
E0 :

0 0 0 0 0
1 0 0 0 0
0 1 1 0 0
0 1 0 0 0
 , E1 :

0 0 1 0 0
0 0 0 0 1
0 0 0 1 0
0 0 0 0 1

From U(4, 2) +U(3, 3) to U(4, 3) +U(3, 4):
F0 :

0 1 0 2
0 0 0 [2]2q
0 0 1 0
0 0 0 0
0 0 0 0
 , F1 :

[2]q 0 0 0
[2]q 0 0 0
[4]q 0 0 0
0 0 [2]q 0
0 [2]q 0 0

On U(4, 3) +U(3, 4):
K0 : diag(q
−1, q−1, q−1, q3, q3), K1 : diag(q
2, q2, q2, q−2, q−2),
D : diag(q−4, q−4, q−4, q−3, q−3).
From U(5, 3) +U(4, 4) to U(4, 3) +U(3, 4):
E0 :

2 0 0 0 0 0
[3]q 0 0 0 0 0
1 0 0 0 0 0
0 0 1 0 0 0
0 0 0 0 0 1
 , E1 :

0 1 0 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 0
0 0 0 0 0 0

From U(4, 3) +U(3, 4) to U(5, 3) +U(4, 4):
F0 :

0 1 0 0 0
0 0 0 0 3
0 0 0 [3]q 0
0 0 0 0 [2]2q
0 0 0 1 0
0 0 0 0 [3]q
 , F1 :

0 0 0 0 0
[2]q 0 [2]q 0 0
0 [2]q [2]q 0 0
0 [2]q [2]q 0 0
0 0 [2]q[3]q 0 0
[2]q 0 0 0 0

On U(5, 3) +U(4, 4):
K0 : diag(q
−3, q, q, q, q, q), K1 : diag(q
4, 1, 1, 1, 1, 1),
D : diag(q−5, q−4, q−4, q−4, q−4, q−4).
676 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
From U(5, 4) +U(4, 5) to U(5, 3) +U(4, 4):
E0 :

0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
1 0 0 0 1 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
 , E1 :

1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 1
0 0 0 0 0 1 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 [3]q
0 0 0 0 0 0 1 0

From U(5, 3) +U(4, 4) to U(5, 4) +U(4, 5):
F0 :

0 0 0 1 0 0
0 1 0 0 0 3
0 0 [2]2q 0 0 0
0 0 1 0 1 0
0 0 0 0 0 [2]2q
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0

, F1 :

[4]q 0 0 0 0 0
3[2]q 0 0 0 0 0
[2]3q 0 0 0 0 0
[2]q[3]q 0 0 0 0 0
2[2]q 0 0 0 0 0
0 0 0 [2]q [2]q 0
0 [2]q 0 0 0 0
0 0 0 0 [2]q 0

On U(5, 4) +U(4, 5):
K0 : diag(q
−1, q−1, q−1, q−1, q−1, q3, q3, q3),K1 : diag(q
2, q2, q2, q2, q2, q−2, q−2, q−2),
D : diag(q−5, q−5, q−5, q−5, q−5, q−4, q−4, q−4).
From U(6, 4) +U(5, 5) +U(4, 6) to U(5, 4) +U(4, 5):
E0 :

1 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0
0 [3]q 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0
[3]q 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0
0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0

,
E1 :

0 0 0 0 0 0 0 0 1 0
0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1 1 0
0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 [2]q[3]q
0 0 0 0 0 0 0 0 0 [2]q
0 0 0 0 0 0 0 0 0 [4]q

P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 677
From U(5, 4) +U(4, 5) to U(6, 4) +U(5, 5) +U(4, 6):
F0 :

0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 0 4 1
0 0 0 0 0 0 [3]q 0
0 0 0 0 0 0 [2]2q 0
0 0 0 0 0 [3]q 0 0
0 0 0 0 0 [2]2q 0 0
0 0 0 0 0 0 0 [3]q
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0

,
F1 :

0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
3[2]q [2]q 0 0 0 0 0 0
0 [2]q 0 0 0 0 0 0
2[2]q 0 0 0 [2]q 0 0 0
[2]q 0 0 [2]q [2]q 0 0 0
[2]3q 0 [2]q 0 0 0 0 0
0 0 0 [2]q 0 0 0 0
[2]q[3]q 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1

