Volume 21, Number 2, Fall/Winter 2021, Pages 151–317
Covered by:
Mathematical Reviews
zbMATH (formerly Zentralblatt MATH)
COBISS
SCOPUS
Science Citation Index-Expanded (SCIE)
Web of Science
ISI Alerting Service
Current Contents/Physical, Chemical & Earth Sciences (CC/PC & ES)
dblp computer science bibliography
The University of Primorska
The Society of Mathematicians, Physicists and Astronomers of Slovenia
The Institute of Mathematics, Physics and Mechanics
The Slovenian Discrete and Applied Mathematics Society
The publication is partially supported by the Slovenian Research Agency from the Call for
co-financing of scientific periodical publications.

Contents
Classification of skew morphisms of cyclic groups which are square roots of
automorphisms
Kan Hu, Young Soo Kwon, Jun-Yang Zhang . . . . . . . . . . . . . . . . . 151
Trivalent dihedrants and bi-dihedrants
Mi-Mi Zhang, Jin-Xin Zhou . . . . . . . . . . . . . . . . . . . . . . . . . 175
A generalization of balanced tableaux and marriage problems with unique
solutions
Brian Tianyao Chan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Enumerating symmetric peaks in non-decreasing Dyck paths
Sergi Elizalde, Rigoberto Flórez, José Luis Ramírez . . . . . . . . . . . . . 219
Density results for Graovac-Pisanski’s distance number
Lowell Abrams, Lindsey-Kay Lauderdale . . . . . . . . . . . . . . . . . . 243
Decompositions of the automorphism groups of edge-colored graphs into the
direct product of permutation groups
Mariusz Grech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Efficient proper embedding of a daisy cube
Aleksander Vesel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
The polynomial method for list-colouring extendability of outerplanar graphs
Przemysław Gordinowicz, Paweł Twardowski . . . . . . . . . . . . . . . . 283
On generalized strong complete mappings and mutually orthogonal Latin
squares
Amela Muratović-Ribić . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
Point-primitive generalised hexagons and octagons and projective linear groups
Stephen P. Glasby, Emilio Pierro, Cheryl E. Praeger . . . . . . . . . . . . . 309
Volume 21, Number 2, Fall/Winter 2021, Pages 151–317
xi

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P2.01 / 151–173
https://doi.org/10.26493/1855-3974.2129.ac1
(Also available at http://amc-journal.eu)
Classification of skew morphisms of
cyclic groups which are square roots of
automorphisms*
Kan Hu †
Department of Mathematics, Zhejiang Ocean University,
Zhoushan, Zhejiang 316022, P.R. China, and
Key Laboratory of Oceanographic Big Data Mining & Application of Zhejiang Province,
Zhoushan, Zhejiang 316022, P.R. China
Young Soo Kwon ‡
Department of Mathematics, Yeungnam University,
Gyeongsan, 712-749, Republic of Korea
Jun-Yang Zhang §
School of Mathematical Sciences, Chongqing Normal University,
Chongqing 401331, P.R. China
Received 29 September 2019, accepted 28 April 2021, published online 18 September 2021
Abstract
The auto-index of a skew morphism φ of a finite group A is the smallest positive integer
h such that φh is an automorphism of A. In this paper we develop a theory of auto-index
of skew morphisms, and as an application we present a complete classification of skew
morphisms of finite cyclic groups which are square roots of automorphisms.
Keywords: Skew morphism, auto-index, period, square root.
Math. Subj. Class. (2020): 20B25, 05C10, 14H57
*The authors would like to thank Marston Conder for his suggestion of the concept of ‘auto-index’, and Kai
Yuan for his help in verifying our examples by the Magma program.
†Supported by Natural Science Foundation of Zhejiang Province (LY16A010010, LQ17A010003).
‡Supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF)
funded by the Ministry of Education (2018R1D1A1B05048450).
§Corresponding author. Supported by Basic Research and Frontier Exploration Project of Chongqing
(No. cstc2018jcyjAX0010), Science and Technology Research Program of Chongqing Municipal Education Com-
mission (No.KJQN201800512) and National Natural Science Foundation of China (11671276).
E-mail addresses: hukan@zjou.edu.cn (Kan Hu), ysookwon@ynu.ac.kr (Young Soo Kwon),
jyzhang@cqnu.edu.cn (Jun-Yang Zhang)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
152 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
1 Introduction
Throughout the paper, groups considered are all finite. A skew morphism of a group A is
a permutation φ on A fixing the identity element of A and for which there is a function
π : A → Z|φ| on A, called the power function of φ, such that φ(ab) = φ(a)φπ(a)(b) for all
a, b ∈ A. It is apparent the notion of skew morphism is a generalization of that of group
automorphism. A skew morphism of A is called proper if it is not an automorphism. Two
skew morphisms φ and φ′ of A are conjugate if there exists an automorphism θ of A such
that φ′ = θφθ−1.
The concept of skew morphism was first introduced by Jajcay and Širáň in [13] as an
algebraic tool to study regular Cayley maps, which are regular embeddings of graphs on
orientable closed surfaces admitting a regular subgroup of automorphisms on the vertices
of the embedded graph. In this direction, regular Cayley maps of cyclic groups and dihedral
groups have been classified, see [8, 21] and [14, 15, 16, 19, 28, 27]. In contrast, classifi-
cation of regular Cayley maps of non-cyclic abelian groups and other metacyclic groups is
still in progress; see [4, 5, 7, 20, 22, 26] for details.
The connection between skew morphisms and regular Cayley maps reveals a deep re-
lationship between skew morphisms and group factorizations with cyclic complements.
Indeed, if a group G is expressible as a product A⟨y⟩ of a subgroup A and a cyclic sub-
group ⟨y⟩ with A ∩ ⟨y⟩ = 1, then left multiplication of elements of A by y gives rise to a
skew morphism φ of A, determined by ya = φ(a)yπ(a) for all a ∈ A. Conversely, if φ is
a skew morphism of a group A, then for any a, b ∈ A, we have
φLa(b) = φ(ab) = φ(a)φ
π(a)(b) = Lφ(a)φ
π(a)(b),
so ⟨φ⟩LA ⊆ LA⟨φ⟩, where LA = {La | a ∈ A} is the left regular representation of A.
Since ⟨φ⟩∩LA = 1, we have |⟨φ⟩LA| = |LA⟨φ⟩|, and hence ⟨φ⟩LA = LA⟨φ⟩. Therefore,
G = LA⟨φ⟩ is a factorization of a transitive permutation group with a cyclic complement,
which is often referred to as the skew-product group of φ. The interested reader is referred
to [6, 17] for more details.
A prominent problem in this field is the classification of skew morphisms of cyclic
groups, which is closely related to regular Cayley maps [8] as well as edge-transitive
embeddings of complete bipartite graphs [11]. Kovács and Nedela [17] showed that if
n = n1n2 such that gcd(n1, n2) = 1 and gcd(n1, ϕ(n2)) = gcd(ϕ(n1), n2) = 1, then
every skew morphism φ of the cyclic additive group Zn is a direct product φ = φ1×φ2 of
skew morphisms φi of Zni , i = 1, 2. In a subsequent paper [18] the authors classified all
skew morphisms of the cyclic groups Zpe , where p is an odd prime. As for the case p = 2,
the associated skew product groups are classified by Du and Hu in [9].
Recently, Bachratý and Jajcay introduced the notion of period of skew morphisms [1].
More precisely, the period of a skew morphism φ is the smallest positive integer d such
that π
(
φd(a)
)
= π(a) for all a ∈ A. In particular, if d = 1 then the skew morphism
is said to be smooth (or coset-preserving). In [1, 23], it was shown that if φ is a skew
morphism of period d, then φd is a smooth skew morphism. The smooth skew morphisms
of cyclic groups and of dihedral groups were classified in [2] and [23] respectively. Let φ
be a skew morphism of a group A with power function π. If for any a ∈ A either π(a) =
π(φ(a)) = · · · = π(φ|φ|−1(a)) = 1 or π(a) = π(φ(a)) = · · · = π(φ|φ|−1(a)) = t where
|φ| is the order of φ and t is a fixed integer with 1 ≤ t < |φ|, then φ is called t-balanced.
Observe that every t-balanced skew morphism φ of a group A is necessarily smooth, and
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 153
in particular φt+1 is an automorphism of A (see [10] and Remark 3.2 in Section 3). Thus,
any t-balanced skew morphism is a (t+ 1)-th root of a group automorphism.
Inspired by those results above, we propose the following two related problems:
Problem 1.1. Let A be a given group, and d a given positive integer.
(a) Classify all skew morphisms of A which are d-th roots of automorphisms of A.
(b) Classify all skew morphisms of A which have period d.
For A = Zn and d = 2, the following main result of this paper is a solution to the first
problem, and by Theorem 3.8 (a) in Section 4 it is also a partial solution to the second one
(skew morphisms of period 2 of Zn whose square is an automorphism are determined).
Theorem 1.2. Every proper skew morphism of the cyclic additive group Zn which is a
square root of an automorphism is conjugate to a skew morphism of the form
φ(x) ≡ sx− x(x− 1)n
2k
(mod n),
where the pair (k, s) of positive integers satisfy the following conditions:
(a) k2 divides n and s ∈ Z∗n if k is odd, and 2k2 divides n and s ∈ Z∗n/2 if k is even,
(b) s ≡ −1 (mod k), s has multiplicative order 2ℓ in Zn/k and gcd(w, k) = 1 where
w =
k
n
(s2ℓ − 1)− s(s− 1)
2
ℓ.
The power function of φ is given by π(x) ≡ 1+2xw′ℓ (mod m), where w′w = 1 (mod k)
and m = 2kℓ is the order of φ. Moreover, two such skew morphisms corresponding to
distinct integer pairs are not conjugate.
The paper is organized as follows. After a summary of preliminary results in Section 2,
we develop a more comprehensive theory of powers of skew morphisms by defining a new
notion called auto-index in Section 3. In Section 4 we show that if φ is a proper skew
morphism of a group A which is a square root of an automorphism, then its power function
has the property π(xy) ≡ π(x) + π(y) − 1 (mod |φ|) for all x, y ∈ A; in particular, if
A = Zn, then π(x) ≡ (π(1)− 1)x+1 (mod |φ|) for all x ∈ Zn. As an application of the
theory, we present a proof of Theorem 1.2 in Section 5. Finally, for the special case when
n = pe is a prime power, we enumerate proper skew morphisms of Zn which are square
roots of automorphisms in Section 6.
2 Preliminaries
In this section we summarize some preliminary results on skew morphisms for future ref-
erence.
Proposition 2.1 ([1, 13]). Let φ be a skew morphism of a group A, and let π : A → Zm be
the power function of φ, where m is the order of φ. Then for any positive integer k,
φk(ab) = φk(a)φσ(a,k)(b), for all a, b ∈ A,
where σ(a, k) =
k∑
i=1
π(φi−1(a)); moreover, φk is a skew morphism if and only if the
congruence kx ≡ σ(a, k) (mod m) is solvable for every a ∈ A.
154 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
Proposition 2.2 ([13]). Let φ be a skew morphism of a group A, and let π : A → Zm be
the power function of φ, where m is the order of φ. Then for any a, b ∈ A,
π(ab) ≡
π(a)∑
i=1
π(φi−1(b)) (mod m).
Proposition 2.3 ([23]). Let φ be a skew morphism of a group A, and let π : A → Zm be
the power function of φ, where m is the order of φ. Then for any automorphism θ of A,
φ′ = θφθ−1 is a skew morphism of A with power function π′ = πθ−1.
It follows that the automorphism group Aut(A) of A acts by conjugation on the set
Skew(A) of all skew morphisms of A. Two skew morphisms of A are conjugate if they
belong to the same orbit under such action.
An important subgroup related to skew morphisms is the kernel of φ defined by
Kerφ = {a ∈ A | π(a) ≡ 1 (mod m)}.
It is well known that, for any a, b ∈ A, π(a) ≡ π(b) (mod m) if and only if ab−1 ∈ Kerφ,
so π takes exactly |A : Kerφ| distinct values in Zm. The index |A : Kerφ| is called the
skew-type of φ. It is obvious that φ is an automorphism if and only if it has skew-type 1. A
skew morphism which is not an automorphism will be called proper.
The subset
Fixφ = {a ∈ A | φ(a) = a}
of fixed-points of φ forms a subgroup of A. A subgroup N of A is φ-invariant if φ(N) =
N . Clearly, Fixφ is φ-invariant, but Kerφ may not be. However, the subset
Coreφ =
m⋂
i=1
φi(Kerφ)
forms the largest φ-invariant subgroup of A contained in Kerφ, and in particular, it is
normal in A [28]. Thus Kerφ is φ-invariant if and only if Kerφ = Coreφ, in which case
the skew morphism is called kernel-preserving. It is apparent that if φ is kernel-preserving,
then the restriction of φ to Kerφ is an automorphism of Kerφ. The following result is
well known.
Proposition 2.4 ([5]). Every skew morphism of an abelian group is kernel-preserving.
The importance of φ-invariant normal subgroups is reflected by the following result.
Proposition 2.5 ([29]). Let φ be a skew morphism of a group A, and let π : A → Zm be
the power function of φ, where m is the order of φ. If N a φ-invariant normal subgroup of
A, then φ defined by φ(x) = φ(x) is a skew morphism of the quotient group A := A/N . In
particular, the order m1 of φ is a divisor of m, and the power function π of φ is determined
by π(a) ≡ π(a) (mod m1) for all a ∈ A.
Since Coreφ is a normal subgroup of A, φ induces a skew morphism φ of the quotient
group A = A/Coreφ. Define
Smoothφ = {a ∈ A | φ(a) = a},
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 155
which is the preimage of the fixed-point subgroup Fixφ of φ under the natural epimor-
phism of A onto A/Coreφ. Since Fixφ is a φ-invariant subgroup of A, Smoothφ is a
φ-invariant subgroup of A.
In the extremal case that Smoothφ = A, the skew morphism φ is called smooth.
In [23] it is shown that a skew morphism φ of A is smooth if and only if π(a) ≡ π(φ(a))
(mod m) for all a ∈ A. More generally, the period of φ is the smallest positive integer d
such that π(φd(a)) ≡ π(a) (mod m) for all a ∈ A. Thus, φ is smooth if and only if it has
period 1. The following properties on the periodicity of skew morphisms are fundamental,
see [23] for details.
Proposition 2.6 ([23]). Let φ be a skew morphism of a group A, and let π : A → Zm be
the power function of φ, where m is the order of φ. If φ has period d, then the following
hold:
(a) d is equal to the order of the induced skew morphism φ of A = A/Coreφ;
(b) d is the smallest positive integer such that φd is a smooth skew morphism of A;
(c) for any a ∈ A,
d∑
i=1
π(φi−1(a)) ≡ 0 (mod d);
(d) conjugate skew morphisms have identical periods.
Note that for any positive integer k, by Proposition 2.6 (a), if φk is a smooth skew
morphism, then the period d of φ divides k.
3 Skew morphisms and automorphisms
Lemma 3.1. Let φ be a skew morphism of a group A, and let π : A → Zm be the power
function of φ, where m is the order of φ. Then for any positive integer k, φk is a group
automorphism if and only if
k∑
i=1
π
(
φi−1(a)
)
≡ k (mod m)
for all a ∈ A. In particular, if φ is smooth, then φk is an automorphism if and only if
kπ(a) ≡ k (mod m) for all a ∈ A.
Proof. By Proposition 2.1, φk is a skew morphism of A if and only if the congruences
kx ≡ σ(a, k) (mod m) (3.1)
are solvable for all a ∈ A, where
σ(a, k) =
k∑
i=1
π
(
φi−1(a)
)
.
Note that if πµ is the power function of µ := φk, then πµ(a) is the solution of (3.1), and
therefore µ is an automorphism if and only if σ(a, k) ≡ k (mod m) for all a ∈ A. In
addition, if φ is smooth, then σ(a, k) = kπ(a), so µ is an automorphism if and only if
kπ(a) ≡ k (mod m) for all a ∈ A.
156 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
Remark 3.2. If φ is a t-balanced skew morphism of a group A, then φ is smooth and for all
a ∈ A\Kerφ, π(a) ≡ t (mod m) where t2 ≡ 1 (mod m) [5]. Therefore (t+1)t ≡ t+1
(mod m). By Lemma 3.1, φt+1 is a group automorphism. This is a generalization of [10,
Lemma 3.4].
Definition 3.3. For a skew morphism φ of a group A, the auto-index of φ is defined to be
the smallest positive integer h such that φh is a group automorphism of A.
Clearly, φ is an automorphism if and only if it has auto-index 1. Lower and upper
bounds of the auto-index of a skew morphism are given as follows.
Lemma 3.4. Let φ be a skew morphism of a group A. Suppose that φ has order m, period
d and auto-index h, then d divides h and h divides m.
Proof. Note that d is the smallest positive integer such that φd is a smooth skew morphism.
Since φh is an automorphism which is necessarily smooth, the minimality of d implies that
d | h. Since φm = 1 is the identity automorphism, the minimality of h implies that h | m,
as required.
Corollary 3.5. If φ is a proper skew morphism of prime order, then it is smooth with auto-
index equal to its order.
Proof. Let d and h denote the period and auto-index of φ, respectively. As φ is proper,
d ≤ |A : Kerφ| < |φ| and h > 1. By Lemma 3.4, d divides h and h divides |φ|. Since
|φ| = p is prime, we obtain d = 1 and h = p, as required.
As an example of Corollary 3.5, φ = (0)(153)(2)(4) is a proper skew morphism of the
cyclic group Z6. It is smooth, and both its order and auto-index are equal to 3.
Lemma 3.6. Let φ be a skew morphism of the cyclic group Zn and let π : Zn → Zm be
the associated power function, where m is the order of φ. If φ has period 2 and auto-index
h, then h is an even positive divisor of m and there exists some u ∈ Zh such that
π(x) ≡
(
π(1)− 1
) x∑
i=1
(
1 +
um
h
)i−1
+ 1 (mod m), for all x ∈ Zn. (3.2)
Proof. Since φ has period 2, by Proposition 2.6 (c), π(x) + π(φ(x)) ≡ 0 (mod 2) for all
x ∈ Zn. By Lemma 3.4, h is an even positive divisor of m. By Lemma 3.1, we have
h ≡
h∑
i=1
π(φi−1(1)) ≡ 1
2
(
π(1) + π
(
φ(1)
))
h (mod m),
and then
1
2
(
π(1) + π
(
φ(1)
))
= 1 + um/h,
for some u ∈ Zh.
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 157
Moreover, since φ has period 2, by Proposition 2.6 (a), φ is an automorphism of order
2. Thus, π(1) ≡ π(1) ≡ 1 (mod 2). Consequently, by Proposition 2.1, we have
π(2) ≡
π(1)∑
i=1
π
(
φi−1(1)
)
≡π(1) + π(1)− 1
2
(
π(1) + π
(
φ(1)
))
≡π(1) +
(
π(1)− 1
)
(1 + um/h)
≡
(
π(1)− 1
)(
1 + (1 + um/h)
)
+ 1 (mod m).
By induction, we obtain (3.2), as required.
In what follows we study skew morphisms of auto-index 2. These skew morphisms are
all square roots of automorphisms. Clearly, every permutation of order 2 on A is a square
root of the identity automorphism of A. Generally, a square root of an automorphism of
A maybe not a skew morphism of A. It seems too difficult to determine all square roots
of automorphisms for a family of groups. In the following example, all square roots of
nonidentity automorphisms of Z8 are determined.
Example 3.7. The cyclic group Z8 has three nonidentity automorphisms as follows:
σ1 = (0)(2)(4)(6)(1, 5)(3, 7), σ2 = (0)(4)(2, 6)(1, 3)(5, 7), σ3 = (0)(4)(2, 6)(1, 7)(5, 3).
Since the square of every permutation of order 4 on Z8 either fixes no element or fixes 4
elements, σ2 and σ3 have no square roots. Set µ = (0)(2)(4)(6)(1, 3, 5, 7) and use Cµ to
denote the set of all square roots of the identity automorphism of Z8 which commute with
µ. Then every square root of σ1 can be represented as a product τµ where τ ∈ Cµ. It is
straightforward to check that µ and µ3 are the only two square roots of σ1 which are skew
morphisms. Since µ3 = σ−13 µσ3, Z8 has a unique conjugate class of skew morphism of
auto-index 2.
We are only concerned with square roots of automorphisms which are also skew mor-
phisms. For convenience, skew morphisms of auto-index 2 are called proper square roots
of automorphisms throughout this paper.
Theorem 3.8. Let φ be a skew morphism of a group A, and let π : A → Zm be the power
function of φ, where m is the order of φ. If φ is a proper square root of an automorphism,
then
(a) φ is kernel-preserving of period at most 2;
(b) π(x) is odd for all x ∈ A;
(c) π(xy) ≡ π(x) + π(y)− 1 (mod m) for all x, y ∈ A;
Proof. Take an arbitrary element x ∈ A. Since φ2 is an automorphism and φ is not an
automorphism, by Lemma 3.1, we have
π(x) + π(φ(x)) ≡ 2 (mod m) and π
(
φ(x)
)
+ π
(
φ2(x)
)
≡ 2 (mod m). (3.3)
158 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
(a) From (3.3) we deduce π(x) ≡ π
(
φ2(x)
)
(mod m), so the period of φ is at most
2. In particular, we see that π
(
φ(x)
)
= 1 whenever π(x) = 1. It follows that φ is kernel-
preserving.
(b) If φ has period 1, then π(x) ≡ π
(
φ(x)
)
(mod m), and hence 2π(x) ≡ π(x) +
π
(
φ(x)
)
≡ 2 (mod m). Since φ is not an automorphism, m must be even. Since π is
a group homomorphism from A to Z∗m [23, Theorem 4.9], π(x) is an odd integer. Now
assume φ has period 2. Since φ is kernel-preserving, Kerφ = Coreφ is normal in A.
By Proposition 2.6 (a), the induced skew morphism φ of A/Kerφ is an automorphism of
order 2. Thus, π(x) ≡ π(x) ≡ 1 (mod 2), and π(x) is also odd.
(c) By Proposition 2.2, we have
π(xy) ≡
π(x)∑
i=1
π(φi−1(y))
≡π(y) + π(x)− 1
2
(
π(y) + π(φ(y))
)
≡π(x) + π(y)− 1 (mod m)
for all x, y ∈ A.
Corollary 3.9. Let φ be a proper square root of an automorphism of a group A, and let
π : A → Zm be the power function of φ, where m is the order of φ. Then
(a) if φ is smooth, then it has skew-type two, 4 divides m, and π(x) = 1 +m/2 for all
x ∈ A \Kerφ;
(b) if φ is not smooth, then it has skew-type at least 3.
Proof. If φ is smooth, then from the proof of Theorem 3.8, we see that m is even and
2π(x) ≡ 2 (mod m) for any x ∈ A. Hence π(x) = 1 or 1 +m/2. Since φ is proper and
π(x) is odd, 4 divides m. If φ is not smooth, then the skew-type of φ is at least 3 since φ
is kernel-preserving of period 2.
Example 3.10 ([25]). The cyclic group Z9 has four skew morphisms of period 2:
φ1 = (0)(1, 2, 7, 5, 4, 8)(3, 6), π1 = [1][3, 5, 3, 5, 3, 5][1, 1];
φ2 = (0)(1, 5, 4, 2, 7, 8)(3, 6), π2 = [1][3, 5, 3, 5, 3, 5][1, 1];
φ3 = (0)(1, 8, 4, 5, 7, 2)(3, 6), π3 = [1][5, 3, 5, 3, 5, 3][1, 1];
φ4 = (0)(1, 8, 7, 2, 4, 5)(3, 6), π4 = [1][5, 3, 5, 3, 5, 3][1, 1].
It can be directly verified that φ2i (i = 1, 2, 3, 4) are automorphisms of Z9, so that all of
these skew morphisms are proper square roots of automorphisms. Note that up to conjuga-
tion by automorphisms they are divided into two classes {φ1, φ4} and {φ2, φ3}.
Example 3.11. Define two functions φ and π on the cyclic group Z8n where n is a positive
integer as follows:
φ(x) ≡
{
2i (mod 8n), if x = 2i;
2(n+ i) + 1 (mod 8n), if x = 2i+ 1
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 159
and
π(x) =
{
1, if x = 2i;
3, if x = 2i+ 1.
It is straightforward to check that φ is a skew morphism of Z8n with power function π
whose square is an involutory automorphism.
4 Technical lemmas
In what follows we restrict our discussion to proper square roots of automorphisms of the
cyclic groups.
Lemma 4.1. Let φ be a skew morphism of the cyclic group Zn, and let π : Zn → Zm be
the power function of φ, where m is the order of φ. If φ is a proper square root of an
automorphism and it has skew-type k, then the following hold:
(a) there is some integer ℓ ≥ 1 such that m = 2kℓ;
(b) there is some integer u ∈ Z∗k such that π(x) ≡ 1 + 2xuℓ (mod m) for all x ∈ Zn;
(c) the number r = φ2(1) is coprime to n and there exists some integer v ∈ Z∗k such
that rℓ ≡ 1 + vn/k (mod n);
(d) k2 is a divisor of n;
(e) the multiplicative order of r in Zn/k is equal to ℓ.
Proof. By Theorem 3.8, φ has period 1 or 2 and
π(x+ y) ≡ π(x) + π(y)− 1 (mod m)
for all x, y ∈ Zn. Thus π(2) ≡ 2π(1)− 1 ≡ 2
(
π(1)− 1
)
+ 1 (mod m) and by induction
π(x) ≡ x
(
π(1)− 1
)
+ 1 (mod m), ∀x ∈ Zn.
In particular, π(m) ≡ m
(
π(1) − 1
)
+ 1 ≡ 1 (mod m), and therefore m ∈ Kerφ. Since
φ is of skew-type k, Kerφ = ⟨k⟩, and hence k | m. Noting that
1 ≡ π(k) ≡ k
(
π(1)− 1
)
+ 1 (mod m),
we get π(1) = 1+um/k for some u ∈ Zk. Consequently, π(x) ≡ 1+xum/k (mod m).
Since π takes k distinct values of the form 1 + im/k (i = 0, 1, . . . , k − 1) in Zm, we
have u ∈ Z∗k. By Theorem 3.8, 1 +m/k is odd, that is, m/k is even. Thus we can write
m = 2kℓ, where ℓ is a positive integer. Then π(x) ≡ 1 + 2xuℓ (mod m).
Set r = φ2(1). Since φ2 ∈ Aut(Zn), r is coprime to n and φ2(x) ≡ rx (mod n) for
all x ∈ Zn. In particular, φ2ℓ(k) ≡ rℓk (mod n). On the other hand, there exists u′ ∈ Zn
such that π(u′) ≡ 1 + 2ℓ (mod m). Therefore
φ(k) + φ(u′) ≡ φ(k + u′) ≡ φ(u′ + k) ≡ φ(u′) + φ1+2ℓ(k) (mod n)
and then φ2ℓ(k) = k. Thus, rℓ ≡ 1 (mod n/k). Write rℓ = 1 + vn/k. Recalling that
φ has period at most 2, we have π
(
φ2ℓ(1)
)
≡ π(1) (mod m) and hence φ2ℓ(1) ≡ 1
160 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
(mod k). It follows that 1 + vn/k ≡ rℓ ≡ φ2ℓ(1) ≡ 1 (mod k), and hence k is a divisor
of vn/k. Note that
φ2ℓj(1) ≡ rℓj ≡
(
1 +
vn
k
)j
≡ 1 + jvn
k
+
j∑
i=2
(
j
i
)(vn
k
)i
≡ 1 + jvn
k
(mod n)
for any positive integer j. By [29, Lemma 3.1], the length of the orbit of 1 under φ is equal
to the order m = 2kℓ of φ. If 0 < j < k, then 1 ̸≡ φ2jℓ(1) ≡ 1 + jvn/k (mod n).
Consequently, v ∈ Z∗k and k2 divides n.
If the multiplicative order of r in Zn/k is i, then ri = 1+tn/k for some positive integer
t. Since rℓ ≡ 1 (mod n/k), we have i | ℓ. On the other hand, since k2 | n for all x ∈ Zn,
we have
φ2ik(x) ≡ rikx ≡ (1 + tn/k)kx ≡ x (mod n).
Since the order of φ is 2kℓ, we get ℓ | i, and therefore ℓ = i.
Corollary 4.2. Let φ be a skew morphism of the cyclic group Zn. If φ is a proper square
root of an automorphism, then the induced skew morphism φ of Zn/Kerφ maps each x to
−x.
Proof. Let m and k be the order and the skew-type of φ, respectively. By Lemma 4.1,
m = 2kℓ for some positive integer ℓ, and
2 ≡ π(x) + π
(
φ(x)
)
≡ 2 + 2
(
x+ φ(x)
)
uℓ (mod 2kℓ)
for all x ∈ Zn, where u ∈ Z∗k. Thus 2
(
x+ φ(x)
)
uℓ ≡ 0 (mod 2kℓ) and then φ(x) ≡ −x
(mod k), as required.
The converse of Corollary 4.2 is generally not true, see [6, Theorem 6.5] for a coun-
terexample. However, we have the following result.
Lemma 4.3. Let φ be a proper skew morphism of the cyclic group Zn. If the induced skew
morphism φ of Zn/Kerφ maps each x to −x, then φ2 is a skew morphism of skew-type at
most 2. In particular, if the skew-type of φ is odd, then φ2 is an automorphism of Zn.
Proof. Throughout the proof, we denote the order and the skew-type of φ by m and k, and
the power functions of φ and φ by π and π, respectively.
If k = 2, then the result is obviously true. In what follows we assume k > 2. Since
φ maps each x to −x, φ is an automorphism of order 2. By Proposition 2.6 (a), φ has
period 2. It follows that m is even, π
(
φ2(x)
)
≡ π(x) (mod m) and π
(
φ(x)
)
≡ π(−x)
(mod m) for all x ∈ Zn. Since π(x) ≡ π(x) ≡ 1 (mod 2), π(x) is odd.
Take two arbitrary elements x, y ∈ Zn. By Proposition 2.2, we have
π(x+ y) ≡
π(x)∑
i=1
π(φi−1(y)) ≡ π(y) + π(x)− 1
2
(
π(y) + π(−y)
)
(mod m).
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 161
In particular,
1 = π(x− x) ≡ π(−x) + π(x)− 1
2
(
π(x) + π(−x)
)
(mod m), (4.1)
1 = π(−x+ x) ≡ π(x) + π(−x)− 1
2
(
π(x) + π(−x)
)
(mod m), (4.2)
π(2x) ≡ π(x) + π(x)− 1
2
(
π(x) + π(−x)
)
(mod m), (4.3)
π(−2x) ≡ π(−x) + π(−x)− 1
2
(
π(x) + π(−x)
)
(mod m), (4.4)
π(2x+ 1) ≡ π(2x) + π(1)− 1
2
(
π(2x) + π(−2x)
)
(mod m), (4.5)
π(−2x− 1) ≡ π(−2x) + π(−1)− 1
2
(
π(2x) + π(−2x)
)
(mod m). (4.6)
Adding (4.1) to (4.2) and (4.3) to (4.4), we get
1
2
(
π(x) + π(−x)
)2 ≡ 2 (mod m)
and
1
2
(
π(x) + π(−x)
)2 ≡ π(2x) + π(−2x) (mod m).
Thus,
π(2x) + π(−2x) ≡ 2 (mod m). (4.7)
Substituting 2 for π(2x) + π(−2x) in (4.5) and (4.6) we obtain
π(2x+ 1) ≡ π(2x) + π(1)− 1 (mod m)
and
π(−2x− 1) ≡ π(−2x) + π(−1)− 1 (mod m).
It follows that
π(2x+ 1) + π(−2x− 1) ≡ π(1) + π(−1) (mod m). (4.8)
From (4.7) and (4.8) we deduce that
φ2(x+ y) = φ2(x) + φ2(y)
if x is even, and
φ2(x+ y) = φ2(x) + φπ(1)+π(−1)(y)
if x is odd. Thus, φ2 is a skew morphism of skew-type at most 2. In particular, if the
skew-type k of φ is an odd number, then
π(1) + π(−1) ≡ π(k + 1) + π(k − 1) ≡ 2 (mod m)
and therefore φ2 is an automorphism, as claimed.
162 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
5 Classification
In this section, we classify proper square roots of automorphisms of Zn.
Theorem 5.1. Define a quadratic polynomial over the ring (Zn,+,×) by
φ(x) ≡ sx− x(x− 1)n
2k
(mod n), x ∈ Zn, (5.1)
where k and s are positive integers satisfying the following conditions:
(a) k2 divides n and s ∈ Z∗n if k is odd, and 2k2 divides n and s ∈ Z∗n/2 if k is even,
(b) s ≡ −1 (mod k), s has multiplicative order 2ℓ in Zn/k and gcd(w, k) = 1 where
w =
k
n
(s2ℓ − 1)− s(s− 1)
2
ℓ.
Then φ is a proper square root of an automorphism of the cyclic additive group Zn whose
skew-type is k and power function is given by
π(x) ≡ 1 + 2xw′ℓ (mod m),
where w′w ≡ 1 (mod k) and m = 2kℓ is the order of φ. Moreover, up to conjugation φ
is uniquely determined by the parameters k and s.
Proof. First, we show that φ is a permutation on Zn. Assume φ(x) ≡ φ(y) (mod n)
where x, y ∈ Zn. Then it suffices to prove that x ≡ y (mod n). Since
sx− x(x− 1)n
2k
≡ sy − y(y − 1)n
2k
(mod n),
we get
s(x− y) ≡ (x− y)(x+ y − 1)n
2k
(mod n).
By (a) and (b) we have s ∈ Z∗n. Thus, from the above equation we deduce that x − y ≡ 0
(mod n/k). By (a) again we obtain
(x− y)(x+ y − 1)n
2k
≡ 0 (mod n),
and hence x ≡ y (mod n).
Second, we show that φ2 is an automorphism of Zn. By (a) and (b), we derive from
formula (5.1) that
φ
(jn
k
)
≡ sjn
k
− jn(jn− k)n
2k3
≡ −jn
k
(mod n) (5.2)
for all positive integers j. Now for any x, y ∈ Zn,
φ(x+ y) ≡ s(x+ y)− (x+ y)(x+ y − 1)n
2k
≡ sx− x(x− 1)n
2k
+ sy − y(y − 1)n
2k
− xyn
k
≡ φ(x) + φ(y)− xyn
k
(mod n).
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 163
It follows that
φ2(x) ≡ φ
(
sx− x(x− 1)n
2k
)
≡ φ(sx) + φ
(
− x(x− 1)n
2k
)
+
n
k
sx2(x− 1)n
2k
≡ φ(sx) + φ
(
− x(x− 1)n
2k
)
(5.2)
≡ s2x− sx(sx− 1)n
2k
+
x(x− 1)n
2k
≡
(
s2 − s(s− 1)n
2k
)
x− (s
2 − 1)x(x− 1)n
2k
(b)
≡
(
s2 − s(s− 1)n
2k
)
x (mod n).
Since s ∈ Z∗n and k2 | n, we have gcd
(
s2− s(s−1)n2k , n
)
= 1. Thus, φ2 is an automorphism
of Zn.
Next we show that φ is a skew morphism of Zn with associated power function π
defined by π(x) ≡ 1 + 2w′ℓ (mod m) for any x ∈ Zn, where w′w ≡ 1 (mod k). Take
arbitrary x, y ∈ Zn. By the conditions (a) and (b), we have
φ(x) + φπ(x)(y) ≡ φ(x) + φ1+2xw
′ℓ(y) ≡ φ(x) + φ2xw
′ℓ
(
φ(y)
)
≡ φ(x) + φ(y)
(
s2 − s(s− 1)n
2k
)ℓw′x
≡ φ(x) + φ(y)
(
s2ℓ − s(s− 1)ℓn
2k
)w′x
≡ φ(x) + φ(y)
(
1 +
wn
k
)w′x
≡ φ(x) + φ(y)
(
1 +
nx
k
)
(mod n)
and
φ(x+ y) ≡ φ(x) + φ(y)− nxy
k
≡ φ(x) +
(
sy − y(y − 1)n
2k
)
− nxy
k
≡ φ(x) +
(
sy − y(y − 1)n
2k
)
+
snxy
k
≡ φ(x) +
(
sy − y(y − 1)n
2k
)(
1 +
nx
k
)
≡ φ(x) + φ(y)
(
1 +
nx
k
)
(mod n).
Therefore, φ(x+ y) ≡ φ(x) + φπ(x)(y) and thus φ is a skew morphism of Zn.
Finally, we prove that up to conjugation φ is uniquely determined by the parameters k
and s. It is evident that if two such skew morphism are conjugate, then they must have the
same skew-type k. Suppose now that φi (i = 1, 2) are two conjugate skew morphisms of
Zn defined by
φi(x) ≡ six−
x(x− 1)n
2k
(mod n),
164 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
where n, k and si satisfy the stated conditions. Then there exists an automorphism θ of Zn
such that φ1θ = θφ2. Set r = θ(1). Then
s1rx−
rx(rx− 1)n
2k
≡ φ1θ(x) ≡ θφ2(x) ≡ s2rx−
rx(x− 1)n
2k
(mod n).
Since gcd(r, n) = 1, this is reduced to
s1x−
x(rx− 1)n
2k
≡ s2x−
x(x− 1)n
2k
(mod n),
or equivalently,
(s1 − s2)x ≡
x(rx− 1)n
2k
− x(x− 1)n
2k
≡ x
2(r − 1)n
2k
(mod n).
If we choose x = ±1, then ±(s1−s2) ≡ (r−1)n/2k (mod n). Therefore 2(s1−s2) ≡ 0
(mod n) and r ≡ 1 (mod k). If k is even, so is n, and hence s1 ≡ s2 (mod n/2). If both
k and n are odd, then s1 ≡ s2 (mod n). If k is odd but n is even, then r is odd. Since
r ≡ 1 (mod k), we obtain r − 1 ≡ 0 (mod 2k). Thus, we also get s1 ≡ s2 (mod n), as
required.
Now we are ready to prove the main result of the paper.
Proof of Theorem 1.2. By Theorem 5.1, the quadratic polynomial of the stated form is
a proper square root of an automorphism of Zn, and distinct pairs (k, s) correspond to
disconjugate skew morphisms.
Conversely, suppose that φ is a proper square root of an automorphism of Zn of skew-
type k > 1. By Lemma 4.1, k2 | n, |φ| = 2kℓ for some positive integer ℓ, and the power
function of φ is given by π(x) ≡ 1 + 2xuℓ (mod 2kℓ) for some u ∈ Z∗k. Set s = φ(1).
By Lemma 3.1, we have
2 ≡ π(1) + π(φ(1)) ≡ (1 + 2uℓ) + (1 + 2suℓ) ≡ 2 + 2(1 + s)uℓ (mod 2kℓ),
which implies 2(1 + s)uℓ ≡ 0 (mod 2kl). Since u ∈ Z∗k, we obtain s ≡ −1 (mod k).
Since φ2 is an automorphism of Zn, φ2(x) ≡ rx (mod n) for some r coprime to n.
By Lemma 4.1, rℓ ≡ 1 + vn/k (mod n) for some v ∈ Z∗k. Then
φ(x) ≡φ(x− 1) + φπ(x−1)(1) ≡ φ(x− 1) + φ2ℓu(x−1)+1(1)
≡φ(x− 1) + φ2ℓu(x−1)(s) ≡ φ(x− 1) + srℓu(x−1)
≡φ(x− 1) + s
(
1 +
vn
k
)u(x−1)
(mod n).
By induction we obtain
φ(x) ≡ s
x∑
i=1
(
1 +
vn
k
)u(i−1)
(mod n), x ∈ Zn.
Since k2 | n, for any positive integer j, we have
(
1 +
vn
k
)j
≡ 1 + jvn
k
+
j∑
i=2
(
j
i
)(vn
k
)i
≡ 1 + jvn
k
(mod n).
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 165
Thus,
φ(x) ≡s
x∑
i=1
(
1 +
vn
k
)u(i−1)
≡ s
x∑
i=1
(
1 +
uvn(i− 1)
k
)
≡s
(
x+
uvnx(x− 1)
2k
)
≡ sx− uvnx(x− 1)
2k
(mod n).
It follows that
r = φ2(1) = φ(s) ≡ s2 − uvns(s− 1)
2k
(mod n). (5.3)
Hence, r ≡ s2 (mod n/k) and by Lemma 4.1 (e), s has multiplicative order 2ℓ in Zn/k.
Since
1 +
vn
k
≡rℓ ≡
(
s2 − s(s− 1)uvn
2k
)ℓ
≡s2ℓ −
(
ℓ
1
)
s2(ℓ−1)
s(s− 1)uvn
2k
+
ℓ∑
i=2
(
ℓ
i
)
s2(ℓ−i)
(
− s(s− 1)uvn
2k
)i
≡s2ℓ − s
2(ℓ−1)s(s− 1)ℓuvn
2k
≡ s2ℓ − s(s− 1)ℓuvn
2k
(mod n),
we have
s2ℓ ≡ 1 +
(
1 +
s(s− 1)ℓu
2
)vn
k
(mod n/k).
By [12, Lemma 1], there exists c ∈ Z∗n such that c ≡ uv (mod k). Define φ′ := θcφθ−1c ,
where θc is the automorphism of Zn taking 1 to c. By Proposition 2.3, φ′ is a skew mor-
phism of Zn. For all x ∈ Zn, we have
φ′(x) =θcφθ
−1
c (x) = θcφ(c
−1x) ≡ c
(
sc−1x− c
−1x(c−1x− 1)cn
2k
)
≡sx− x(x− c)n
2k
≡
(
s+
(c− 1)n
2k
)
x− x(x− 1)n
2k
(mod n).
Let s′ = s + (c−1)n2k , then it is easily seen that s
′ ≡ −1 (mod k), s′ ∈ Z∗n, and s′ has
multiplicative order 2ℓ in Zn/k. Therefore, up to conjugation we can assume
φ(x) ≡ sx− x(x− 1)n
2k
(mod n) and π(x) ≡ 1 + 2w′ℓx (mod 2kℓ),
where s ≡ −1 (mod k), s ∈ Z∗n, w′ ∈ Z∗k, and 2ℓ is the multiplicative order of s in Zn/k.
We show that ww′ ≡ 1 (mod k), that is, w′ is the modular inverse of w in Zk. Noting
that the congruence
w ≡ k
n
(s2ℓ − 1)− s(s− 1)
2
ℓ (mod k)
is equivalent to
s2ℓ − s(s− 1)ℓn
2k
≡ 1 + nw
k
(mod n),
166 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
we have
2s− n
k
≡ φ(2) ≡ φ(1) + φπ(1)(1)
≡ s+ φ2w
′ℓ(s)
≡ s+ s
(
s2 − s(s− 1)n
2k
)ℓw′
≡ s+ s
(
s2ℓ − s(s− 1)ℓn
k
)w′
≡ s+ s
(
1 +
nw
k
)w′
≡ 2s+ sww
′n
k
≡ 2s− nww
′
k
(mod n),
which is reduced to ww′ ≡ 1 (mod k).
In what follows we consider the particular case that k is even. We have
φ2(2) = 2φ2(1) ≡ 2s2 − s(s− 1)n
k
≡ 2s2 − 2n
k
(mod n)
and
φ2(2) ≡ φ
(
2s− n
k
)
≡ s
(
2s− n
k
)
−
(
2s− n
k
)(
2s− n
k
− 1
) n
2k
≡ 2s2 − sn
k
−
(
s− n
2k
)
(2s− 1)n
k
≡ 2s2 − sn
k
−
(
2s2 − s− sn
k
+
n
2k
)n
k
≡ 2s2 − 2s
2n
k
− n
2
2k2
≡ 2s2 − 2n
k
− n
2
2k2
(mod n).
Thus,
2s2 − 2n
k
≡ 2s2 − 2n
k
− n
2
2k2
(mod n),
and therefore 2k2 | n. Moreover, if s > n/2, then we write s′ = s− n/2 and define
φ′(x) ≡ s′x− x(x− 1)n
2k
(mod n), x ∈ Zn.
It is easily seen that φ′ is also a square root of an automorphism of Zn. We show that φ′ is
conjugate to φ. Since 2k2 | n, n = 2ekn1 where e ≥ 1 and 2 ∤ n1. Note that the number
c := kn1 + 1 is coprime to n. Let θc be the automorphism of Zn taking x to cx. Then, for
any x ∈ Zn,
φ′θc(x) ≡ s′cx−
cx(cx− 1)n
2k
≡ (s− n
2
)cx−
(
cx(x− 1) + c(c− 1)x2
)
n
2k
≡ scx− cx(x− 1)n
2k
+
nx
2
− c(c− 1)x
2n
2k
≡ scx− cx(x− 1)n
2k
≡ θcφ(x) (mod n).
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 167
Thus, φ is conjugate to φ′, as required.
Corollary 5.2. Every smooth proper square root of an automorphism of the cyclic group
Zn is conjugate to a skew morphism of the form
φ(x) ≡ sx− x(x− 1)n
4
(mod n), x ∈ Zn,
with the associated power function given by
π(x) ≡ 1 + 2ℓx (mod 4ℓ), x ∈ Zn,
where 8 | n, both s and 2n (s
2ℓ−1)− s(s−1)2 ℓ are odd numbers, and the multiplicative order
of s in Zn/2 is equal to 2ℓ. In particular, φ has order 4ℓ and skew-type 2.
Proof. By Corollary 3.9, every smooth proper square root of an automorphism has skew-
type 2. The result follows immediately from Theorem 1.2.
Remark 5.3. Note that if φ is proper skew morphism of Zn and φ2 is an involutory auto-
morphism, then |φ| = 4, and by Theorem 1.2, k = 2, ℓ = 1 and φ is smooth.
Corollary 5.4. Let φ be a non-smooth skew morphism of the cyclic group Zn. If φ has
skew-type 3, then it is conjugate to a skew morphism of the form
φ(x) ≡ sx− n
6
x(x− 1) (mod n), x ∈ Zn,
where 9 | n, s ∈ Z∗n has multiplicative order 2ℓ in Zn/3, s ≡ −1 (mod 3) and
3
n
(s2ℓ − 1)− ℓ ≡ w′ ̸≡ 0 (mod 3).
Moreover, the order of φ is m = 6ℓ and the power function of φ is given by
π(x) ≡ 1 + m
3
w′x (mod m).
Proof. Since φ is a non-smooth skew morphism of Zn of skew-type 3, the induced skew
morphism φ of Zn/Kerφ is an automorphism of the form φ = (0)(1,−1). By Lemma 4.3,
φ2 is an automorphism. The result then follows from Theorem 1.2.
By Theorem 1.2, we have the following special property of a square root of an auto-
morphism of the cyclic group Zn.
Corollary 5.5. Let φ be a proper square root of an automorphism of the cyclic group Zn.
Then every subgroup of Zn is φ-invariant.
Proof. Let H = ⟨h⟩ be a subgroup of Zn. If φ and φ′ are conjugate by an automorphism
of Zn and H is φ-invariant, then H is also φ′-invariant. So it suffices to consider the skew
morphisms φ given by Theorem 1.2. Let k be the skew-type of φ. For any integer j,
φ(jh) ≡ sjh− jh(jh− 1)n
2k
≡ h
(
sj − j(jh− 1)n
2k
)
(mod n).
If n is even, n2k is a positive integer, and if n is odd, then h is also odd and
j(jh−1)n
2k is a
positive integer. This means that φ(jh) ∈ H , and hence H is φ-invariant.
168 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
6 The prime power case
In this section, for the case where n = pe is a prime power, we enumerate the conjugacy
classes of proper square roots of automorphisms of Zn.
We need a technical result from number theory.
Proposition 6.1 ([3, 24]). Suppose that n = pe, where p is a prime and e ≥ 1. Then
(a) if p > 2, then Z∗pe ∼= Zp−1 × Zpe−1 is cyclic of order pe−1(p− 1). In particular, for
each i, 1 ≤ i ≤ e − 1, an element of the form 1 + upe−i in Z∗pe has order pi if and
only if p ∤ u,
(b) if p = 2, then Z∗2e is trivial if e = 1, Z∗2e ∼= Z2 if e = 2, and Z∗2e ∼= Z2 × Z2e−2 if
e ≥ 3. In particular, in the last case for each i, 2 ≤ i ≤ e − 1, an element of the
form ±1 + u2i in Z∗2e has order 2e−i if and only if 2 ∤ u.
Let N(pe) denote the number of conjugacy classes of proper square roots of automor-
phisms of Zpe . Then N(pe) is determined in the following theorem.
Theorem 6.2. Suppose that p is a prime and e ≥ 1. If p ̸= 2, then
N(pe) =
{
1
p−1 (p
e
2 − 1)2, if e is even
1
p−1 (p
e+1
2 − 1)(p e−12 − 1), if e is odd,
while if p = 2, then
N(2e) =