On U(6, 4) +U(5, 5) +U(4, 6):
K0 : diag(q
−3, q−3, q, q, q, q, q, q, q, q5), K1 : diag(q
4, q4, 1, 1, 1, 1, 1, 1, 1, q−4),
D : diag(q−6, q−6, q−5, q−5, q−5, q−5, q−5, q−5, q−5, q−4).
17 Appendix E: Some linear algebra
In this appendix we consider the following situation. Let V denote an infinite-dimensional
vector space. Let S : V → V and T : V → V denote F-linear maps. Assume that S is
locally nilpotent and
ST − q2TS = I. (17.1)
We will show that S is surjective and T is injective. We remark that the surjectivity of S is
used in the proof of Lemma 11.10, and the injectivity of T is used to obtain the surjectivity
of S.
Lemma 17.1. The map T is injective.
Proof. Let v ∈ V such that Tv = 0. We show that v = 0. For n ≥ 1, use (17.1) and
induction on n to obtain
TSnv = −q−n−1[n]qSn−1v. (17.2)
Since S is locally nilpotent, there exists n ≥ 1 such that Snv = 0. By applying (17.2)
repeatedly, we see that each of Snv, Sn−1v, . . . Sv, v is equal to 0. In particular v = 0.
678 Ars Math. Contemp. 23 (2023) #P4.10 / 649–682
For n ∈ N, we adjust (17.1) to obtain
ST − qn[n+ 1]qI = q2
(
TS − qn−1[n]qI
)
. (17.3)
Therefore, the kernel of ST − qn[n + 1]qI is equal to the kernel of TS − qn−1[n]qI . Let
Vn denote this common kernel. By construction(
ST − qn[n+ 1]qI
)
Vn = 0,
(
TS − qn−1[n]qI
)
Vn = 0. (17.4)
Note that the sum
∑
n∈N Vn is direct. For notational convenience define V−1 = 0.
Lemma 17.2. We have ker(S) = V0.
Proof. By the discussion below (17.3), we obtain ker(TS) = V0. The map T is injective
by Lemma 17.1, so ker(S) = ker(TS). Therefore ker(S) = V0.
Lemma 17.3. For n ∈ N we have
SVn ⊆ Vn−1, TVn ⊆ Vn+1.
Proof. First we verify SVn ⊆ Vn−1. For n = 0 this holds by Lemma 17.2. For n ≥ 1 we
use (17.4) to obtain(
ST − qn−1[n]qI
)
SVn = S
(
TS − qn−1[n]qI
)
Vn = S0 = 0,
so SVn ⊆ Vn−1. Next we verify TVn ⊆ Vn+1. For n ≥ 0 we have(
TS − qn[n+ 1]qI
)
TVn = T
(
ST − qn[n+ 1]qI
)
Wn = T0 = 0,
so TVn ⊆ Vn+1.
Lemma 17.4. For n ≥ 1 the following maps are inverses:
S : Vn → Vn−1, q1−n[n]−1q T : Vn−1 → Vn. (17.5)
Proof. By (17.4) we have (STn − I)Vn−1 = 0 and (TnS − I)Vn = 0, where Tn =
q1−n[n]−1q T .
Lemma 17.5. For n ≥ 1 the maps
S : Vn → Vn−1, T : Vn−1 → Vn
are bijections.
Proof. By Lemma 17.4.
Lemma 17.6. For n ∈ N,
ker(Sn+1) = V0 + V1 + · · ·+ Vn. (17.6)
Proof. We use induction on n. First assume that n = 0. Then (17.6) holds by Lemma 17.2.
Next assume that n ≥ 1. The inclusion ⊇ in (17.6) holds by Lemma 17.3. We next obtain
the inclusion ⊆ in (17.6). Let v ∈ ker(Sn+1). We will show that v ∈ V0 + V1 + · · ·+ Vn.
We have 0 = Sn+1v = SnSv, so by induction Sv ∈ V0+V1+· · ·+Vn−1. By Lemma 17.5,
there exists w ∈ V1 + V2 + · · · + Vn such that Sw = Sv. Therefore S(w − v) = 0, so
w − v ∈ V0. By these comments v ∈ V0 + V1 + · · ·+ Vn.
P. Terwilliger : Using a q-shuffle algebra to describe the basic module V (Λ0) . . . 679
Lemma 17.7. We have V =
∑
n∈N Vn.
Proof. Since S is locally nilpotent, we have V = ∪n∈Nker(Sn+1). The result follows
from this and Lemma 17.6.
Lemma 17.8. The map S is surjective.
Proof. By Lemma 17.7 and since Vn = SVn+1 for n ∈ N.
ORCID iDs
Paul Terwilliger https://orcid.org/0000-0003-0942-5489
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