0, if e < 3
1, if e = 3
2e−1 − 3 · 2 e−22 , if e > 3 is even
2e−1 − 2 e+12 , if e > 3 is odd.
Proof. Denote n = pe and k = pf . Then for fixed prime p and integer e ≥ 1, by Theo-
rem 1.2, N(pe) is equal to the number of pairs (f, s) which satisfy the following conditions:
(a) 2 ≤ 2f ≤ e and s ∈ Z∗pe if p ̸= 2, and 2 ≤ 2f ≤ e− 1 and s ∈ Z∗2e−1 if p = 2,
(b) s ≡ −1 (mod pf ), s has multiplicative order 2ℓ in Zpe−f and p ∤ w, where
w = pf−e(s2ℓ − 1)− 1
2
s(s− 1)ℓ.
For each admissible value of the parameter f , let N(pe, pf ) denote the number of admis-
sible values of the parameter s. In what follows, we first determine N(pe, pf ), and then
determine N(pe). We divide the proof into two cases according to the parity of p.
Case (A). p ̸= 2.
Since s ≡ −1 (mod pf ), we may write s = tph − 1 where 1 ≤ f ≤ h ≤ e and
t ∈ Z∗pe−h . Then s
2 = 1 + tph(tph − 2). According to the multiplicative order 2ℓ of s in
Zpe−f , we distinguish two subcases as follows.
If h < e− f , by Proposition 6.1 we have ℓ = pe−f−h. Since s has multiplicative ordr
2ℓ in Zpe−f , we have pe−f ∥ s2ℓ − 1. Since p | 12s(s− 1)ℓ, we have p ∤ w.
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 169
If h ≥ e− f , then ℓ = 1. Recalling that 1 ≤ f ≤ h ≤ e, we have
w ≡ tpf+h−e(tph − 2)− 1
2
(tph − 1)(tph − 2) ≡ −1− 2tpf+h−e (mod p).
Thus, p | w if and only if h = e − f and p | 1 + 2t, where t ∈ Z∗pf , in which case the
number of such t is equal to pf−1.
Consequently,
N(pe, pf ) =
e∑
h=f
ϕ(pe−h)− pf−1 = 1 +
e−1∑
h=f
pe−h−1(p− 1)− pf−1 = pe−f − pf−1,
where ϕ is the Euler’s totient function. Therefore,
N(pe) =
⌊e/2⌋∑
f=1
N(pe, pf ) =
⌊e/2⌋∑
f=1
(pe−f − pf−1) = 1
p− 1
(p⌊e/2⌋ − 1)(pe−⌊e/2⌋ − 1).
Note that ⌊e/2⌋ = e/2 if e is even, and ⌊e/2⌋ = (e− 1)/2 if e is odd. The stated formula
follows from substitution.
Case (B). p = 2.
It is straightforward to check that N(22) = 0, N(23) = N(23, 21) = 1 and N(24) =
N(24, 21) = 2. In what follows, we assume e ≥ 5 and distinguish two subcases.
Subcase (a). s ≡ 1 (mod 4).
Since s ≡ −1 (mod 2f ), we have f = 1. Since s ∈ Z∗2e−1 , we may write s = 1+ 2
ht
where 2 ≤ h ≤ e − 2 and t ∈ Z∗2e−h−1 . By Proposition 6.1 (b), s has multiplicative order
2e−h−1 in Z2e−1 , and so ℓ = 2e−h−2. We have 2 ∤ w since
2e−1 ∥ (s2ℓ − 1) and 2 | 1
2
s(s− 1)ℓ.
Subcase (b). s ≡ −1 (mod 4).
We may write s = −1 + 2ht, where 2 ≤ h ≤ e − 1 and t ∈ Z∗2e−h−1 . Since s ≡ −1
(mod 2f ), we have f ≤ h. Recall that s has multiplicative order 2ℓ in Z2e−f .
If h < e − f − 1, then e > f + h + 1 ≥ 4. By Proposition 6.1, s has multiplicative
order 2e−f−h in Z2e−f , and hence ℓ = 2e−f−h−1. We also have 2 ∤ w since
2e−f ∥ (s2ℓ − 1) and 2 | 1
2
s(s− 1)ℓ.
If h ≥ e− f − 1, then ℓ = 1 and hence
w ≡ 2f−e
(
(−1 + 2ht)2 − 1
)
− (−1 + 2ht)(−1 + 2h−1t)
≡ (−1 + 2h−1t)(2f−e+h+1t− 2ht+ 1)
≡ 2f−e+h+1t+ 1 (mod 2).
It follows that 2 ∤ w if and only if h > e− f − 1. Therefore the case h = e− f − 1 should
be excluded.
170 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
From the above discussion, we obtain
N(2e, 21) =
e−2∑
h=2
ϕ(2e−h−1) +
e−1∑
h=2
ϕ(2e−h−1)− ϕ(2) = 2e−2 − 2,
and for f > 1,
N(2e, 2f ) =
e−f−2∑
h=f
ϕ(2e−h−1) +
e−1∑
h=e−f
ϕ(2e−h−1) = 2e−f−1 − 2f−1.
Consequently, for e ≥ 5, we get
N(2e) =
⌊ e−12 ⌋∑
f=1
N(2e, 2f ) = 2e−2 − 2 +
⌊ e−12 ⌋∑
f=2
(2e−f−1 − 2f−1)
= 2e−2 − 2 + (2⌊
e−1
2 ⌋−1 − 1)(2e−1−⌊
e−1
2 ⌋) − 2).
Note that ⌊ e−12 ⌋ = (e − 2)/2 if e if even, and ⌊
e−1
2 ⌋ = (e − 1)/2 if e is odd. The result
follows from substitution for ⌊ e−12 ⌋ in the above formula, as required.
Remark 6.3. By Theorem 1.2, one can enumerate the conjugacy classes of proper square
roots of automorphisms of Zn for any positive integer n in the following steps:
(a) Find the set of all positive integers k satisfying that k2 divides n if k is odd, and 2k2
divides n if k is even. Denote this set by A(n).
(b) For any k ∈ A(n), find the set of all s satisfying (i) s ≡ −1 (mod k) and (ii) s ∈ Z∗n
if k is odd, and s ∈ Z∗n/2 if k is even. Denote this set by S(n, k).
(c) For any s ∈ S(n, k), calculate the smallest positive integer ℓ such that s2ℓ ≡ 1
(mod n/k) and check whether kn (s
2ℓ − 1)− 12s(s− 1)ℓ is coprime to k or not. Let
A(n, k) be the set of all s ∈ S(n, k) satisfying that kn (s
2ℓ − 1) − 12s(s − 1)ℓ is
coprime to k.
(d) Now (k, s) is admissible for proper square root of automorphism of Zn if and only
if k ∈ A(n) and s ∈ A(n, k). The number N(n) of the conjugacy classes of proper
square roots of automorphisms of Zn is
∑
k∈A(n) |A(n, k)|.
Using the method above, we obtain N(18) = 2, N(24) = 2, N(40) = 2 and N(72) =
16. In each case the parameters (n, k, s) are given below (details are omitted):
(n, k) (18, 3) (24, 2) (40, 2) (72, 2) (72, 3) (72, 6)
s 11, 17 7, 11 11, 19 7, 11, 19, 23, 31, 35 11, 17, 29, 35, 47, 53, 65, 71 23, 35
We close the paper by attaching a full list of conjugacy classes of proper square roots
of automorphisms of Zn for some small values of n.
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 171
Table 1: Proper square roots of automorphisms of Zn.
n φ(x) π(x) φ2(x)
8 6x2 + 5x (mod 8) 1 + 2x (mod 4) 5x (mod 8)
9 3x2 + 2x (mod 9) 1 + 2x (mod 6) 4x (mod 9)
9 3x2 + 4x (mod 9) 1 + 2x (mod 6) 4x (mod 9)
16 12x2 + 9x (mod 16) 1 + 2x (mod 4) 9x (mod 16)
16 12x2 + 11x (mod 16) 1 + 2x (mod 4) 9x (mod 16)
18 15x2 + 2x (mod 18) 1 + 2x (mod 6) 13x (mod 18)
18 15x2 + 14x (mod 18) 1 + 2x (mod 6) 7x (mod 18)
24 18x2 + 13x (mod 24) 1 + 2x (mod 4) 23x (mod 24)
24 18x2 + 17x (mod 24) 1 + 2x (mod 4) 13x (mod 24)
27 9x2 + 2x (mod 27) 1 + 6x (mod 18) 4x (mod 27)
27 9x2 + 5x (mod 27) 1 + 6x (mod 18) 25x (mod 27)
27 9x2 + 8x (mod 27) 1 + 2x (mod 6) 10x (mod 27)
27 9x2 + 11x (mod 27) 1 + 6x (mod 18) 13x (mod 27)
27 9x2 + 14x (mod 27) 1 + 12x (mod 18) 7x (mod 27)
27 9x2 + 17x (mod 27) 1 + 4x (mod 6) 19x (mod 27)
27 9x2 + 20x (mod 27) 1 + 6x (mod 18) 22x (mod 27)
27 9x2 + 23x (mod 27) 1 + 12x (mod 18) 16x (mod 27)
32 24x2 + 11x (mod 32) 1 + 4x (mod 8) 25x (mod 32)
32 24x2 + 13x (mod 32) 1 + 4x (mod 8) 25x (mod 32)
32 24x2 + 17x (mod 32) 1 + 2x (mod 4) 17x (mod 32)
32 24x2 + 19x (mod 32) 1 + 4x (mod 8) 9x (mod 32)
32 24x2 + 21x (mod 32) 1 + 4x (mod 8) 9x (mod 32)
32 24x2 + 23x (mod 32) 1 + 2x (mod 4) 17x (mod 32)
32 28x2 + 11x (mod 32) 1 + 2x (mod 8) 9x (mod 32)
32 28x2 + 19x (mod 32) 1 + 6x (mod 8) 25x (mod 32)
40 30x2 + 21x (mod 40) 1 + 2x (mod 4) 31x (mod 40)
40 30x2 + 29x (mod 40) 1 + 2x (mod 4) 21x (mod 40)
64 48x2 + 19x (mod 64) 1 + 8x (mod 16) 41x (mod 64)
64 48x2 + 21x (mod 64) 1 + 8x (mod 16) 25x (mod 64)
64 48x2 + 23x (mod 64) 1 + 4x (mod 8) 17x (mod 64)
64 48x2 + 25x (mod 64) 1 + 4x (mod 8) 17x (mod 64)
64 48x2 + 27x (mod 64) 1 + 8x (mod 16) 25x (mod 64)
64 48x2 + 29x (mod 64) 1 + 8x (mod 16) 41x (mod 64)
64 48x2 + 33x (mod 64) 1 + 2x (mod 4) 33x (mod 64)
64 48x2 + 35x (mod 64) 1 + 8x (mod 16) 9x (mod 64)
64 48x2 + 37x (mod 64) 1 + 4x (mod 16) 57x (mod 64)
64 48x2 + 39x (mod 64) 1 + 4x (mod 8) 49x (mod 64)
64 48x2 + 41x (mod 64) 1 + 4x (mod 8) 49x (mod 64)
64 48x2 + 43x (mod 64) 1 + 8x (mod 16) 57x (mod 64)
64 48x2 + 45x (mod 64) 1 + 8x (mod 16) 9x (mod 64)
64 48x2 + 47x (mod 64) 1 + 2x (mod 4) 33x (mod 64)
64 56x2 + 11x (mod 64) 1 + 12x (mod 16) 25x (mod 64)
64 56x2 + 19x (mod 64) 1 + 4x (mod 16) 9x (mod 64)
64 56x2 + 23x (mod 64) 1 + 2x (mod 8) 17x (mod 64)
64 56x2 + 27x (mod 64) 1 + 12x (mod 16) 57x (mod 64)
64 56x2 + 35x (mod 64) 1 + 4x (mod 16) 41x (mod 64)
64 56x2 + 39x (mod 64) 1 + 6x (mod 8) 49x (mod 64)
172 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
ORCID iDs
Kan Hu https://orcid.org/0000-0003-4775-7273
Young Soo Kwon https://orcid.org/0000-0002-1765-0806
Jun-Yang Zhang https://orcid.org/0000-0002-0871-2059
References
[1] M. Bachratý and R. Jajcay, Powers of skew-morphisms, in: Symmetries in Graphs, Maps,
and Polytopes, Springer International Publishing, volume 159, pp. 1–25, 2016, doi:10.1007/
978-3-319-30451-9.
[2] M. Bachratý and R. Jajcay, Classification of coset-preserving skew-morphisms of finite cyclic
groups, Australas. J. Comb. 67 (2017), 259–280, https://ajc.maths.uq.edu.au/
?page=get_volumes&volume=67.
[3] B. G. Basmaji, On the ismorphisms of two metacyclic groups, Proc. Amer. Math. Soc. 22
(1969), 175–182, doi:10.2307/2036947.
[4] M. Conder, R. Jajcay and T. Tucker, Regular Cayley maps for finite abelian groups, J. Algebraic
Combin. 25 (2007), 259–283, doi:10.1007/s10801-006-0037-0.
[5] M. Conder, R. Jajcay and T. Tucker, Regular t-balanced Cayley maps, J. Combin. Theory Ser.
B 97 (2007), 453–473, doi:10.1016/j.jctb.2006.07.008.
[6] M. D. E. Conder, R. Jajcay and T. W. Tucker, Cyclic complements and skew morphisms of
groups, J. Algebra 453 (2016), 68–100, doi:10.1016/j.jalgebra.2015.12.024.
[7] M. D. E. Conder, Y. S. Kwon and J. Širáň, Reflexibility of regular Cayley maps for abelian
groups, J. Combin. Theory Ser. B 99 (2009), 254–260, doi:10.1016/j.jctb.2008.07.002.
[8] M. D. E. Conder and T. W. Tucker, Regular Cayley maps for cyclic groups, Trans. Amer. Math.
Soc. 366 (2014), 3585–3609, doi:10.1090/s0002-9947-2014-05933-3.
[9] S. Du and K. Hu, Skew-morphisms of cyclic 2-groups, J. Group Theory 22 (2019), 617–635,
doi:10.1515/jgth-2019-2046.
[10] R. Feng, R. Jajcay and Y. Wang, Regular t-balanced Cayley maps for abelian groups, Discrete
Math. 311 (2011), 2309–2316, doi:10.1016/j.disc.2011.04.012.
[11] Y.-Q. Feng, K. Hu, R. Nedela, M. Škoviera and N.-E. Wang, Complete regular dessins and
skew-morphisms of cyclic groups, Ars Math. Contemp. 18 (2020), 289–307, doi:10.26493/
1855-3974.1748.ebd.
[12] K. Hu, R. Nedela and N.-E. Wang, Nilpotent groups of class two which underly a unique regular
dessin, Geom. Dedicata 179 (2015), 177–186, doi:10.1007/s10711-015-0074-8.
[13] R. Jajcay and J. Širáň, Skew-morphisms of regular Cayley maps, Discrete Math. 244 (2002),
167–179, doi:10.1016/s0012-365x(01)00081-4.
[14] I. Kovács and Y. S. Kwon, Regular Cayley maps on dihedral groups with the smallest kernel,
J. Algebraic Combin. 44 (2016), 831–847, doi:10.1007/s10801-016-0689-3.
[15] I. Kovács and Y. S. Kwon, Classification of reflexible Cayley maps for dihedral groups, J.
Combin. Theory Ser. B 127 (2017), 187–204, doi:10.1016/j.jctb.2017.06.002.
[16] I. Kovács and Y. S. Kwon, Regular Cayley maps for dihedral groups, J. Comb. Theory Ser. B
148 (2021), 84–124, doi:10.1016/j.jctb.2020.12.002.
[17] I. Kovács and R. Nedela, Decomposition of skew-morphisms of cyclic groups, Ars Math. Con-
temp. 4 (2011), 329–349, doi:10.26493/1855-3974.157.fc1.
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 173
[18] I. Kovács and R. Nedela, Skew-morphisms of cyclic p-groups, J. Group Theory 20 (2017),
1135–1154, doi:10.1515/jgth-2017-0015.
[19] J. H. Kwak, Y. S. Kwon and R. Feng, A classification of regular t-balanced Cayley maps on
dihedral groups, European J. Combin. 27 (2006), 382–393, doi:10.1016/j.ejc.2004.12.002.
[20] J. H. Kwak and J.-M. Oh, A classification of regular t-balanced Cayley maps on dicyclic
groups, European J. Combin. 29 (2008), 1151–1159, doi:10.1016/j.ejc.2007.06.023.
[21] Y. S. Kwon, A classification of regular t-balanced Cayley maps for cyclic groups, Discrete
Math. 313 (2013), 656–664, doi:10.1016/j.disc.2012.12.012.
[22] J.-M. Oh, Regular t-balanced Cayley maps on semi-dihedral groups, J. Combin. Theory Ser. B
99 (2009), 480–493, doi:10.1016/j.jctb.2008.09.006.
[23] N.-E. Wang, K. Hu, K. Yuan and J.-Y. Zhang, Smooth skew morphisms of dihedral groups, Ars
Math. Contemp. 16 (2019), 527–547, doi:10.26493/1855-3974.1475.3d3.
[24] M. Xu and Q. Zhang, A classification of metacyclic 2-groups, Algebra Colloq. 13 (2006), 25–
34, doi:10.1142/s1005386706000058.
[25] K. Yuan, Y. Wang and J. H. Kwak, Enumeration of skew-morphisms of groups of small
orders and their corresponding Cayley maps, Adv. Math. (China) 45 (2016), 21–36, doi:
10.1103/physrevd.45.21.
[26] J.-Y. Zhang, Regular Cayley maps of skew-type 3 for abelian groups, European J. Combin. 39
(2014), 198–206, doi:10.1016/j.ejc.2014.01.006.
[27] J.-Y. Zhang, A classification of regular Cayley maps with trivial Cayley-core for dihedral
groups, Discrete Math. 338 (2015), 1216–1225, doi:10.1016/j.disc.2015.01.036.
[28] J.-Y. Zhang, Regular Cayley maps of skew-type 3 for dihedral groups, Discrete Math. 338
(2015), 1163–1172, doi:10.1016/j.disc.2015.01.038.
[29] J.-Y. Zhang and S. Du, On the skew-morphisms of dihedral groups, J. Group Theory 19 (2016),
993–1016, doi:10.1515/jgth-2016-0027.

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P2.02 / 175–200
https://doi.org/10.26493/1855-3974.2373.c02
(Also available at http://amc-journal.eu)
Trivalent dihedrants and bi-dihedrants*
Mi-Mi Zhang
School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China
Jin-Xin Zhou †
Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China
Received 3 July 2020, accepted 22 February 2021, published online 21 October 2021
Abstract
A Cayley (resp. bi-Cayley) graph on a dihedral group is called a dihedrant (resp. bi-
dihedrant). In 2000, a classification of trivalent arc-transitive dihedrants was given by
Marušič and Pisanski, and several years later, trivalent non-arc-transitive dihedrants of or-
der 4p or 8p (p a prime) were classified by Feng et al. As a generalization of these results,
our first result presents a classification of trivalent non-arc-transitive dihedrants. Using this,
a complete classification of trivalent vertex-transitive non-Cayley bi-dihedrants is given,
thus completing the study of trivalent bi-dihedrants initiated in our previous paper [Dis-
crete Math. 340 (2017) 1757–1772]. As a by-product, we generalize a theorem in [The
Electronic Journal of Combinatorics 19 (2012) #P53].
Keywords: Cayley graph, non-Cayley, bi-Cayley, dihedral group, dihedrant, bi-dihedrant.
Math. Subj. Class. (2020): 05C25, 20B25
1 Introduction
In this paper we describe an investigation of trivalent Cayley graphs on dihedral groups
as well as vertex-transitive trivalent bi-Cayley graphs over dihedral groups. To be brief,
we shall say that a Cayley (resp. bi-Cayley) graph on a dihedral group a dihedrant (resp.
bi-dihedrant).
Cayley graphs are usually defined in the following way. Given a finite group G and an
inverse closed subset S ⊆ G\{1}, the Cayley graph Cay(G,S) on G with respect to S is a
graph with vertex set G and edge set {{g, sg} | g ∈ G, s ∈ S}. For any g ∈ G, R(g) is the
*This work was supported by the National Natural Science Foundation of China (11671030, 12101181) and
the Natural Science Foundation of Hebei Province (A2019205180)
†Corresponding author.
E-mail addresses: mmzhang@hebtu.edu.cn (Mi-Mi Zhang), jxzhou@bjtu.edu.cn (Jin-Xin Zhou)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
176 Ars Math. Contemp. 21 (2021) #P2.02 / 175–200
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 subgroup of Aut (Cay(G,S)). We say that the Cayley graph
Cay(G,S) is normal if R(G) is normal in Aut (Cay(G,S)) (see [19]).
In 2000, Marušič and Pisanski [13] initiated the study of automorphisms of dihedrants,
and they gave a classification of trivalent arc-transitive dihedrants. Following this work,
highly symmetrical dihedrants have been extensively studied, and one of the remarkable
achievements is the complete classification of 2-arc-transitive dihedrants (see [7, 12]). In
contrast, however, relatively little is known about the automorphisms of non-arc-transitive
dihedrants. In [1], the authors claimed that every trivalent non-arc-transitive dihedrant is
normal. However, this is not true. There exist non-arc-transitive and non-normal dihe-
drants. Actually, in [22, 26], the automorphism groups of trivalent dihedrants of order 4p
and 8p are determined for each prime p, and the result reveals that every non-arc-transitive
trivalent dihedrant of order 4p or 8p is either a normal Cayley graph, or isomorphic to the
so-called cross ladder graph. For an integer m ≥ 2, the cross ladder graph, denoted by
CL4m, is a trivalent graph of order 4m with vertex set V0∪V1∪ . . . V2m−2∪V2m−1, where
Vi = {x0i , x1i }, and edge set {{xr2i, xr2i+1}, {xr2i+1, xs2i+2} | i ∈ Zm, r, s ∈ Z2} (see Fig. 1
for CL4m). It is worth mentioning that the cross ladder graph plays an important role in the
· · ·




S
S
S
S
S 



S
S
S
S
S 



S
S
S
S
S· · ·
• • • • • • • •
• • • • • • • •
x02m−1 x
0
0 x
0
1 x
0
2 x
0
3 x
0
2m−4 x
0
2m−3 x
0
2m−2
x12m−1 x
1
0 x
1
1 x
1
2 x
1
3 x
1
2m−4 x
1
2m−3 x
1
2m−2
Figure 1: The cross ladder graph CL4m
study of automorphisms of trivalent graphs (see, for example, [5, 21, 26]). Motivated by
the above mentioned facts, we shall focus on trivalent non-arc-transitive dihedrants. Our
first theorem generalizes the results in [22, 26] to all trivalent dihedrants.
Theorem 1.1. Let Σ = Cay(H,S) be a connected trivalent Cayley graph, where H =
⟨a, b | an = b2 = 1, bab = a−1⟩(n ≥ 3). If Σ is non-arc-transitive and non-normal, then
n is even and Σ ∼= CL4·n2 and S
α = {b, ba, ban2 } for some α ∈ Aut (H).
Recall that for an integer m ≥ 2, the cross ladder graph CL4m has vertex set V0 ∪
V1 ∪ . . . V2m−2 ∪ V2m−1, where Vi = {x0i , x1i }. The multi-cross ladder graph, denoted
by MCL4m,2, is the graph obtained from CL4m by blowing up each vertex xri of CL4m
into two vertices xr,0i and x
r,1
i . The edge set is {{x
r,s
2i , x
r,t
2i+1}, {x
r,s
2i+1, x
s,r
2i+2} | i ∈
Zm, r, s, t ∈ Z2} (see Fig. 2 for MCL20,2).
Note that the multi-cross ladder graph MCL4m,2 is just the graph given in [23, Def-
inition 7]. From [6, Proposition 3.3] we know that every MCL4m,2 is vertex-transitive.
However, not all multi-cross ladder graphs are Cayley graphs. Actually, in [23, Theo-
rem 9], it is proved that MCL4p,2 is a vertex-transitive non-Cayley graph for each prime
p > 7. Our second theorem generalizes this result to all multi-cross ladder graphs.
Theorem 1.2. The multi-cross ladder graph MCL4m,2 is a Cayley graph if and only if
either m is even, or m is odd and 3 | m.
Mi-Mi Zhang and Jin-Xin Zhou: Trivalent dihedrants and bi-dihedrants 177
t

t t

t t

t t

t t

t
t
@
@
@@
t

t
@
@
@@
t

t
@
@
@@
t

t
@
@
@@
t

t
@
@
@@
t
t

t
@
@
@@
t

t
@
@
@@
t

t
@
@
@@
t

t
@
@
@@
t

t
t
@
@
@@
t t
@
@
@@
t t
@
@
@@
t t
@
@
@@
t t
@
@
@@
t
x0,00 x
0,0
1 x
0,0
2 x
0,0
3 x
0,0
4 x
0,0
5 x
0,0
6 x
0,0
7 x
0,0
8 x
0,0
9
x1,10 x
1,1
1 x
1,1
2 x
1,1
3 x
1,1
4 x
1,1
5 x
1,1
6 x
1,1
7 x
1,1
8 x
1,1
9
x1,00 x
1,0
1 x
1,0
2 x
1,0
3 x
1,0
4 x
1,0
5 x
1,0
6 x
1,0
7 x
1,0
8 x
1,0
9
x0,10 x
0,1
1 x
0,1
2 x
0,1
3 x
0,1
4 x
0,1
5 x
0,1
6 x
0,1
7 x
0,1
8 x
0,1
9
Figure 2: The multi-cross ladder graph MCL20,2
Both of the above two theorems are crucial in attacking the problem of classification
of trivalent vertex-transitive non-Cayley bi-dihedrants. Before proceeding, we give some
background to this topic, and set some notation.
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)
over H relative to R,L, S is a 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}. If |R| = |L| = s, then BiCay(H, R, L, S) is said to be an s-type
bi-Cayley graph.
In [20] we initiated a program to investigate the automorphism groups of the trivalent
vertex-transitive bi-dihedrants. This was partially motivated by the following facts. As
one of the most important finite graphs, the Petersen graph is a bi-circulant, but it is not
a Cayley graph. Note that a bi-circulant is a bi-Cayley graph over a cyclic group. The
Petersen graph is the initial member of a family of graphs P (n, t), known now as the
generalized Petersen graphs (see [17]), which can be also constructed as bi-circulants. Let
n ≥ 3, 1 ≤ t < n/2 and set H = ⟨a⟩ ∼= Zn. The generalized Petersen graph P (n, t)
is isomorphic to the bi-circulant BiCay(H, {a, a−1}, {at, a−t}, {1}). The complete
classification of vertex-transitive generalized Petersen graphs has been worked out in [8,
14]. Latter, this was generalized by Marušič et al. in [13, 15] where all trivalent vertex-
transitive bi-circulants were classified, and more recently, all trivalent vertex-transitive bi-
Cayley graphs over abelian groups were classified in [24]. The characterization of trivalent
vertex-transitive bi-dihedrants is the next natural step.
Another motivation for us to consider trivalent vertex-transitive bi-dihedrants comes
from the excellent work in a highly cited article [16], where the authors give a census of
trivalent vertex-transitive graphs of order up to 1280. This is very important in the study of
trivalent vertex-transitive graphs. Actually, by checking this census of graphs of order up
to 1000, we find out that there are 981 non-Cayley graphs, and among these graphs, 233
graphs are non-Cayley bi-dihedrants. This may suggest bi-dihedrants form an important
class of trivalent vertex-transitive non-Cayley graphs.
In [20], we gave a classification of trivalent arc-transitive bi-dihedrants, and we also
proved that every trivalent vertex-transitive 0- or 1-type bi-dihedrant is a Cayley graph, and
gave a classification of trivalent vertex-transitive non-Cayley bi-dihedrants of order 4n with
178 Ars Math. Contemp. 21 (2021) #P2.02 / 175–200
n odd. The goal of this paper is to complete the classification of trivalent vertex-transitive
non-Cayley bi-dihedrants.
Before stating the main result, we need the following concepts. For a bi-Cayley graph
Γ = BiCay(H, R, L, S) over a group H , we can assume that the identity 1 of H is in S
(see Proposition 2.3 (2)). 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 . The bi-Cayley graphs corresponding to two equivalent bi-Cayley triples of the same
group are isomorphic (see Proposition 2.3 (3)-(4)).
Theorem 1.3. Let Γ = BiCay(R,L, S) be a trivalent vertex-transitive bi-dihedrant where
H = ⟨a, b | an = b2 = 1, bab = a−1⟩ is a dihedral group. Then either Γ is a Cayley graph
or one of the following occurs:
(1) (R,L, S) ≡ ({b, ba}, {a, a−1}, {1}), where n = 5.
(2) (R,L, S) ≡ ({b, baℓ+1}, {ba, baℓ2+ℓ+1}, {1}), where n ≥ 5, ℓ3 + ℓ2 + ℓ + 1 ≡
0 (mod n), ℓ2 ̸≡ 1 (mod n).
(3) (R,L, S) ≡ ({ba−ℓ, baℓ}, {a, a−1}, {1}), where n = 2m and ℓ2 ≡ −1 (mod m).
Furthermore, Γ is also a bi-Cayley graph over an abelian group Zn × Z2.
(4) (R,L, S) ≡ ({b, ba}, {b, ba2m}, {1}), where n = 2(2m + 1), m ̸≡ 1 (mod 3), and
the corresponding graph is isomorphic the multi-cross ladder graph MCL4m,2.
(5) (R,L, S) ≡ ({b, ba}, {ba24ℓ, ba12ℓ−1}, {1}), where n = 48ℓ and ℓ ≥ 1.
Moreover, all of the graphs arising from (1)-(4) are vertex-transitive non-Cayley.
2 Preliminaries
All groups considered in this paper are finite, and all graphs are finite, connected, simple
and undirected. For the group-theoretic and graph-theoretic terminology not defined here
we refer the reader to [3, 18].
2.1 Definitions and notations
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. For two groups M and N , N ⋊ M
denotes a semidirect product of N by M . For a subgroup H of a group G, denote CG(H)
the centralizer of H in G and by NG(H) the normalizer of H of G. Let G be a permutation
group on a set Ω and α ∈ Ω. Denote by Gα the stabilizer of α in G. We say that G is
semiregular on Ω if Gα = 1 for every α ∈ Ω and regular if G is transitive and semiregular.
For a finite, simple and undirected graph Γ, we use V (Γ), E(Γ), A(Γ), Aut (Γ) to
denote its vertex set, edge set, arc set and full automorphism group, respectively. For any
subset B of V (Γ), the subgraph of Γ induced by B will be denoted by Γ[B]. For any
v ∈ V (Γ) and a positive integer i no more than the diameter of Γ, denote by Γi(v) be the
set of vertices at distance i from v. Clearly, Γ1(v) is just the neighborhood of v. We shall
often abuse the notation by using Γ(v) to replace Γ1(v).
A graph Γ is said to be vertex-transitive, and arc-transitive (or symmetric) if Aut (Γ)
acts transitively on V (Γ) and A(Γ), respectively. Let Γ be a connected vertex-transitive
Mi-Mi Zhang and Jin-Xin Zhou: Trivalent dihedrants and bi-dihedrants 179
graph, and let G ≤ Aut (Γ) be vertex-transitive on Γ. For a G-invariant partition B of
V (Γ), the quotient graph ΓB is defined as the graph with vertex set B such that, for any
two different vertices B,C ∈ B, B is adjacent to C if and only if there exist u ∈ B and
v ∈ C which are adjacent in Γ. Let N be a normal subgroup of G. Then the set B of
orbits of N in V (Γ) is a G-invariant partition of V (Γ). In this case, the symbol ΓB will be
replaced by ΓN . The original graph Γ is said to be a N -cover of ΓN if Γ and ΓN have the
same valency.
2.2 Cayley graphs
Let Γ = Cay(G,S) be a Cayley graph on G with respect to S. Then Γ is vertex-transitive
due to R(G) ≤ Aut (Γ). In general, we have the following proposition.
Proposition 2.1 ([2, Lemma 16.3]). A vertex-transitive graph Γ is isomorphic to a Cayley
graph on a group G if and only if its automorphism group has a subgroup isomorphic to
G, acting regularly on the vertex set of Γ.
In 1981, Godsil [9] proved that the normalizer of R(G) in Aut (Cay(G,S)) is R(G)⋊
Aut (G,S), where Aut (G,S) is the group of automorphisms of G fixing the set S set-
wise. This result has been successfully used in characterizing various families of Cayley
graphs Cay(G,S) such that R(G) = Aut (Cay(G,S)) (see, for example, [9, 10]). Recall
that a Cayley graph Cay(G,S) is said to be normal if R(G) is normal in Aut (Cay(G,S))
(see [19]).
Proposition 2.2 ([19, Proposition 1.5]). The Cayley graph Γ = Cay(G,S) is normal if
and only if A1 = Aut (G,S), where A1 is the stabilizer of the identity 1 of G in Aut (Γ).
2.3 Basic properties of bi-Cayley graphs
In this subsection, we let Γ be a connected bi-Cayley graph BiCay(H,R,L, S) over a group
H . It is easy to prove some basic properties of such a Γ, as in [24, Lemma 3.1].
Proposition 2.3. 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).
Next, we collect several results about the automorphisms of bi-Cayley graph Γ =
BiCay(H, R, L, S). For each g ∈ H , define a permutation as follows:
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.
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.
(2.2)
180 Ars Math. Contemp. 21 (2021) #P2.02 / 175–200
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)
Proposition 2.4 ([25, Theorem 1.1]). Let Γ = BiCay(H,R,L, S) be a connected 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 Cross ladder graphs
The goal of this section is to prove Theorem 1.1.
Proof of Theorem 1.1. Suppose that Σ = Cay(H,S) is a connected trivalent Cayley graph
which is neither normal nor arc-transitive, where H = ⟨a, b | an = b2 = 1, bab =
a−1⟩(n ≥ 3). Then S is a generating subset of H and |S| = 3. So S must contain an
involution of H outside ⟨a⟩. As Aut (H) is transitive on the coset b⟨a⟩, we may assume
that S = {b, x, y} for x, y ∈ H \ ⟨b⟩.
Suppose first that x is not an involution. Then we must have y = x−1. Since S
generates H , one has ⟨a⟩ = ⟨x⟩, and so bxb = x−1. Then there exists an automorphism of
H sending b, x to b, a respectively. So we may assume that S = {b, a, a−1}. Now it is easy
to check that Σ is isomorphic to the generalized Petersen graph P (n, 1). Since Σ is not
arc-transitive, by [8, 14], we have |Aut (Σ)| = 2|H|, and so Σ would be a normal Cayley
graph of H , a contradiction.
Therefore, both x and y must be involutions. Suppose that x ∈ ⟨a⟩. Then n is even
and x = an/2. Again since S generates H , one has y = baj , where 1 ≤ j ≤ n − 1 and
either (j, n) = 1 or (j, n) = 2 and n2 is odd. Note that the subgroup of Aut (H) fixing b
is transitive on the set of generators of ⟨a⟩ and that ⟨an/2⟩ is the center of H . There exists
α ∈ Aut (H) such that
Sα = {b, ba, an2 } or {b, ba2, an2 }.
Without loss of generality, we may assume that S = {b, ba, an2 } or {b, ba2, an2 }. If S =
{b, ba2, an2 }, we shall prove that Σ ∼= P (n, 1). Note that the generalized Petersen graph
P (n, 1) has vertex set {ui, vi | i ∈ Zn} and edge set {{ui, ui+1}, {vi, vi+1}, {ui, vi} | i ∈
Zn}. Define a map from V (Σ) to V (P (n, 1)) as follows:
φ : a2i 7→ u2i, a2i+
n
2 7→ v2i,
ba2i 7→ u2i−1, ba2i+
n
2 7→ v2i−1,
where 0 ≤ i ≤ n2 − 1. It is easy to see that φ is an isomorphism form Σ to P (n, 1). Since
Σ is not arc-transitive, by [8, 14], we have |Aut (Σ)| = 2|H|, and so Σ would be a normal
Cayley graph of H , a contradiction. If S = {b, ba, an2 }, then Σ has a connected subgraph
Σ1 = Cay(H, {b, ba}) which is a cycle of length 2n, and Σ is just the graph obtained from
Mi-Mi Zhang and Jin-Xin Zhou: Trivalent dihedrants and bi-dihedrants 181
Σ1 by adding a 1-factor such that each vertex g of Σ1 is adjacent to its antipodal vertex
a
n
2 g. Then R(H) ⋊ Z2 ∼= Aut (Σ1) ≤ Aut (Σ), and then since Σ is assumed to be not
arc-transitive, Aut (Σ) will fix the 1-factor {{g, an2 g} | g ∈ H} setwise. This implies that
Aut (Σ) ≤ Aut (Σ1) and so Aut (Σ) = Aut (Σ1). Consequently, we have Σ is a normal
Cayley graph of H , a contradiction.
Similarly, we have y /∈ ⟨a⟩. Then we may assume that x = bai and y = baj for some
1 ≤ i, j ≤ n − 1 and i ̸= j. Then S = {b, bai, baj} ⊆ b⟨a⟩. This implies that Σ is
a bipartite graph with ⟨a⟩ and b⟨a⟩ as its two partition sets. Since Σ is not arc-transitive,
Aut (Σ)1 is intransitive on the neighbourhood S of 1, and since Σ is not a normal Cayley
graph of H , there exists a unique element, say s ∈ S, such that Aut (Σ)1 = Aut (Σ)s.
Considering the fact that Aut (H) is transitive on b⟨a⟩, without loss of generality, we may
assume that Aut (Σ)1 = Aut (Σ)b and Aut (Σ)1 swaps bai and baj . Then for any h ∈ H ,
we have
Aut (Σ)h = (Aut (Σ)1)R(h) = (Aut (Σ)b)R(h) = Aut (Σ)bh.
Direct computation shows that
Σ2(1) = {a−i, a−j , ai, ai−j , aj , aj−i},
Σ3(1) = {ba−i, baj−i, ba−j , bai−j , ba2i, baj+i, ba2i−j , ba2j , ba2j−i}.
Let Aut (Σ)∗1 be the kernel of Aut (Σ)1 acting on S. Take an α ∈ Aut (Σ)∗1. Then α
fixes every element in S. As Aut (Σ)h = Aut (Σ)bh for any h ∈ H , α will fix b(bai) = ai
and b(baj) = aj . Note that Σ(bai) \ {1, ai} = {ai−j} and Σ(baj) \ {1, aj} = {aj−i}.
Then α also fixes ai−j and aj−i, and then α also fixes bai−j and baj−i.
If |Σ2(1)| = 6, then it is easy to check that a−i is the unique common neighbor of b
and baj−i. So α also fixes a−i. Now one can see that α fixes every vertex in Σ2(1). If
|Σ2(1)| < 6 and either |Σ1(b) ∩ Σ1(bai)| > 1 or |Σ1(b) ∩ Σ1(baj)| > 1, then α also fixes
every vertex in Σ2(1). In the above two cases, by the connectedness and vertex-transitivity
of Σ, α would fix all vertices of Σ, implying that α = 1. Hence, Aut (Σ)∗1 = 1 and
Aut (Σ)1 ∼= Z2. This forces that Σ is a normal Cayley graph of H , a contradiction.
Thus, we have |Σ2(1)| < 6 and |Σ1(b) ∩ Σ1(bai)| = |Σ1(b) ∩ Σ1(baj)| = 1. This
implies that Σ1(bai) ∩ Σ1(baj) = {1, ai−j} = {1, aj−i}, and so ai−j = aj−i. It follows
that ai−j is an involution, and hence n is even and ai−j = an/2. So S = {b, bai, bai+n/2}.
As S generates H , one has ⟨ai, an/2⟩ = ⟨a⟩. So either (i, n) = 1 or (i, n) = 2 and n2 is
odd. Note that the subgroup of Aut (H) fixing b is transitive on the set of generators of ⟨a⟩
and that ⟨an/2⟩ is the center of H . There exists α ∈ Aut (H) such that
Sα = {b, ba, ba1+n2 } or {b, ba2, ba2+n2 }.
Let βϵ be the automorphism of H induced by the map a 7→ a−1, b 7→ baϵ, where ϵ ∈ Z2.
Then
{b, ba, ba1+n2 }β1 = {b, ba, ban2 }, and {b, ba2, ba2+n2 }β2 = {b, ba2, ban2 }.
If n2 is odd, then the map η : a 7→ a
2+n2 , b 7→ ban2 induces an automorphism of H ,
and {b, ba, ban2 }η = {b, ba2, ban2 }. So there always exists γ ∈ Aut (H) such that Sγ =
{b, ba, ban2 }, completing the proof of the first part of our theorem.
Finally, we shall prove Σ ∼= CL4·n2 . Without loss of generality, assume that S =
{b, ba, ban2 }. Recall that V (CL4·n2 ) = {x
r
i | i ∈ Z2n, r ∈ Z2} and E(CL4·n2 ) =
182 Ars Math. Contemp. 21 (2021) #P2.02 / 175–200
{{xri , xri+1}, {xr2i, x
r+1
2i+1}, | i ∈ Z2n, r ∈ Z2}. Let ϕ be a map from V (Σ) to V (CL4·n2 )
as following:
ϕ : ai 7→ x02i, ai+
n
2 7→ x12i,
baj 7→ x02j−1, baj+
n
2 7→ x12j−1,
where 0 ≤ i ≤ n2 − 1 and 1 ≤ j ≤
n
2 . It is easy to check that ϕ is an isomorphism from Σ
and X(CL4·n2 ), as desired.
4 Multi-cross ladder graphs
The goal of this section is to prove Theorem 1.2. We first show that each MCL4m,2 is a
bi-Cayley graph.
Lemma 4.1. The multi-cross ladder graph MCL4m,2 is isomorphic to the bi-Cayley graph
BiCay(H, {c, ca}, {ca, ca2b}, {1}), where
H = ⟨a, b, c | am = b2 = c2 = 1, ab = a, ac = a−1, bc = b⟩.
Proof. For convenience, let Γ be the bi-Cayley graph given in our lemma, and let X =
MCL4m,2. Let ϕ be a map from V (X) to V (Γ) defined by the following rule:
ϕ : x1,12t 7→ (at)0, x
1,1
2t+1 7→ (cat+1)0, x
1,0
2t 7→ (cat+1)1, x
1,0
2t+1 7→ (at)1,
x0,12t 7→ (cat+1b)1, x
0,1
2t+1 7→ (atb)1, x
0,0
2t 7→ (atb)0, x
0,0
2t+1 7→ (cat+1b)0,
where t ∈ Zm.
It is easy to see that ϕ is an adjacency preserving isomorphism from X to Γ.
Remark 1 Let m be odd, let e = ab and f = ca. Then the group given in Lemma 4.1 has
the following presentation:
H = ⟨e, f | e2m = f2 = 1, ef = e−1⟩.
Clearly, in this case, H is a dihedral group. Furthermore, the corresponding bi-Cayley
graph given in Lemma 4.1 will be
BiCay(H, {f, fe}, {f, fem−1}, {1}).
Proof of Theorem 1.2. By Lemma 4.1, we may let Γ = MCL4m,2 be just the bi-Cayley
graph BiCay(H,R,L, S), where
H = ⟨a, b, c | am = b2 = c2 = 1, ab = a, ac = a−1, bc = b⟩,
R = {c, ca}, L = {ca, ca2b}, S = {1}.
We first prove the sufficiency. Assume first that m is even. Then the map
a 7→ ab, b 7→ b, c 7→ cb
induces an automorphism, say α of H of order 2. Furthermore, Rα = {c, ca}α = caLca,
Lα = {ca, ca2b}α = caRca and Sα = {1}α = ca{1}ca = S−1. By Proposition 2.4,
δα,ca,ca ∈ Aut (Γ) and R(H)⋊⟨δα,ca,ca⟩ acts regularly on V (Γ). Consequently, by Propo-
sition 2.1, Γ is a Cayley graph.
Mi-Mi Zhang and Jin-Xin Zhou: Trivalent dihedrants and bi-dihedrants 183
Assume now that m is odd and 3 | m. In this case, we shall use the bi-Cayley presen-
tation for Γ as in Remark 5.1, that is,
Γ = BiCay(H, {f, fe}, {f, fem−1}, {1}),
where
H = ⟨e, f | e2m = f2 = 1, ef = e−1⟩.
Let β be a permutation of V (Γ) defined as following:
β : (f ie3t+1)i ↔ (f iem+3t+1)i, (f i+1e3t+1)i ↔ (f iem+3t+1)i+1,
(f i+1e3t+2)i ↔ (f i+1em+3t+2)i, (f ie3t+2)i ↔ (f i+1em+3t+2)i+1,
(e3t)i ↔ (fe3t)i+1, (em+3t)i ↔ (fem+3t)i+1,
where t ∈ Zm
3
and i ∈ Z2. It is easy to check that β is an automorphism of Γ of order 2.
Furthermore, R(e),R(f) and β satisfy the following relations:
R(e)2m = R(f)2 = β2 = 1, R(f)−1R(e)R(f) = R(e)−1, R(f)−1βR(f) = β,
R(e)6β = βR(e)6, R(e)2β = βR(e)4βR(e)−2.
Let G = ⟨R(e2),R(f), β⟩ and P = ⟨R(e2), β⟩. Then R(f) /∈ P and G = P ⟨R(f)⟩.
Since R(e)6β = βR(e)6, we have R(e6) ∈ Z(P ). Since R(e)2β = βR(e)4βR(e)−2, it
follows that
(R(e)2β)3 = R(e)2β[βR(e)4βR(e)−2]R(e)2β = R(e6).
Let N = ⟨R(e6)⟩. Clearly, N is a normal subgroup of G. Furthermore,
P/N = ⟨R(e2)N, βN | R(e2)3N = β2N = (R(e2)β)3N = N⟩ ∼= A4.
Therefore, |P | = 4m and |G| ≤ 8m.
Let
∆00 = {x0 | x ∈ ⟨e2, f⟩}, ∆10 = {(ex)0 | x ∈ ⟨e2, f⟩},
∆01 = {x1 | x ∈ ⟨e2, f⟩}, ∆11 = {(ex)1 | x ∈ ⟨e2, f⟩}.
Then ∆ij’s (i, j ∈ Z2) are four orbits of ⟨R(e2),R(f)⟩. Moreover,
1
βR(f)
0 = 11 ∈ ∆01, e
β
0 = (e
m+1)0 ∈ ∆00, eβ1 = (fem+1)0 ∈ ∆00.
This implies that G is transitive on V (Γ). Hence, |G| = 8m and so G is regular on V (Γ),
and by Proposition 2.1, Γ is a Cayley graph.
To prove the necessity, it suffices to prove that if m is odd and 3 ∤ m, then Γ is a non-
Cayley graph. In this case, we shall use the original definition of Γ = MCL4m,2. Suppose
that m is odd and 3 ∤ m. We already know from [6, Proposition 3.3] that Γ is vertex-
transitive. Let A = Aut (Γ). For m = 5 or 7, using Magma [4], Γ is a non-Cayley graph.
In what follows, we assume that m ≥ 11.
For each j ∈ Zm, C0j = (x
0,0
2j , x
0,0
2j+1, x
0,1
2j , x
0,1
2j+1) and C
1
j = (x
1,1
2j , x
1,1
2j+1, x
1,0
2j , x
1,0
2j+1)
are two 4-cycles. Set F = {Cij | i ∈ Z2, j ∈ Zm}. From the construction of Γ =
MCL4m,2, it is easy to see that in Γ = MCL4m,2 passing each vertex there is exactly one 4-
cycle, which belongs to F . Clearly, any two distinct 4-cycles in F are vertex-disjoint. This
implies that ∆ = {V (Cij) | i ∈ Z2, j ∈ Zm} is an A-invariant partition of V (Γ). Consider
184 Ars Math. Contemp. 21 (2021) #P2.02 / 175–200
the quotient graph Γ∆, and let T be the kernel of A acting on ∆. Then Γ∆ ∼= Cm[2K1],
the lexicographic product of a cycle of length m and an empty graph of order 2. Hence
A/T ≤ Aut (Cm[2K1]) ∼= Zm2 ⋊ D2m. Note that between any two adjacent vertices of
Γ∆ there is exactly one edge of Γ = MCL4m,2. Then T fixes each vertex of Γ and hence
T = 1. So we may view A as a subgroup of Aut (Γ∆) ∼= Aut (Cm[2K1]) ∼= Zm2 ⋊D2m.
For convenience, we will simply use the Cij’s to represent the vertices of Γ∆. Then Γ∆
has vertex set
{C0j ,C1j | j ∈ Zm}
and edge set
{{C0j ,C0j+1}, {C1j ,C1j+1}, {C0j ,C1j+1}, {C1j ,C0j+1} | j ∈ Zm}.
Let B = {{C0j ,C1j} | j ∈ Zm}. Then B is an Aut (Γ∆)-invariant partition of V (Γ∆).
Let K be the kernel of Aut (Γ∆) acting on B. Then K = ⟨k0⟩ × ⟨k2⟩ × · · · × ⟨km−1⟩,
where we use ki to denote the transposition (C0j C
1
j ) for j ∈ Zm. Clearly, K is the maximal
normal 2-subgroup of Aut (Γ∆).
Suppose to the contrary that Γ = MCL4m,2 is a Cayley graph. By Proposition 2.1, A
has a subgroup, say G acting regularly on V (Γ). Then G has order 8m, and
G/(G ∩K) ∼= GK/K ≤ Aut (Γ∆)/K ≲ D2m.
Since m odd, it follows that |G ∩K| = 4 or 8, and so G ∩K ∼= Z22 or Z32.
If G ∩ K ∼= Z22, then |GK/K| = 2m and GK/K = Aut (Γ∆)/K ∼= D2m. So
GK = Aut (Γ∆) ∼= Zm2 ⋊ D2m. Let M be a Hall 2′-subgroup of G. Then M ∼= Zm
and M is also a Hall 2′-subgroup of Aut (Γ∆). Clearly, Aut (Γ∆) is solvable, so all Hall
2′-subgroups of Aut (Γ∆) are conjugate. Without loss of generality, we may let M = ⟨α⟩,
where α is the following permutation on V (Γ∆):
α = (C00 C
0
1 . . .C
0
m−1)(C
1
0 C
1
1 . . .C
1
m−1).
Then K ⋊ ⟨α⟩ acts transitively on V (Γ∆). Clearly, CK(α) is contained in the center of
K ⋊ ⟨α⟩. So CK(α) is semiregular on V (Γ∆). This implies that
CK(α) = ⟨k0k1 . . . km−1⟩ ∼= Z2.
On the other hand, let L = (G ∩ K)M . Clearly, G ∩ K ⊴ G, so L is a subgroup of G
of order 4m. For any odd prime factor p of m, let P be a Sylow p-subgroup of M . Then
P is also a Sylow p-subgroup of L, and since M is cyclic, one has M ≤ NL(P ). By
Sylow theorem, we have |L : NL(P )| = kp + 1 | 4 for some integer k. Since 3 ∤ m, one
has L = NL(P ). It follows that M ⊴ L and so L = M × (G ∩ K). This implies that
G ∩K ≤ CK(M) = CK(α) ∼= Z2, a contradiction.
If G ∩ K ∼= Z32, then |GK/K| = m. Furthermore, GK/K ∼= Zm and GK/K acts
on B regularly. Since G is transitive on V (Γ), there exists g ∈ G such that (x1,10 )g =
x1,11 , where x
1,1
0 , x
1,1
1 ∈ C
1
0. As V (Γ∆) = {Cij | i ∈ Z2, j ∈ Zm}, g fixes the 4-cycle
C10 = (x
1,1
0 , x
1,1
1 , x
1,0
0 , x
1,0
1 ). Since B = {{C
0
j ,C
1
j} | j ∈ Zm} is also A-invariant, g
fixes {C00,C10} setwise. Since GK/K acts on B regularly, g fixes {C0j ,C1j} setwise for
every j ∈ Zm. Observe that {x1,10 , x
1,1
2m−1} and {x
1,1
1 , x
1,1
2 } are the unique edges of Γ
between C10 and C
1
m−1, C
1
0 and C
1
2, respectively. This implies that g will map C
1
m−1 to C
1
2,
contradicting that g fixes {C0j ,C1j} setwise for every j ∈ Zm.
Mi-Mi Zhang and Jin-Xin Zhou: Trivalent dihedrants and bi-dihedrants 185
5 A family of trivalent VNC bi-dihedrants
The goal of this section is to prove the following lemma which gives a new family of
trivalent vertex-transitive non-Cayley bi-dihedrants. To be brief, a vertex-transitive non-
Cayley graph is sometimes simply called a VNC graph.
Lemma 5.1. Let H = ⟨a, b | an = b2 = 1, ab = a−1⟩ be a dihedral group, where n = 48ℓ
and ℓ ≥ 1. Then Γ = BiCay(H, {b, ba}, {ba24ℓ, ba12ℓ−1}, {1}) is a VNC dihedrant.
Proof. We first define a permutation on V (Γ) as follows:
g : (a3r)0 7→ (a3r)0, (a3r)1 7→ (ba3r)0, (a3r+1)0 7→ (ba3r+1)1,
(a3r+1)1 7→ (a24ℓ+3r+1)1, (a3r+2)i 7→ (ba12ℓ+3r+2)i+1, (ba3r)0 7→ (a3r)1,
(ba3r)1 7→ (ba24ℓ+3r)1, (ba3r+1)0 7→ (ba3r+1)0, (ba3r+1)1 7→ (a3r+1)0,
(ba3r+2)i 7→ (a−12ℓ+3r+2)i+1,
where r ∈ Z16ℓ, i ∈ Z2.
It is easy to check that g is an involution, and furthermore, for any t ∈ Z16ℓ, we have
Γ((a3r)0)
g = {(a3r)1, (ba3r)0, (ba3r+1)0} = Γ((a3r)0),
Γ((a3r)1)
g = {(ba3r)1, (a3r)0, (a3r−1)0} = Γ((ba3r)0),
Γ((ba3r)1)
g = {(ba24ℓ+3r)0, (a3r)1, (a12ℓ+3r+1)1} = Γ((ba24ℓ+3r)1),
Γ((a3r+1)0)
g = {(ba3r+1)0, (a24ℓ+3r+1)1, (a36ℓ+3r+2)1} = Γ((ba3r+1)1),
Γ((a3r+1)1)
g = {(a24ℓ+3r+1)0, (ba3r+1)1, (ba36ℓ+3r)1} = Γ((a24ℓ+3r+1)1),
Γ((ba3r+1)0)
g = {(ba3r+1)1, (a3r+1)0, (a3r)0} = Γ((ba3r+1)0),
Γ((a3r+2)0)
g = {(ba12ℓ+3r+2)0, (a36ℓ+3r+2)1, (a3r+3)1} = Γ((ba12ℓ+3r+2)1),
Γ((a3r+2)1)
g = {(ba12ℓ+3r+2)1, (a12ℓ+3r+2)0, (a12ℓ+3r+1)0} = Γ((ba12ℓ+3r+2)0).
This implies that g is an automorphism of Γ. Observing that g maps 11 to b0, it follows that
⟨R(H), g⟩ is transitive on V (Γ), and so Γ is a vertex-transitive graph.
Below, we shall first prove the following claim.
Claim. Aut (Γ)10 = ⟨g⟩.
Let A = Aut (Γ). It is easy to see that g fixes 10, and so g ∈ A10 . To prove the Claim,
it suffices to prove that |A10 | = 2.
Note that the neighborhood Γ(10) of 10 in Γ is = {11, b0, (ba)0}. By a direct com-
putation, we find that in Γ there is a unique 8-cycle passing through 10, 11 and b0, that
is,
C0 = (10, 11, (ba24ℓ)1, (ba24ℓ)0, (a24ℓ)0, (a24ℓ)1, b1, b0, 10).
Furthermore, in Γ there is no 8-cycle passing through 10 and (ba)0. So A10 fixes (ba)0.
If A10 also fixes 11 and b0, then A10 will fix every neighbor of 10, and the connected-
ness and vertex-transitivity of Γ give that A10 = 1, a contradiction. Therefore, A10 swaps
11 and b0, and (ba)0 is the unique neighbor of 10 such that A10 = A(ba)0 . It follows that
{10, (ba)0} is a block of imprimitivity of A acting on V (Γ). Since Γ is vertex-transitive,
every v ∈ V (Γ) has a unique neighbor, say u such that Au = Av . Then the set
B = {{u, v} ∈ E(Γ) | Au = Av}
forms an A-invariant partition of V (Γ). Clearly, {10, (ba)0} ∈ B. Similarly, since C0
is also the unique 8-cycle of Γ passing through 10, 11 and b0, A11 swaps 10 and b0, and
186 Ars Math. Contemp. 21 (2021) #P2.02 / 175–200
(ba12ℓ−1)1 is the unique neighbor of 11 such that A11 = A(ba12ℓ−1)1 . So {11, (ba12ℓ−1)1} ∈
B. Set
B0 = {{10, (ba)0}R(h) | h ∈ H} and B1 = {{11, (ba12ℓ−1)1}R(h) | h ∈ H}.
Clearly, B = B0 ∪ B1.
Now we consider the quotient graph ΓB of Γ relative to B. It is easy to see that ⟨R(a)⟩
acts semiregularly on B with B0 and B1 as its two orbits. So ΓB is isomorphic to a bi-
Cayley graph over ⟨a⟩. Set B0 = {10, (ba)0} and B1 = {11, (ba12ℓ−1)1}. Then one
may see that the neighbors of B0 in ΓB are: B
R(a)
0 , B
R(a−1)
0 , B1, B
R(a−12ℓ+2)
1 , and the
neighbors of B1 in ΓB are: B
R(a12ℓ+1)
1 , B
R(a−12ℓ−1)
1 , B0, B
R(a12ℓ−2)
0 . So
ΓB ∼= Γ′ = BiCay(⟨a⟩, {a, a−1}, {a12ℓ+1, a−12ℓ−1}, {1, a−12ℓ+2}).
Observe that there is one and only one edge of Γ between B0 and any one of its neighbors in
ΓB. Clearly, A acts transitively on V (ΓB), so there is one and only one edge of Γ between
every two adjacent blocks of B. It follows that A acts faithfully on V (ΓB), and hence we
may view A as a subgroup of Aut (ΓB). Recall that g ∈ A10 = A(ba)0 . Moreover, g swaps
the two neighbors 11 and b0 of 10. Clearly, 11 ∈ B1 and b0 ∈ BR(a
−1)
0 , so g swaps the
two blocks B1 and B
R(a−1)
0 . Similarly, g swaps the two neighbors (ba)1 and a0 of (ba)0.
Clearly, (ba)1 ∈ BR(a
−12ℓ+2)
1 and a0 ∈ B
R(a)
0 , so g swaps the two blocks B
R(a−12ℓ+2)
1 and
B
R(a)
0 . Note that R(ab) swaps the two vertices in B0. So ⟨g,R(ab)⟩ acts transitively on the
neighborhood of B0 in ΓB. This implies that A acts transitively on the arcs of ΓB, and so
Γ′ is a tetravalent arc-transitive bi-circulant. In [11], a characterization of tetravalent edge-
transitive bi-circulants is given. It is easy to see that our graph Γ′ belongs to Class 1(c) of
[11, Theorem 1.1]. By checking [11, Theorem 4.1], we see that the stabilizer Aut (Γ′)u of
u ∈ V (Γ′) has order 4. This implies that |A| = 4|V (ΓB)| = 8n. Consequently, |A10 | = 2
and so our claim holds.
Now we are ready to finish the proof. Suppose to the contrary that Γ is a Cayley graph.
By Proposition 2.1, A contains a subgroup, say J acting regularly on V (Γ). By Claim,
J has index 2 in A, and since g ∈ A10 , one has A = J ⋊ ⟨g⟩. It is easy to check that
R(a),R(b) and g satisfy the following relations:
(gR(b))4 = R(a24ℓ), gR(a3) = R(a3)g, gR(ba) = R(ba)g, g = R(a)(gR(b))2R(a12ℓ−1).
Suppose that R(H) ≰ J . Then A = JR(H). Since |J |/|R(H)| = 2, it follows that
|R(H) : J ∩ R(H)| = 2. Thus, J ∩ R(H) = ⟨R(a)⟩ or ⟨R(a2),R(b)⟩. If R(H) ∩ J =
⟨R(a)⟩, then we have R(b) /∈ J , R(a) ∈ J , and hence A = J∪JR(b) = J∪Jg, implying
that JR(b) = Jg. It follows that gR(b) ∈ J , and then g = R(a)(gR(b))2R(a12ℓ−1) ∈ J
due to R(a) ∈ J , a contradiction. If R(H)∩J = ⟨R(a2),R(b)⟩, then R(a) /∈ J , and again
we have A = J∪JR(a) = J∪Jg, implying that JR(a) = Jg. So, R(a)g, gR(a−1) ∈ J .
Then
g = R(a)gR(b)gR(b)R(a12ℓ−1) = (R(a)g)R(b)(gR(a−1))R(ba12ℓ−2) ∈ J,
a contradiction.
Suppose that R(H) ≤ J . Then |J : R(H)| = 2 and R(H) ⊴ J . Since J is regular
on V (Γ), by Proposition 2.4, there exists a δα,x,y ∈ J such that 1
δα,x,y
0 = 11, where α ∈
Mi-Mi Zhang and Jin-Xin Zhou: Trivalent dihedrants and bi-dihedrants 187
Aut (H) and x, y ∈ H . By the definition of δα,x,y , we have 11 = 1
δα,x,y
0 = (x · 1α)1 = x1,
implying that x = 1. Furthermore, we have the following relations:
Rα = x−1Lx,Lα = y−1Ry, Sα = y−1S−1x,
where R = {b, ba}, L = {ba24ℓ, ba12ℓ−1}, S = {1}. In particular, the last equality implies
that x = y due to S = {1}. So we have x = y = 1. From the proof of Claim we know that
B0 = {10, (ba)0} and B1 = {11, (ba12ℓ−1)1} are two blocks of imprimitivity of A acting
on V (Γ). So we have ((ba)0)δα,1,1 = (ba12ℓ−1)1. It follows that (ba)α = ba12ℓ−1, and
then from Rα = L we obtain that bα = ba24ℓ. Consequently, we have aα = a36ℓ−1. One
the other hand, we have {b, ba} = R = Lα = {b, ba24ℓ+1}. This forces that ba = ba24ℓ+1,
which is clearly impossible.
6 Two families of trivalent Cayley bi-dihedrants
In this section, we shall prove two lemmas which will be used the proof of Theorem 1.3.
Lemma 6.1. Let H = ⟨a, b | a12m = b2 = 1, ab = a−1⟩ be a dihedral group with m odd.
Then for each i ∈ Z12m, Γ = BiCay(H, {b, bai}, {ba6m, ba3m−i}, {1}) is a Cayley graph
whenever ⟨ai, a3m⟩ = ⟨a⟩.
Proof. Let g be a permutation of V (Γ) defined as follows:
g : (a6km+3ri)j 7→ (ba6(k+1)m+3ri)j+1, (ba6km+3ri)j 7→ (a6km+3ri)j+1,
(a3km+(3r+1)i)0 7→ (a3(k+1)m+(3r+1)i)0, (ba3km+(3r+1)i)0 7→ (a3(k+1)m+(3r+1)i)1,
(a3km+(3r+1)i)1 7→ (ba3(k+1)m+(3r+1)i)0, (ba3km+(3r+1)i)1 7→ (ba3(k−1)m+(3r+1)i)1,
(a3km+(3r+2)i)0 7→ (ba3(k+1)m+(3r+2)i)1, (ba3km+(3r+2)i)0 7→ (ba3(k+1)m+(3r+2)i)0,
(a3km+(3r+2)i)1 7→ (a3(k−1)m+(3r+2)i)1, (ba3km+(3r+2)i)1 7→ (a3(k+1)m+(3r+2)i)0,
where r ∈ Zm, k ∈ Z4 and j ∈ Z2.
It is easy to check that g ∈ Aut (Γ). Furthermore, one may check that g and R(a2)
satisfy the following relations:
R(a12m) = g4 = 1, g2 = R(a6m), R(a6)g = gR(a6),
R(b−1)gR(b) = gR(a6m), R(a2)g = gR(a4)gR(a−2).
By the last equality, we have
(R(a2)g)3 = [gR(a4)gR(a−2))]R(a2)gR(a2)g = gR(a4)g2R(a2)g.
It then follows from the second and third equalities that
gR(a4)g2R(a2)g = gR(a6+6m)g = g2R(a6+6m) = R(a6).
Therefore, (R(a2)g)3 = R(a6).
Let G = ⟨R(a2),R(b), g⟩ and T = ⟨R(a6)⟩. Then T ⊴G and
G/T = ⟨R(a2)T,R(b)T, gT ⟩
= ⟨R(a2)T, gT | R(a2)3T = g2T = (R(a2)g)3T = T ⟩⋊ ⟨R(b)T ⟩
∼= A4 ⋊ Z2.
188 Ars Math. Contemp. 21 (2021) #P2.02 / 175–200
So |G| = 48m.
Let
Ω00 = {t0 | t ∈ ⟨a2, b⟩}, Ω01 = {t1 | t ∈ ⟨a2, b⟩},
Ω10 = {(at)0 | t ∈ ⟨a2, b⟩}, Ω11 = {(at)1 | t ∈ ⟨a2, b⟩}.
Then Ωij’s (0 ≤ i, j ≤ 1) are orbits of T and V (Γ) =
⋃
0≤i,j≤1
Ωij . Since 1
g
0 = (ba
6m)1 ∈
Ω01, a
g
0 = (a
3m+1)0 ∈ Ω00 and ag1 = (ba3m+1)1 ∈ Ω01, it follows that G is transitive,
and so regular on V (Γ). By Proposition 2.1, Γ is a Cayley graph on G, as required.
Lemma 6.2. Let H = ⟨a, b | a12m = b2 = 1, ab = a−1⟩ be a dihedral group with m even
and 4 ∤ m. Then the following two bi-Cayley graphs:
Γ1 = BiCay(H, {b, ba}, {ba6m, ba3m−1}, {1}),
Γ2 = BiCay(H, {b, ba}, {ba6m, ba9m−1}, {1})
are both Cayley graphs.
Proof. Let V = H0 ∪H1. Then V (Γ1) = V (Γ2) = V . We first define two permutations
on V as follows:
g1 : (a
4r)i 7→ (ba6m+4r)i+1, (ba4r)i 7→ (a4r)i+1,
(a4r+1)i 7→ (ba9m+4r+1)i+1, (ba4r+1)i 7→ (a3m+4r+1)i+1,
(a4r+2)i 7→ (ba4r+2)i+1, (ba4r+2)i 7→ (a6m+4r+2)i+1,
(a4r+3)i 7→ (ba3m+4r+3)i+1, (ba4r+3)i 7→ (a9m+4r+3)i+1,
g2 : (a
4r)i 7→ (ba6m+4r)i+1, (ba4r)i 7→ (a4r)i+1,
(a4r+1)i 7→ (ba3m+4r+1)i+1, (ba4r+1)i 7→ (a9m+4r+1)i+1,
(a4r+2)i 7→ (ba4r+2)i+1, (ba4r+2)i 7→ (a6m+4r+2)i+1,
(a4r+3)i 7→ (ba9m+4r+3)i+1, (ba4r+3)i 7→ (a3m+4r+3)i+1,
where r ∈ Z3m and i ∈ Z2.
It is easy to check that gj ∈ Aut (Γj) for j = 1 or 2. Furthermore, R(a2),R(b) and gj
(j = 1 or 2) satisfy the following relations:
R(a12m) = R(b2) = g4j = 1,R(b)R(a2)R(b) = R(a−2),
g2j = R(a6m),R(b)gjR(b) = g
−1
j ,
g−11 R(a)g1 = R(a3m+1), g
−1
2 R(a)g2 = R(a9m+1).
For j = 1 or 2, let Gj = ⟨R(a),R(b), gj⟩. From the above relations it is east to see
that
Gj = (⟨R(a)⟩⟨gj⟩)⋊ ⟨R(b)⟩
has order at most 48m. Observe that 1gj0 = (ba
6m)1 ∈ H1 for j = 1 or 2. It follows that
Gj is transitive on V (Γj), and so Gj acts regularly on V (Γj). By Proposition 2.1, each Γj
is a Cayley graph.
7 Vertex-transitive trivalent bi-dihedrants
In this section, we shall give a complete classification of trivalent vertex-transitive non-
Cayley bi-dihedrants. For convenience of the statement, throughout this section, we shall
make the following assumption.
Mi-Mi Zhang and Jin-Xin Zhou: Trivalent dihedrants and bi-dihedrants 189
Assumption I.
• H: the dihedral group D2n = ⟨a, b | an = b2 = 1, bab = a−1⟩(n ≥ 3),
• Γ = BiCay(H, R, L, {1}): a connected trivalent 2-type vertex-transitive bi-Cayley
graph over the group H (in this case, |R| = |L| = 2),
• G: a minimum group of automorphisms of Γ subject to that R(H) ≤ G and G is
transitive on the vertices but intransitive on the arcs of Γ.
The following lemma given in [20] shows that the group G must be solvable.
Lemma 7.1 ([20, Lemma 6.2]). G = R(H)P is solvable, where P is a Sylow 2-subgroup
of G.
7.1 H0 and H1 are blocks of imprimitivity of G
The case where H0 and H1 are blocks of imprimitivity of G has been considered in [20],
and the main result is the following proposition.
Proposition 7.2 ([20, Theorem 1.3]). If H0 and H1 are blocks of imprimitivity of G on
V (Γ), then either Γ is Cayley or one of the following occurs:
(1) (R,L, S) ≡ ({b, baℓ+1}, {ba, baℓ2+ℓ+1}, {1}), where n ≥ 5, ℓ3 + ℓ2 + ℓ + 1 ≡
0 (mod n), ℓ2 ̸≡ 1 (mod n);
(2) (R,L, S) ≡ ({ba−ℓ, baℓ}, {a, a−1}, {1}), where n = 2k and ℓ2 ≡ −1 (mod k).
Furthermore, Γ is also a bi-Cayley graph over an abelian group Zn × Z2.
Furthermore, all of the graphs arising from (1)-(2) are vertex-transitive non-Cayley.
In particular, it is proved in [20] that if n is odd and Γ is not a Cayley graph, then H0
and H1 are blocks of imprimitivity of G on V (Γ). Consequently, we can get a classification
of trivalent vertex-transitive non-Cayley bi-Cayley graphs over a dihedral group D2n with
n odd.
Proposition 7.3 ([20, Proposition 6.4]). If n is odd, then either Γ is a Cayley graph, or H0
and H1 are blocks of imprimitivity of G on V (Γ).
7.2 H0 and H1 are not blocks of imprimitivity of G
In this subsection, we shall consider the case where H0 and H1 are not blocks of imprimi-
tivity of G on V (Γ). We begin by citing a lemma from [20].
Lemma 7.4 ([20, Lemma 6.3]). Suppose that H0 and H1 are not blocks of imprimitivity
of G on V (Γ). Let N be a normal subgroup of G, and let K be the kernel of G acting on
V (ΓN ). Let ∆ be an orbit of N . If N fixes H0 setwise, then one of the following holds:
(1) Γ[∆] has valency 1, |V (ΓN )| ≥ 3 and Γ is a Cayley graph;
(2) Γ[∆] has valency 0, ΓN has valency 3, and K = N is semiregular.
The following lemma deals with the case where CoreG(R(H)) = 1, and in this case
we shall see that Γ is just the cross ladder graph.
190 Ars Math. Contemp. 21 (2021) #P2.02 / 175–200
Lemma 7.5. Suppose that H0 and H1 are not blocks of imprimitivity of G on V (Γ). If
CoreG(R(H)) =
⋂
g∈G R(H) = 1, then Γ is isomorphic to the cross ladder graph CL4n
with n odd, and furthermore, for any minimal normal subgroup N of G, we have the
following:
(1) N is a 2-group which is non-regular on V (Γ);
(2) N does not fix H0 setwise;
(3) every orbit of N consists of two non-adjacent vertices.
Proof. Let N be a minimal normal subgroup of G. By Lemma 7.1, G is solvable. It follows
that N is an elementary abelian r-subgroup for some prime divisor r of |G|. Clearly,
N ≰ R(H) due to CoreG(R(H)) = 1. Then |NR(H)|/|R(H)| | |G|/|R(H)|. From
Lemma 7.1 it follows that |G|/|R(H)| is a power of 2, and hence N is a 2-group.
Suppose that N is regular on V (Γ). Then NR(H) is transitive on V (Γ) and R(H)
is also a 2-group. Therefore, NR(H) is not transitive on the arcs of Γ. The minimality
of G gives that G = NR(H). Since n is even, R(an2 ) is in the center of R(H). Set
Q = N⟨R(an2 )⟩. Then Q⊴G and then 1 ̸= N ∩Z(Q)⊴G. Since N is a minimal normal
subgroup of G, one has N ≤ Z(Q), and hence Q is abelian. It follows that ⟨R(an2 )⟩⊴G,
contrary to the assumption that CoreG(R(H)) = 1. Thus, N is not regular on V (Γ). (1) is
proved.
For (2), by way of contradiction, suppose that N fixes H0 setwise. Consider the quo-
tient graph ΓN of Γ relative to N , and let K be the kernel of G acting on V (ΓN ). Take ∆
to be an orbit of N on V (Γ). Then either (1) or (2) of Lemma 7.4 happens.
For the former, Γ[∆] has valency 1 and |V (ΓN )| ≥ 3. Then ΓN is a cycle. Moreover,
any two neighbors of u ∈ ∆ are in different orbits of N . It follows that the stabilizer
Nv of v in N fixes every neighbor of u. The connectedness of Γ implies that Nv = 1.
Thus, K = N is semiregular and ΓN is a cycle of length ℓ = 2|R(H)|/|N |. So G/N ≤
Aut (ΓN ) ∼= D2ℓ. If G/N < Aut (ΓN ), then |G : N | = ℓ and so |G| = 2|R(H)|.
This implies that R(H) ⊴ G, contrary to the assumption that CoreG(R(H)) = 1. If
G/N = Aut (ΓN ), then |G : R(H)| = 4. Since N ̸≤ R(H) and since N fixes H0
setwise, one has |G : R(H)N | = 2. It follows that R(H)N ⊴G. Clearly, H0 and H1 are
just two orbits of R(H)N , and they are also two blocks of imprimitivity of G on V (Γ), a
contradiction.
For the latter, Γ[∆] has valency 0, ΓN has valency 3 and N = K is semiregular. Let
H̄i be the set of orbits of N contained in Hi with i = 1, 2. Then ΓN [H̄0] and ΓN [H̄1] are
of valency 2 and the edges between H̄0 and H̄1 form a perfect matching. Without loss of
generality, we may assume that 10 ∈ ∆. Since R(H) acts on H0 by right multiplication,
we have the subgroup of R(H) fixing ∆ setwise is just R(H)∆ = {R(h) | h0 ∈ ∆}. If
R(H)∆ ≤ ⟨R(a)⟩, then R(H)∆ ⊴ R(H), and the transitivity of R(H) on H0 implies
that R(H)∆ will fix all orbits of N contained in H0. Since the edges between H̄0 and
H̄1 are independent, R(H)∆ fixes all orbits of N . It follows that R(H)∆ ≤ N , namely,
R(H)N/N acts regularly on H̄0. Then |R(H)/(R(H)∩N)| = |R(H)N/N | = |H0/N |,
and so |N | = |R(H) ∩ N |, forcing N ≤ R(H), a contradiction. Thus, R(H)∆ ≰
⟨R(a)⟩, and so ⟨R(a)⟩R(H)∆ = R(H). This implies that ⟨R(a), N⟩/N is transitive and
so regular on H̄0. Similarly, ⟨R(a), N⟩/N is also regular on H̄1. Thus, ΓN is a trivalent
2-type bi-Cayley graph over ⟨R(a), N⟩/N . By [24, Lemma 5.3], H̄0 and H̄0 are blocks of
imprimitivity of G/N , and so H0 and H1 are blocks of imprimitivity of G, a contradiction.
Mi-Mi Zhang and Jin-Xin Zhou: Trivalent dihedrants and bi-dihedrants 191
So far, we have completed the proof of (2). Then N does not fix H0 setwise, and then
NR(H) is transitive on V (Γ). The minimality of G gives that G = NR(H). Let P and
P1 be Sylow 2-subgroups of G and R(H), respectively, such that P1 ≤ P . Then N ≤ P
and P = NP1.
If n is even, then by a similar argument to the second paragraph, a contradiction occurs.
Thus, n is odd. As H ∼= D2n, P1 ∼= Z2 and P1 is non-normal in R(H). So N∩R(H) = 1.
Clearly, |V (Γ)| = 4n. If N is semiregular on V (Γ), then N ∼= Z2 or Z2 × Z2, and then
|G| = |R(H)||N | = 2|R(H)| or 4|R(H)|. Since CoreG(R(H)) = 1, we must have
|G : R(H)| = 4 and G ≲ Sym(4). Since n is odd, one has n = 3 and H ∼= Sym(3). So
G ∼= Sym(4) and hence G10 ∼= Z2. Then all involutions of G(∼= Sym(4)) not contained
in N are conjugate. Take 1 ̸= g ∈ G10 . Then g is an involution which is not contained in
N because N is semiregular on V (Γ). Since R(H) ∩ N = 1, every involution in R(H)
would be conjugate to g. This is clearly impossible because R(H) is semiregular on V (Γ).
Thus, N is not semiregular on V (Γ). (3) is proved.
Since n is odd, we have |V (ΓN )| > 2. Since N is not semiregular on V (Γ), ΓN has
valency 2 and Γ[∆] has valency 0. This implies that the subgraph induced by any two
adjacent two orbits of N is either a union of several cycles or a perfect matching. Thus,
ΓN has even order. As Γ has order 4n with n odd, every orbit of N has length 2. It is easy
to see that Γ is isomorphic to the cross ladder graph CL4n.
The following is the main result of this section.
Theorem 7.6. Suppose that H0 and H1 are not blocks of imprimitivity of G on V (Γ). Then
Γ = BiCay(H,R,L, S) is vertex-transitive non-Cayley if and only if one of the followings
occurs:
(1) (R,L, S) ≡ ({b, ba}, {b, ba2m}, {1}), where n = 2(2m + 1), m ̸≡ 1 (mod 3), and
the corresponding graph is isomorphic the multi-cross ladder graph MCL4m,2;
(2) (R,L, S) ≡ ({b, ba}, {ba24ℓ, ba12ℓ−1}, {1}), where n = 48ℓ and ℓ ≥ 1.
Proof. The sufficiency can be obtained from Theorem 1.2 and Lemma 5.1. We shall prove
the necessity in the following subsection by a series of lemmas.
7.3 Proof of the necessity of Theorem 7.6
The purpose of this subsection is to prove the necessity of Theorem 7.6. Throughout this
subsection, we shall always assume that H0 and H1 are not blocks of imprimitivity of G on
V (Γ) and that Γ = BiCay(H,R,L, S) is vertex-transitive non-Cayley. In this subsection,
we shall always use the following notation.
Assumption II. Let N = CoreG(R(H)).
Our first lemma gives some properties of the group N .
Lemma 7.7. 1 < N < ⟨R(a)⟩, |⟨R(a)⟩ : N | = n/|N | is odd and the quotient graph ΓN
of Γ relative to N is isomorphic to the cross ladder graph CL4n/|N |.
Proof. If N = 1, then from Lemma 7.5 it follows that Γ ∼= CL4n which is a Cayley
graph by Theorem 1.1, a contradiction. Thus, N > 1. Since H0 and H1 are not blocks of
imprimitivity of G on V (Γ), one has N < R(H).
192 Ars Math. Contemp. 21 (2021) #P2.02 / 175–200
Consider the quotient graph ΓN . Clearly, N fixes H0 setwise. Recall that H0 and
H1 are not blocks of imprimitivity of G on V (Γ) and that Γ is non-Cayley. Applying
Lemma 7.4, we see that ΓN is a trivalent 2-type bi-Cayley graph over R(H)/N . This
implies that |R(H) : N | > 2, and since H is a dihedral group, one has N < ⟨R(a)⟩.
Again, by Lemma 7.4, R(H)/N acts semiregularly on V (ΓN ) with two orbits, H̄0
and H̄1, where H̄i is the set of orbits of N contained in Hi with i = 1, 0. Furthermore,
N is just the kernel of G acting on V (ΓN ) and N acts semiregularly on V (Γ). Then
G/N is also a minimal vertex-transitive automorphism group of ΓN containing R(H)/N .
If H̄0 and H̄1 are blocks of imprimitivity of G/N on V (ΓN ), then H0 and H1 will be
blocks of imprimitivity of G on V (Γ), which is impossible by our assumption. Thus, H̄0
and H̄1 are not blocks of imprimitivity of G/N on V (ΓN ). Since N = CoreG(R(H)),
CoreG/N (R(H)/N) is trivial. Then from Lemma 7.5 it follows that ΓN ∼= CL 4n|N| , where
n
|N | is odd.
Next, we introduce another notation which will be used in the proof.
Assumption III. Take M/N to be a minimal normal subgroup of G/N .
We shall first consider some basic properties of the quotient graph ΓM of Γ relative to
M .
Lemma 7.8. The quotient graph ΓM of Γ relative to M is a cycle of length n/|N |. Fur-
thermore, every orbit of M on V (Γ) is a union of an orbit of N on H0 and an orbit of N
on H1, and these two orbits of N are non-adjacent.
Proof. Applying Lemma 7.5 to ΓN and G/N , we obtain the following facts:
(a) M/N is an elementary abelian 2-group which is not regular on V (ΓN ),
(b) M/N does not fix H̄0 setwise,
(c) every orbit of M/N on V (ΓN ) consists of two non-adjacent vertices of ΓN .
From (b) and (c) it follows that every orbit of M on V (Γ) is just a union of an orbit of N on
H0 and an orbit of N on H1, and these two orbits are non-adjacent. Since every orbit of N
on V (Γ) is an independent subset of V (Γ), each orbit of M on V (Γ) is also an independent
subset.
Recall that ΓN ∼= CL4m where m = n|N | is odd. The quotient graph of ΓN relative to
M/N is just a cycle of length m, and so the quotient graph ΓM of Γ relative to M is also
a cycle of length m.
By Lemma 7.8, each orbit of M on V (Γ) is an independent subset. It follows that the
subgraph induced by any two adjacent orbits of M is either a perfect matching or a union
of several cycles. For convenience of the statement, the following notations will be used in
the remainder of the proof:
Assumption IV.
(1) Let ∆ and ∆′ be two adjacent orbits of M on V (Γ) such that Γ[∆ ∪∆′] is a union
of several cycles.
(2) Let ∆ = ∆0 ∪∆1 and ∆′ = ∆′0 ∪∆′1, where ∆0,∆′0 ⊆ H0 and ∆1,∆′1 ⊆ H1 are
four orbits of N on V (Γ).
Mi-Mi Zhang and Jin-Xin Zhou: Trivalent dihedrants and bi-dihedrants 193
(3) 10 ∈ ∆0.
Since Γ[∆] and Γ[∆′] are both null graphs and since Γ[∆ ∪ ∆′] is a union of several
cycles, we have the following easy observation.
Lemma 7.9. Γ[∆i ∪∆′j ] is a perfect matching for any 0 ≤ i, j ≤ 1.
The following lemma tells us the possibility of R (Recall that we assume that Γ =
BiCay(H,R,L, {1})).
Lemma 7.10. Up to graph isomorphism, we may assume that R = {b, bai} with i ∈
Zn \ {0} and that b0 ∈ ∆′0. Furthermore, we have
∆0 = {h0 | R(h) ∈ N},∆′0 = {(bh)0 | R(h) ∈ N},
∆′1 = {h1 | R(h) ∈ N},∆1 = {(bh)1 | R(h) ∈ N},
and 11 is adjacent to (bal)1 ∈ ∆1 for some R(al) ∈ N .
Proof. Recall that N is a proper subgroup of ⟨R(a)⟩ and that n/|N | is odd. Since n is
even by Proposition 7.3, it follows that N is of even order, and so the unique involution
R(an/2) of ⟨R(a)⟩ is contained in N . As 10 ∈ ∆0 and N ≤ ⟨R(a)⟩ acts on H0 by right
multiplication, one has ∆0 = {h0 | h ∈ N}. Since Γ[∆0] is an empty graph, one has
an/2 /∈ R. By Proposition 2.3 (1), we have ⟨R ∪ L⟩ = H , and since R and L are both
self-inverse, either R ⊆ b⟨a⟩ or L ⊆ b⟨a⟩. By Proposition 2.3 (4), we may assume that
R ⊆ b⟨a⟩.
Recall that Γ[∆i ∪∆′j ] is a perfect matching for any 0 ≤ i, j ≤ 1. Then 10 is adjacent
to r0 ∈ ∆′0 for some r ∈ R. Since R ⊆ b⟨a⟩ and Aut (H) is transitive on b⟨a⟩, by
Proposition 2.3 (3), we may assume that r = b. So 10 is adjacent to b0 ∈ ∆′0. Since
N ≤ ⟨R(a)⟩ acts on Hi with i = 0 or 1 by right multiplication, we see that the two orbits
∆0,∆
′
0 of N are just the form as given in the lemma. Since S = {1}, the edges between
H0 and H1 form a perfect matching. This enables us to obtain another two orbits ∆1,∆′1
of N which have the form as given in the lemma.
By Lemma 7.9, Γ[∆1∪∆′1] is a perfect matching. So we may assume that 11 is adjacent
to (bal)1 ∈ ∆1 for some R(al) ∈ N .
Now we shall introduce some new notations which will be used in the following.
Assumption V.
(1) Let T = ⟨R(al)⟩ be of order t, where al is given in the above lemma.
(2) Let
Ω0 = {(ai
n
t )0 | 0 ≤ i ≤ t− 1}, Ω1 = {(bai
n
t )1 | 0 ≤ i ≤ t− 1},
Ω′0 = {(bai
n
t )0 | 0 ≤ i ≤ t− 1}, Ω′1 = {(ai
n
t )1 | 0 ≤ i ≤ t− 1}.
(3) B = {BR(h) | h ∈ H}, where B = Ω0 ∪ Ω1.
(4) Let B′ = Ω′0 ∪ Ω′1. Then B′ = BR(b).
Lemma 7.11. The followings hold.
(1) T ≤ N .
194 Ars Math. Contemp. 21 (2021) #P2.02 / 175–200
(2) Ω0,Ω1,Ω′0,Ω
′
1 are four orbits of T .
(3) Γ[Ω0 ∪ Ω1 ∪ Ω′0 ∪ Ω′1] is a cycle of length 4t.
(4) B is a G-invariant partition of V (Γ).
Proof. By Lemma 7.10, we see that R(al) ∈ N , and so T ≤ N . (1) holds. Since T =
⟨R(al)⟩ is assumed to be of order t, one has T = ⟨R(an/t)⟩, and then one can obtain (2).
By the adjacency rule of bi-Cayley graph, we can obtain (3).
Set Ω = Ω0 ∪ Ω1 ∪ Ω′0 ∪ Ω′1 and B = Ω0 ∪ Ω1. By Lemma 7.8, Γ[∆] is a null
graph, and so B = ∆ ∩ Ω. Since Γ has valency 3, it follows that ∆ ∪ ∆′ is a block of
imprimitivity of G on V (Γ), and hence Ω is also a block of imprimitivity of G on V (Γ)
since Γ[Ω] is a component of Γ[∆ ∪∆′]. Since ∆ is also a block of imprimitivity of G on
V (Γ), B(= ∆∩Ω) is a block of imprimitivity of G on V (Γ). Then B = {BR(h) | h ∈ H}
is a G-invariant partition of V (Γ).
Lemma 7.12. T < N and the quotient graph ΓB of Γ relative to B is isomorphic to the
cross ladder graph CL 4n
2t
. Moreover, T is the kernel of G acting on B.
Proof. Let KB be the kernel of G acting on B. Clearly, T ≤ KB. Let B′ = Ω′0 ∪Ω′1. Then
B′ = BR(b) ∈ B. Let BR(h) ∈ B be adjacent to B and BR(h) ̸= B′.
Suppose that Γ[B ∪ BR(h)] is a perfect matching. Since G is transitive on B, ΓB is a
cycle of length 2nt . Clearly, G/KB is vertex-transitive but not edge-transitive on ΓB, so
G/KB ∼= D2n/t. If t = 1, then it is easy to see that Γ ∼= CL4n which is a Cayley graph by
Theorem 1.1, a contradiction. If t > 1, then since Γ[Ω] = Γ[B∪B′] is a cycle of length 4t,
KB acts faithfully on B, and so KB ≤ Aut (Γ[B ∪B′]) ∼= D8t. Since KB fixes B, one has
|KB| | 4t, implying that |G| = |KB| · 2nt | 8n. As |R(H)| = 2n and R(H) is non-normal
in G, one has |KB| = 4t due to T ≤ KB. In view of the fact that KB ≲ D8t, KB has a
characteristic cyclic subgroup, say J , of order 2t. Then we have J ⊴G because KB ⊴G.
Clearly, J is regular on B and J ∩N = T , so JR(H) is regular on V (Γ). It follows from
Proposition 2.1 that Γ is a Cayley graph, a contradiction.
Therefore, Γ[B∪BR(h)] is not a perfect matching. If N = T , then B = ∆ and B′ = ∆′
are orbits of M , and then Γ[B ∪BR(h)] will be a perfect matching, a contradiction. Thus,
N > T.
Now we are going to prove that ΓB ∼= CL n2t . Since B is adjacent to B
R(h), Ωi is
adjacent to ΩR(h)j for some i, j ∈ {0, 1}. Then because Ωi and Ω
R(h)
j are orbits of T ,
Γ[Ωi ∪ ΩR(h)j ] is a perfect matching. This implies that ΓB is of valency 3, and so KB is
intransitive on B. As every Bh ∈ B is a union of two orbits of T on V (Γ), KB fixes every
orbit of T . Since N is cyclic, the normality of N in G implies that T ⊴G. Clearly, Ω0 is
adjacent to three pair-wise different orbits of T , so the quotient graph ΓT of Γ relative to
T is of valency 3. Consequently, the kernel of G acting on V (ΓT ) is T . Then KB = T .
Now R(H)/T ∼= D2n/t is regular on B, and so ΓB is a Cayley graph over R(H)/T .
Furthermore, G/T is not arc-transitive on ΓB. Since R(H)/T is non-normal in G/T ,
ΓB is a non-normal Cayley graph over R(H)/T . If ΓB is arc-transitive, then by [13,
Theorem 1], either |Aut (ΓB)| = 3k|R(H)/T | with k ≤ 2, or ΓB has order 2 ·p with p = 3
or 7. For the former, since G/T is not arc-transitive on ΓB, one has |G/T : R(H)/T | ≤ 2,
implying R(H)⊴G, a contradiction. For the latter, we have 2nt = 6 or 14, implying
n
t = 3
or 7. It follows that T is a maximal subgroup of ⟨R(a)⟩, and so T = N , a contradiction.
Mi-Mi Zhang and Jin-Xin Zhou: Trivalent dihedrants and bi-dihedrants 195
Therefore, ΓB is not arc-transitive. Since R(H)/T is non-normal in G/T , by Theorem 1.1,
one has ΓB ∼= CL 4n
2t
, as required.
Proof of Theorem 7.6. By Lemma 7.12, we have ΓB ∼= CL 4n
2t
. By the definition of CL 4n
2t
,
we may partition the vertex set of ΓB in the following way:
V (ΓB) = V0 ∪ V1 ∪ · · ·V 2n
2t −2
∪ V 2n
2t −1
, where Vi = {B0i , B1i }, i ∈ Z 2n2t
and
E(ΓB) = {{Br2i, Br2i+1}, {Br2i+1, Bs2i+2} | i ∈ Z n2t , r, s ∈ Z2}.
Assume that B00 = B and B
0
1 = B
′. Recall that B = Ω0 ∪ Ω1 and B′ = Ω′0 ∪ Ω′1 =
BR(b). Moreover, Ω0,Ω1,Ω′0 and Ω
′
1 are four orbits of T . Then every B
j
i ∈ B is just a
union of two orbits of T . For convenience, we may let
Bji = Ω
j
i0 ∪ Ω
j
i1, i ∈ Z 2n2t , j ∈ Z2,
where Ωji0,Ω
j
i1 are two orbits of T . For B = B
0
0 , we let Ω0 = Ω
0
00 and Ω1 = Ω
0
01, and for
B′ = B01 , we let Ω
′
0 = Ω
0
10 and Ω
′
1 = Ω
0
11.
For convenience, in the remainder of the proof, we shall use C4t to denote a cycle of
length 4t, and we also call C4t a 4t-cycle. Recall that Γ[B ∪ B′] = Γ[B00 ∪ B01 ] ∼= C4t,
and that the edges between Ω00i(= Ωi) and Ω
0
1j(= Ω
′
j) form a perfect matching for all
i, j ∈ Z2. Since T ⊴ G, the quotient graph ΓT of Γ relative to T has valency 3. So the
edges between any two adjacent orbits of T form a perfect matching.
From the construction of ΓB, one may see that there exists g ∈ G such that {V0, V1}g =
{V2i, V2i+1} for each i ∈ Z n2t . So for each i ∈ Z n2t , r ∈ Z2, we may assume that Γ[B
r
2i ∪
Br2i+1]
∼= C4t, and Ωr(2i)s ∼ Ω
r
(2i+1)t for all s, t ∈ Z2. (Here Ω
r
(2i)s ∼ Ω
r
(2i+1)t means
that Ωr(2i)s and Ω
r
(2i+1)t are adjacent in ΓB.) Again, from the construction of ΓB, we may
assume that
Ω0(2i+2)0 ∼ Ω
0
(2i+1)0,Ω
0
(2i+2)1 ∼ Ω
1
(2i+1)0,Ω
1
(2i+2)1 ∼ Ω
1
(2i+1)0,Ω
1
(2i+2)1 ∼ Ω
0
(2i+1)0,
for each i ∈ Z n
2t
. We draw a local subgraph of ΓB in Figure 3. Observing that every
H
HHHH


HH
HHH





@
@
@@@
@
@@



◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦




























· · ·
· · ·
· · ·
· · ·
B0−1 B
0
0 B
0
1 B
0
2
B1−1 B
1
0 B
1
1 B
1
2
Ω000
Ω001
Ω010
Ω011
Ω100
Ω101
Ω110
Ω111
Ω1
(−1)0
Ω1
(−1)1
Ω0
(−1)0
Ω0
(−1)1
Ω120
Ω121
Ω020
Ω021
Figure 3: The sketch graph of ΓB
Vi = {B0i , B1i } with i ∈ Z 2n2t is a block of imprimitivity of G/KB acting on V (ΓB). So
every B0i ∪ B1i with i ∈ Z 2n2t is a block of imprimitivity of G acting on V (Γ). Let E be
the kernel of G acting on the block system Λ = {B0i ∪ B1i | i ∈ Z 2n2t }. Then G/E
∼= Dn
t
196 Ars Math. Contemp. 21 (2021) #P2.02 / 175–200
acts regularly on Λ. Clearly, R(H) is also transitive on Ω, so G/E = R(H)E/E. By
Lemma 7.12, T is a the kernel of G acting on B. So E/T is an elementary 2-group.
From R(H)/(R(H) ∩ E) ∼= Dn
t
it follows that R(H) ∩ E = ⟨R(a n2t )⟩ ∼= Z2t, and
so (R(H) ∩ E)/T is a normal subgroup of G/T of order 2. This implies that B1i =
(B0i )
R(a
n
2t ) for i ∈ Z 2n
2t
. We may further assume that Ω101 = (Ω
0
00)
R(a
n
2t ) ⊆ B10 . So
Ω000 ∪ Ω101 is just the orbit of ⟨R(a
n
2t )⟩ containing 10.
Observing that Ω010 ∼ Ω020 and the edges between them are of the form {g0, (baig)0}
with g0 ∈ Ω010, one has Ω020 = baiΩ010 = bai(Ω000)R(b) = (Ω000)R(a
−i). So Ω120 ⊆
(B10)
R(a−i).
Since B01 = B
′ = BR(b) = (B00)
R(b), one has B11 = (B
1
0)
R(b). Recall that 11 ∈
Ω011 = Ω
′
1 and 11 is adjacent to 10 ∈ Ω000 = Ω0 and (bal)1 ∈ Ω001 = Ω1. As we assume
that Ω011 ∼ Ω120, 11 is adjacent to some vertex in Ω120. So Ω120 ⊆ H1 and hence
Ω120 = (Ω
1
00)
R(a−i)
= (Ω001)
R(a
n
2t )R(a−i)
= (Ω001)
R(a
n
2t
−i)
= {(bak nt )1 | 0 ≤ k ≤ t− 1}R(a
n
2t
−i).
So we have the following claim.
Claim 1 L = {bal, bak nt + n2t−i} and R = {bai, b}, where |R(al)| = t, i ∈ Zn and
0 ≤ k ≤ t− 1.
Let G∗10 be the kernel of G10 acting on the neighborhood of 10 in Γ. Then G
∗
10 ≤ E10 .
Recall that for each i ∈ Z n
2t
, r ∈ Z2, Γ[Br2i ∪ Br2i+1] ∼= C4t and the edges between
B02i+1∪B12i+1 and B02i+2∪B12i+2 form a perfect matching. It follows that E acts faithfully
on each B0i ∪B1i . Clearly, G∗10 ≤ E10 , so G
∗
10 acts faithfully on each B
0
i ∪B1i .
Claim 2 If t > 2 then G∗10 = 1, and if t = 2 then G
∗
10 ≤ Z2 and 3 | n.
Assume that t ≥ 2. Since Γ[B00 ∪ B01 ] ∼= C4t, G∗10 fixes every vertex in B
0
0 , and so
fixes every vertex in Ω0(−1)0 since Ω
0
(−1)0 ∼ Ω
0
00 (see Figure 3). This implies that G
∗
10 fixes
Ω0(−1)1 setwise, and so fixes Ω
1
00 setwise since Ω
0
(−1)1 ∼ Ω
1
00. Consequently, G
∗
10 also
fixes Ω101 setwise. Similarly, by considering the edges between B
0
1 ∪B11 and B02 ∪B12 , we
see that G∗10 fixes both Ω
1
10 and Ω
1
11 setwise. Recall that the edges between Ω
1
0i and Ω
1
1j
form a perfect matching for i, j ∈ Z2. As Γ[B01 ∪ B11 ] ∼= C4t, G∗10 acts faithfully on Ω
1
00
(or Ω101), and so G
∗
10 ≤ Z2.
If t > 2, then since Γ[B0−2 ∪B0−1] ∼= C4t, G∗10 will fix every vertex in this cycle, and in
particular, G∗10 will fix every vertex in Ω
0
(−1)1. As Ω
0
(−1)1 ∼ Ω
1
00, G
∗
10 will fix every vertex
in Ω100. Since G
∗
10 acts faithfully on Ω
1
00, one has G
∗
10 = 1.
Let t = 2. We shall show that 3 | n. Then T = ⟨R(an2 )⟩. Recall that (R(H)∩E)/T is
a normal subgroup of G/T of order 2. Let M = R(H)∩E. Then M is a normal subgroup
of G of order 4. Since R(H) is dihedral, one has M = ⟨R(an4 )⟩. Let C = CG(M).
Then R(a) ∈ C and R(b) /∈ C. It follows that C is a proper subgroup of G. Since
G/E acts regularly on Λ, C10 fixes every element in Λ. Since C10 centralizes M , C10
fixes every vertex in the orbit Ω000 ∪ Ω101 of M containing 10. Clearly, C10 ≤ G10 , so
C10/(C10 ∩G∗10) ≤ Z2. As we have shown that G
∗
10 acts faithfully on Ω
1
01, it follows that
Mi-Mi Zhang and Jin-Xin Zhou: Trivalent dihedrants and bi-dihedrants 197
C10 ∩G∗10 = 1 since C10 fixes Ω
1
01 pointwise, and hence C10 ≤ Z2. On the other hand, as
G∗10 ≤ Z2, one has |G| | 4 · 4n = 16n. Since C < G and R(a) ∈ C, one has |C| = kn
with k | 8.
Suppose that 3 ∤ n. For any odd prime divisor p of n, let P be a Sylow p-subgroup
of ⟨R(a)⟩. Then P is also a Sylow p-subgroup of C. If P is not normal in C, then by
Sylow’s theorem, we have |C : NC(P )| = k′p + 1 | 8 for some integer k′. Since p ̸= 3,
one has p = 7 and k′ = 1. This implies that |C| = 8|NC(P )|, and so |C| = 8n due to
R(a) ∈ C and C < G. Since C10 ≤ Z2, one has |C : C10 | ≥ 4n, and so C is transitive
on V (Γ). Moreover, we have CC(P ) = NC(P ) = ⟨R(a)⟩. By Burnside theorem, C
has a normal subgroup M such that C = M ⋊ P . Then the quotient graph ΓM of Γ
relative to M would be a cycle of length |P |, and the subgraph induced by each orbit of M
is just a perfect matching. This implies that M is just the kernel of G acting on V (ΓM ).
Furthermore, C/M is a vertex-transitive subgroup of Aut (ΓM ). Since ΓM is a cycle, C/M
must contain a subgroup, say B/M acting regularly on V (ΓM ). Then B will be regular on
V (Γ), and so by Proposition 2.1, Γ is a Cayley graph, a contradiction. Therefore, P ⊴ C,
and since C ⊴G, one has P ⊴G, implying P ≤ N . By the arbitrariness of P , n/|N | must
be even, contrary to Lemma 7.7. Thus, 3 | n, as claimed.
The following claim shows that t = 1 or 2.
Claim 3 t ≤ 2.
By way of contradiction, suppose that t > 2. Let C = CG(T ). Then ⟨R(a)⟩ ≤ C and
R(H) ≰ C since |T | = t > 2. Clearly, C10 ≤ E10 . As C10 centralizes T , C10 will fixes
every vertex in Ω000 since Ω
0
00 is an orbit of T containing 10. Since Γ[B
0
0 ∪ B01 ] ∼= C4t,
C10 fixes every vertex in this 4t-cycle, and so C10 ≤ G∗10 = 1 (by Claim 2). Thus, C
acts semiregularly on V (Γ). If C = ⟨R(a)⟩, then by N/C-theorem, we have G/⟨R(a)⟩ =
G/C ≤ Aut (T ). Since T ≤ N ≤ ⟨R(a)⟩ is cyclic, Aut (T ) is abelian. It then follows
that R(H)/C ⊴ G/C, and hence R(H) ⊴ G, a contradiction. If C > ⟨R(a)⟩, then
|C| = 2n because Γ is non-Cayley. Since H0 and H1 are not blocks of imprimitivity of
G on V (Γ), C does not fix H0 setwise, and so R(H)C is transitive on V (Γ). Clearly,
R(H) ∩ C = ⟨R(a)⟩, so |R(H)C| = |R(H)||C|/|⟨R(a)⟩| = 4n. It follows that R(H)C
is regular on V (Γ), contradicting that Γ is non-Cayley.
By Claim 3, we only need to consider the following two cases:
Case 1 t = 1.
In this case, by Claim 1, we have R = {b, bai} and L = {b, ban2 −i}. For convenience,
we let n = 2ℓ. Then R = {b, bai} and L = {b, baℓ−i}.
By Proposition 2.3 (1), the connectedness of Γ implies that ⟨ai, aℓ⟩ = ⟨a⟩. Then either
(i, 2ℓ) = 1, or i = 2k with (k, 2ℓ) = 1 and ℓ is odd. Recall that H = ⟨a, b | a2ℓ = b2 =
1, bab = a−1⟩. For any λ ∈ Z∗2ℓ, let αλ be the automorphism of H induced by the map
aλ 7→ a, b 7→ b.
So if (i, 2ℓ) = 1, then we have
(R,L)αi = ({b, ba}, {b, baℓ−1}),
and if i = 2k with (k, 2ℓ) = 2 and ℓ is odd, then we have
(R,L)αk = ({b, ba2}, {b, baℓ−2}).
198 Ars Math. Contemp. 21 (2021) #P2.02 / 175–200
So by Proposition 2.3 (3), we have
(R,L, S) ≡ ({b, ba}, {b, baℓ−1}, {1}) or ({b, ba2}, {b, baℓ−2}, {1})(ℓ is odd).
Suppose that ℓ is even. Then (R,L, S) ≡ ({b, ba}, {b, baℓ−1}, {1}). Since ℓ is even,
one has (2ℓ, ℓ+1) = 1 and (ℓ+1)2 ≡ 1 (mod 2ℓ). Then it is easy to check that αℓ+1 is an
automorphism of H of order 2 that swaps {b, ba} and {b, baℓ−1}. By Proposition 2.4, we
have δαℓ+1,1,1 ∈ I , and then Γ ∼= BiCay(H, {b, ba}, {b, baℓ+1}, {1}) is a Cayley graph, a
contradiction.
Now we assume that n = 2ℓ with ℓ = 2m+ 1 for some integer m. Let
Γ1 = BiCay(H, {b, ba}, {b, ba2m}, {1}),Γ2 = BiCay(H, {b, ba2}, {b, ba2m−1}, {1}).
Direct calculation shows that (n, 2m − 1) = 1, and 2m(2m − 1) ≡ 2 (mod n). Then
the automorphism α2m−1 : a 7→ a2m−1, b 7→ b maps the pair of two subsets ({b, ba},
{b, ba2m}) to ({b, ba2m−1}, {b, ba2}). So, we have (R,L, S) ≡ ({b, ba}, {b, ba2m}, {1}).
By Lemma 4.1 and Theorem 1.2, Γ ∼= MCL(4m, 2) and Γ is non-Cayley if and only if
3 ∤ (2m + 1). Note that 3 ∤ (2m + 1) is equivalent to m ̸≡ 1 (mod 3). So we obtain the
first family of graphs in Theorem 7.6.
Case 2 t = 2.
In this case, by Claim 1, we have R = {b, bai} and L = {ban2 , ba 3n4 −i} or {ban2 ,
ba
n
4 −i}. We still use the following notation: For any λ ∈ Z∗2ℓ, let αλ be the automorphism
of H induced by the map
aλ 7→ a, b 7→ b.
Note that
({b, bai}, {ban2 , ba 3n4 −i})α−1 = ({b, ba−i}, {ban2 , ban4 −(−i)}).
By replacing −i by i, we may always assume that
(R,L) = ({b, bai}, {ban2 , ban4 −i}).
By Claim 2, we have 3 | n. So we may assume that n = 12m for some integer m.
Then we have
(R,L) = ({b, bai}, {ba6m, ba3m−i}).
Since Γ is connected, by Proposition 2.3, we have ⟨ai, a3m⟩ = ⟨a⟩. If m is odd, by
Lemma 6.1, Γ will be a Cayley graph which is impossible. Thus, m is even. It then follows
that ⟨ai⟩ ∩ ⟨a3m⟩ > 1 since ⟨ai, a3m⟩ = ⟨ai⟩⟨a3m⟩ = ⟨a⟩. Since ⟨a3m⟩ ∼= Z4, one has
|⟨ai⟩ ∩ ⟨a3m⟩| = 2 or 4. For the former, we would have |⟨ai⟩| = 6m, and since m is
even, one has 4 | |⟨ai⟩|, and hence a3m ∈ ⟨ai⟩, a contradiction. Thus, we have |⟨ai⟩ ∩
⟨a3m⟩| = 4, that is, ⟨ai⟩ = ⟨a⟩. So (i, 12m) = 1, and then αi ∈ Aut (H) which maps
({b, bai}, {ba6m, ba3m−i}) to ({b, ba}, {ba6m, ba3m−1}) or ({b, ba}, {ba6m, ba−3m−1}).
Then
(R,L, S) ≡ ({b, ba}, {ba6m, ba3m−1}, {1}) or ({b, ba}, {ba6m, ba−3m−1}), {1}.
If m ≡ 2 (mod 4), then by Lemma 6.2, we see that Γ will be a Cayley graph, a con-
tradiction. Thus, m ≡ 0 (mod 4). Clearly, (3m − 1, 12m) = 1, and hence the map
a 7→ a3m−1, b 7→ ba6m induces an automorphism, say β of H . It is easy to check that
({b, ba}, {ba6m, ba3m−1})β = ({ba6m, ba−3m−1}, {b, ba}).
Mi-Mi Zhang and Jin-Xin Zhou: Trivalent dihedrants and bi-dihedrants 199
Thus,
(R,L, S) ≡ ({b, ba}, {ba6m, ba3m−1}, {1}).
By Proposition 5.1, Γ is a non-Cayley graph. Let m = 4ℓ for some integer ℓ. Then n = 48ℓ
and then we get the second family of graphs in Theorem 7.6. This completes the proof of
Theorem 7.6.
7.4 Proof of Theorem 1.3
By [20, Theorem 1.2], if Γ is 0- or 1-type, then Γ is a Cayley graph. Let Γ be of 2-
type. Suppose that Γ is a non-Cayley graph. Let G ≤ Aut (Γ) be minimal subject to that
R(H) ≤ G and G is transitive on V (Γ). If Γ is arc-transitive or H0 and H1 are blocks
of imprimitivity of G on V (Γ), then by [20, Theorem 1.1] and Proposition 7.2, we obtain
the graphs in part (1)–(3) of Theorem 1.3. Otherwise, Γ is not arc-transitive and H0 and
H1 are not blocks of imprimitivity of G on V (Γ), by Theorem 7.6, we obtain the last two
families of graphs of Theorem 1.3.
ORCID iDs
Mi-Mi Zhang https://orcid.org/0000-0002-8719-5888
Jin-Xin Zhou https://orcid.org/0000-0002-8353-896X
References
[1] M. Alaeiyan, M. Ghasemi and G. R. Safakish, The normality of cubic Cayley graphs
for dihedral groups, Vietnam J. Math. 37 (2009), 43–48, http://www.math.ac.vn/
publications/vjm/VJM_37/43.htm.
[2] N. Biggs, Algebraic Graph Theory, Cambridge University Press, 1993.
[3] J. Bondy and U. Murty, Graph Theory with Applications, Elsevier Science Ltd/North-Holland,
1976.
[4] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language,
J. Symbolic Comput. 24 (1997), 235–265, doi:10.1006/jsco.1996.0125, computational algebra
and number theory (London, 1993).
[5] P. Cameron, J. Sheehan and P. Spiga, Semiregular automorphisms of vertex-transitive cubic
graphs, European J. Combin. 27 (2006), 924–930, doi:10.1016/j.ejc.2005.04.008.
[6] E. Dobson, A. Malnič, D. Marušič and L. A. Nowitz, Semiregular automorphisms of vertex-
transitive graphs of certain valencies, J. Combin. Theory Ser. B 97 (2007), 371–380, doi:10.
1016/j.jctb.2006.06.004.
[7] S. Du, A. Malnič and D. Marušič, Classification of 2-arc-transitive dihedrants, J. Combin.
Theory Ser. B 98 (2008), 1349–1372, doi:10.1016/j.jctb.2008.02.007.
[8] R. Frucht, J. E. Graver and M. E. Watkins, The groups of the generalized Petersen graphs, Proc.
Cambridge Philos. Soc. 70 (1971), 211–218, doi:10.1017/s0305004100049811.
[9] C. D. Godsil, On the full automorphism group of a graph, Combinatorica 1 (1981), 243–256,
doi:10.1007/bf02579330.
[10] C. D. Godsil, The automorphism groups of some cubic Cayley graphs, European J. Combin. 4
(1983), 25–32, doi:10.1016/s0195-6698(83)80005-5.
[11] I. Kovács, B. Kuzman, A. Malnič and S. Wilson, Characterization of edge-transitive 4-valent
bicirculants, J. Graph Theory 69 (2012), 441–463, doi:10.1002/jgt.20594.
200 Ars Math. Contemp. 21 (2021) #P2.02 / 175–200
[12] D. Marušič, On 2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003), 162–
196, doi:10.1016/s0095-8956(02)00033-3, dedicated to Crispin St. J. A. Nash-Williams.
[13] D. Marušič and T. Pisanski, Symmetries of hexagonal molecular graphs on the torus, Croat.
Chem. Acta 73 (2000), 969–981, https://hrcak.srce.hr/131972.
[14] R. Nedela and M. Škoviera, Which generalized Petersen graphs are Cayley graphs?, J. Graph
Theory 19 (1995), 1–11, doi:10.1002/jgt.3190190102.
[15] T. Pisanski, A classification of cubic bicirculants, Discrete Math. 307 (2007), 567–578, doi:
10.1016/j.disc.2005.09.053.
[16] P. Potočnik, P. Spiga and G. Verret, Cubic vertex-transitive graphs on up to 1280 vertices, J.
Symbolic Comput. 50 (2013), 465–477, doi:10.1016/j.jsc.2012.09.002.
[17] M. E. Watkins, A theorem on Tait colorings with an application to the generalized Petersen
graphs, J. Combinatorial Theory 6 (1969), 152–164.
[18] H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.
[19] M.-Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182
(1998), 309–319, doi:10.1016/s0012-365x(97)00152-0, graph theory (Lake Bled, 1995).
[20] M.-M. Zhang and J.-X. Zhou, Trivalent vertex-transitive bi-dihedrants, Discrete Math. 340
(2017), 1757–1772, doi:10.1016/j.disc.2017.03.017.
[21] W.-J. Zhang, Y.-Q. Feng and J.-X. Zhou, Cubic vertex-transitive non-Cayley graphs of order
12p, Sci. China Math. 61 (2018), 1153–1162, doi:10.1007/s11425-016-9095-8.
[22] C. Zhou and Y.-Q. Feng, Automorphism groups of connected cubic Cayley graphs of order 4p,
Algebra Colloq. 14 (2007), 351–359, doi:10.1142/s100538670700034x.
[23] J.-X. Zhou and Y.-Q. Feng, Cubic vertex-transitive non-Cayley graphs of order 8p, Electron. J.
Combin. 19 (2012), Paper 53, 13, doi:10.37236/2087.
[24] J.-X. Zhou and Y.-Q. Feng, Cubic bi-Cayley graphs over abelian groups, European J. Combin.
36 (2014), 679–693, doi:10.1016/j.ejc.2013.10.005.
[25] J.-X. Zhou and Y.-Q. Feng, The automorphisms of bi-Cayley graphs, J. Combin. Theory Ser. B
116 (2016), 504–532, doi:10.1016/j.jctb.2015.10.004.
[26] J.-X. Zhou and M. Ghasemi, Automorphisms of a family of cubic graphs, Algebra Colloq. 20
(2013), 495–506, doi:10.1142/s1005386713000461.
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P2.03 / 201–217
https://doi.org/10.26493/1855-3974.2260.c0e
(Also available at http://amc-journal.eu)
A generalization of balanced tableaux and
marriage problems with unique solutions
Brian Tianyao Chan *
KeyFi, First Floor, First St. Vincent Bank Ltd. Building James Street, Kingstown,
St. Vincent and the Grenadines
Received 22 February 2020, accepted 24 February 2021, published online 30 September 2021
Abstract
We consider families of finite sets that we call flagged and that have been character-
ized by Chang as being the families of sets that admit unique solutions to Hall’s mar-
riage problem and we consider generalizations of Edelman and Greene’s balanced tableaux
previously investigated by Viard. In this paper, we introduce a natural generalization of
Edelman and Greene’s balanced tableaux that involves families of sets that satisfy Hall’s
marriage condition and certain words in [m]n, then prove that flagged families can be char-
acterized by a strong existence condition relating to this generalization. As a consequence
of this characterization, we show that the arithmetic mean of the sizes of subclasses of such
generalized tableaux is given by a generalization of the hook-length formula.
Keywords: Balanced tableaux, Hall’s marriage condition, shelling.
Math. Subj. Class. (2020): 05A20, 05C70, 05E45
1 Introduction
Hall’s Marriage Theorem is a combinatorial theorem proved by Hall [11] that asserts that
a finite family of sets has a transversal if and only if this family satisfies the marriage con-
dition. This theorem is known to be equivalent to at least six other theorems which include
Dilworth’s Theorem, Menger’s Theorem, and the Max-Flow Min-Cut Theorem [20]. Hall
Jr. proved [10] that Hall’s Marriage Theorem also holds for arbitrary families of finite sets,
where by arbitrary we mean families of finite sets that do not necessarily have a finite num-
ber of members. Afterwards, Chang [3] noted how Hall Jr.’s work in [10] can be used to
characterize marriage problems with unique solutions.
*The author would like to thank Stephanie van Willigenburg for her guidance and advice during the develop-
ment of this paper. Furthermore, the author gratefully acknowledges the insightful advice and feedback from the
anonymous referees. The author was supported in part by the Natural Sciences and Engineering Research Council
of Canada [funding reference number PGSD2 - 519022 - 2018]. https://sites.google.com/view/btchan
E-mail address: bchan600@gmail.com (Brian Tianyao Chan)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
202 Ars Math. Contemp. 21 (2021) #P2.03 / 201–217
Standard skew tableaux are well-known and intensively studied in algebraic combina-
torics, for example [15, 18, 19, 21]. Moreover, another class of tableaux was introduced by
Edelman and Greene in [5, 4], where they defined balanced tableaux on partition shapes.
In investigating the number of maximal chains in the weak Bruhat order of the symmet-
ric group, Edelman and Greene proved [5, 4] that the number of balanced tableaux of a
given partition shape equals the number of standard Young tableaux of that shape. Since
then, connections to random sorting networks [1], the Lascoux-Schützenberger tree [16],
and a generalization of balanced tableaux pertaining to Schubert polynomials [7] have been
explored.
In this paper we consider a new perspective for marriage problems with unique so-
lutions by interpreting such objects as shapes for generalized tableaux. Specifically we
call the families of finite sets that admit marriage problems with unique solutions flagged
and give a new characterization of these families of sets in Theorem 3.10. In this char-
acterization, we generalize standard skew tableaux and Edelman and Greene’s balanced
tableaux to families with systems of distinct representatives, we generalize hook sets to
members of such families, and we generalize bijective fillings of tableaux to certain words
in [m]n. We then use our characterization of marriage problems with unique solutions to
show in Theorem 3.25 that the arithmetic mean of the sizes of subclasses of such gener-
alized tableaux is given by a generalization of the hook-length formula. The hook-length
formula was discovered by Frame, Robinson, and Thrall and they proved that it enumerates
the number of standard Young tableaux of a given partition shape [8]. The formula consists
of parameters known as hook-lengths. Subsequent to Frame, Robinson and Thrall’s work,
hook-lengths have been shown to be connected to many known properties of tableaux.
They are integral, for instance, in work by Edelman and Greene on balanced tableaux [4]
and in results established by Morales, Pak, and Panova [17, 18]. Properties of Edelman
and Greene’s balanced tableaux and related notions are of interest [6, 7]. Moreover, gen-
eralizations of balanced tableaux were investigated by Viard. In [24, 23], Viard proved
what is equivalent to the following which we state using the terminology in this paper.
If F is a flagged family, if t is a transversal of F , and if f is a configuration of t, then
there exists a permutation σ that satisfies f . Moreover, Viard proved [24] what is equiv-
alent to the following which we also state using the terminology in this paper. Let S
be a finite subset of N2 and let F be the family of hooks {H(i,j) : (i, j) ∈ S} where
H(i,j) = {(i, j)} ∪ {(i, j′) ∈ S : j′ > j} ∪ {(i′, j) ∈ S : i′ > i}. Furthermore, let t be
the transversal of F defined by t(H(i,j)) = (i, j) for all (i, j) ∈ S. Then the average value
of An,n(f) over all configurations f of t satisfying An,n(f) ≥ 1 is given by the hook-
length formula n!/
∏
(i,j)∈S h(i,j) where h(i,j) = |H(i,j)| for all (i, j) ∈ S. Afterwards,
we indicate how our generalization of standard skew tableaux and balanced tableaux can
be analysed using Naruse’s Formula for skew tableaux and how such an approach can be
extended to skew shifted shapes [9, 17, 19] and likely to certain d-complete posets [9, 19].
2 Preliminaries
Throughout this paper, let N denote the set of positive integers and for all n ∈ N, define
[n] = {1, 2, . . . , n}. For all X ′ ⊆ X , let the restriction of f to X ′, which we denote by
f |X′ , be the function g : X ′ → Y defined by g(r) = f(r) for all r ∈ X ′. For all m,n ∈ N,
say that a function f : [n] → [m] is order-preserving if for all 1 ≤ i ≤ j ≤ n, f(i) ≤
f(j). Lastly, we write examples of permutations using one-line notation. When describing
B. T. Chan: A generalization of balanced tableaux and marriage problems with . . . 203
families of sets, call F ∈ F a member of F . We treat families of sets as multisets, so the
members of F are counted with multiplicity. That is, |F| = |I| if F = {Fi : i ∈ I}.
An illustrative class of examples that we use in this paper will come from skew shapes.
Hence, we recall them below and describe the notation we will use. A partition λ is a
weakly decreasing sequence of positive integers. We write λ = (λ1, λ2, . . . , λℓ) to denote
such a partition, where λi ∈ N for all 1 ≤ i ≤ ℓ. If λ is a partition, then we will also rep-
resent it as a Young diagram, which we also denote by λ. Specifically, the Young diagram
of λ = (λ1, λ2, . . . , λℓ) is a subset of N2 defined by
ℓ⋃
i=1
{(i, j) : 1 ≤ j ≤ λi}.
Moreover, if λ and µ are Young diagrams such that µ ⊂ λ, then define a skew shape λ/µ
to be the set λ\µ. We also consider a Young diagram λ as the skew shape λ/µ where µ
is the empty partition. We use the English convention for depicting Young diagrams and
skew shapes. In order to follow this convention, we call the elements of λ/µ the cells of
λ/µ, the non-empty subsets of the form {(i′, j′) ∈ λ/µ : i′ = i} the rows of λ/µ, and the
non-empty subsets of the form {(i′, j′) ∈ λ/µ : j′ = j} the columns of λ/µ.
3 Flagged families of sets and words in [m]n
We investigate families of sets that satisfy Hall’s marriage condition and generalizations of
Edelman and Greene’s balanced tableaux by proving relationships between these classes of
structures. In Section 3.1, we introduce marriage problems with unique solutions as flagged
families and generalizations of balanced tableaux, then we give a new characterization of
marriage problems with unique solutions in terms of these tableaux. In Section 3.2, we
explain how our results relate to tableaux on skew shapes. Lastly, in Section 3.3, we show
that the arithmetic mean of the sizes of subclasses of the above generalized tableaux is
given by a generalization of the hook-length formula.
3.1 A new characterization
A well-known notion for families of sets is the following.
Definition 3.1 (Folklore [14]). Let n ∈ N, and let F be a finite family of subsets of [n].
Then a transversal of F is an injective function t : F → [n] such that t(F ) ∈ F for all
F ∈ F . The set {t(F ) : F ∈ F} is called a system of distinct representatives of F .
Families of sets that have transversals are of great interest. Exemplary of this is Hall’s
Marriage Theorem, which we present below.
Definition 3.2 (Marriage condition, Hall [11]). Let n ∈ N, and let F be a finite family of
subsets of [n]. Then F satisfies the marriage condition if for all subfamilies F ′ of F ,
|F ′| ≤
∣∣∣∣ ⋃
F∈F ′
F
∣∣∣∣.
Theorem 3.3 (Marriage Theorem, Hall [11]). Let n ∈ N, and let F be a family of non-
empty subsets of [n]. Then F has a transversal if and only if F satisfies the marriage
condition.
204 Ars Math. Contemp. 21 (2021) #P2.03 / 201–217
In order to meaningfully use the families of sets in Hall’s Marriage Theorem, we will
define more structure on them.
Definition 3.4. Let n ∈ N, let F be a family of non-empty subsets of [n], and let t be a
transversal of F . Then a configuration f of t is a function f : [n] → N such that for all
F ∈ F ,
f(t(F )) ≤ |F |.
Moreover, for m ∈ [n], a surjective map σ : [n] → [m] satisfies f if for all F ∈ F , the
positive integer σ(t(F )) is the kth smallest element of the set σ(F ), where k = f(t(F )).
Example 3.5. Let F = {F1, F2} be a family of sets on [2] where F1 = [2] and F2 = [2].
The injective function t : F → [2] defined by t(F1) = 1 and t(F2) = 2 is a transversal of
F . Consider three configurations f ′, f ′′, and f ′′′ of t defined by f ′(1) = 1 and f ′(2) = 1,
f ′′(1) = 1 and f ′′(2) = 2, and f ′′′(1) = 2 and f ′′′(2) = 2.
Note σ : [2] → [1] satisfies f ′ because σ(1) = 1 is the smallest element of σ(F1) =
σ([2]) = [1] and because σ(2) = 1 is the smallest element of σ(F2) = σ([2]) = [1]. How-
ever, no permutation σ : [2] → [2] can satisfy f ′. It can also be checked that the surjective
map σ : [2] → [1] and the permutation σ = 21 do not satisfy f ′′ but the permutation
σ = 12 satisfies f ′′. Moreover, for all m ∈ [2] and for all surjective maps σ : [2] → [m], σ
does not satisfy f ′′′.
Now, we define the following stronger form of the marriage condition.
Definition 3.6 (cf. [3]). Let n ∈ N, let F be a finite family of subsets of [n], and write
m = |F|. Then F is flagged if there exists a bijection σF : [m] → F such that for all
k ∈ [m], ∣∣∣∣ k⋃
i=1
σF (i)
∣∣∣∣ = k. (3.1)
Informally, σF maps each k to a subset, such that the union of the first k subsets has
cardinality k.
In [3], Chang noted the following as a simple consequence of Hall Jr.’s work ([10],
Theorem 2).
Proposition 3.7 (Chang [3]). If n ∈ N, then a finite family F of subsets of [n] has exactly
one transversal if and only if F is flagged.
In particular, by Theorem 3.3, all flagged families satisfy the marriage condition. The
families of sets F in Proposition 3.7 are referred to as marriage problems with unique
solutions [13, 12].
Remark 3.8. When describing a flagged family F , we will use total orderings on the
members of this family by fixing orderings F1, F2, . . . , Fn of the members of F that
satisfy
F = {Fi : 1 ≤ i ≤ n},
and, for all 1 ≤ k ≤ n, ∣∣∣∣ k⋃
i=1
Fi
∣∣∣∣ = k.
B. T. Chan: A generalization of balanced tableaux and marriage problems with . . . 205
We observe that no flagged family is a multi-set. Let F be a flagged family and fix an
ordering F1, F2, · · · , Fn of the members of F as described in Remark 3.8. Suppose that
Fj = Fj′ for some j < j′. Then,
j⋃
i=1
Fi =
j′⋃
i=1
Fi,
contradicting Equation (3.1) of Definition 3.6.
Before proving the main result of this paper, we prove the following lemma.
Lemma 3.9. Let F be a flagged family of subsets of [n]. Moreover, let S ⊆ [n] be the set
of elements k ∈ [n] such that k ∈ F for exactly one member F of F . Then S is not empty.
Proof. Let m = |F|. Because F is flagged, Definition 3.6 and Equation (3.1) imply that
there exists a bijection σF : [m] → F and an element k ∈ [n] such that
m−1⋃
i=1
σF (i) =
( m⋃
i=1
σF (i)
)
\{k}.
So as k ∈ σF (m) and as, for all 1 ≤ i < m, k /∈ σF (i), it follows that k ∈ S and that S is
non-empty.
Now, we prove the main result of this paper.
Theorem 3.10. Let n ∈ N, let F be a family of subsets of [n] such that |F| = n, assume
that F satisfies the marriage condition, and let t be a transversal of F . Moreover, let
S ⊆ [n] be the set of elements k ∈ [n] such that k ∈ F for exactly one member F of F .
Lastly, let m be an integer satisfying
min(n, n− |S|+ 1) ≤ m ≤ n.
Then F is flagged if and only if for all configurations f of t, there exists a surjective map
σ : [n] → [m] such that σ satisfies f .
Proof. Let n, F , t, S, and m be as described in the theorem. First assume that for all
configurations f of t, there exists a surjective map σ : [n] → [m] that satisfies f . If n = 1,
then the only family of {1} with a transversal is the family F = {{1}}, which is flagged.
So assume without loss of generality that n ≥ 2. Consider the configuration f1 of t
defined by f1(t(F )) = |F | for all F ∈ F . By assumption, there exists a surjective map
σ′ : [n] → [m] that satisfies f1. Moreover, let k ∈ [n − 1], and assume that we can fix an
ordering F = {F ′i : i ∈ [n]} of F so that the following holds for all integers 0 ≤ j ≤ k−1.∣∣∣∣ n−j⋃
i=1
F ′i
∣∣∣∣ = n− j (3.2)
Note that Equation (3.2) holds if k = 1 because the fact that F has a transversal implies
that
⋃
F∈F F = [n].
Next, let 1 ≤ s ≤ n− k + 1 satisfy
σ′(t(F ′s)) = max
1≤j≤n−k+1
σ′(t(F ′j)). (3.3)
206 Ars Math. Contemp. 21 (2021) #P2.03 / 201–217
Suppose that there exists an element j ∈ [n] such that 1 ≤ j ≤ n − k + 1, j ̸= s,
and t(F ′s) ∈ F ′j . By Equation (3.3), σ′(t(F ′j)) ≤ σ′(t(F ′s)). So as t(F ′s) ∈ F ′j and
t(F ′s) ̸= t(F ′j), it follows that for some 1 ≤ ℓ ≤ |F ′j | − 1, σ′(t(F ′j)) is an ℓth smallest
element of σ′(F ′j). But then, as f1(t(F
′
j)) = |F ′j |, σ′ does not satisfy f1, contradicting the
assumption that σ′ satisfies f1.
Hence, t(F ′s) /∈ F ′i for all 1 ≤ i ≤ n − k + 1 satisfying i ̸= s. In particular, fix an
ordering F = {F ′′i : i ∈ [n]} of the members of F so that F ′′i = F ′i if i > n− k + 1 and
F ′′n−k+1 = F
′
s, where s is as described in the above paragraph. By Equation (3.2) and the
fact that t(F ′s) /∈ F ′i for all 1 ≤ i ≤ n− k+1 satisfying i ̸= s, it follows that this ordering
of the members of F satisfies the following equation for all integers 0 ≤ j ≤ k.∣∣∣∣ n−j⋃
i=1
F ′′i
∣∣∣∣ = n− j
As the choice of k ∈ [n − 1] is arbitrary, it follows that there exists an ordering F =
{F1, F2, . . . , Fn} of F such that ∣∣∣∣ k⋃
i=1
Fi
∣∣∣∣ = k
for all 1 ≤ k ≤ n. Hence, F satisfies Equation (3.1) of Definition 3.6. So, by Defini-
tion 3.6, F is flagged.
Next, assume that F is flagged. We proceed by strong induction on n. Because F is
flagged, we will use the total orderings as described in Remark 3.8 to describe the members
of this family.
If n = 1, then the only family of subsets of {1} with a transversal is the family F =
{{1}}. Moreover, with t being the transversal of F defined by mapping {1} to 1, the
only configuration f of t is the function f : {1} → N defined by f(1) = 1, S = {1},
min(n, n− |S|+ 1) = 1, and the surjective map σ : {1} → {1} satisfies f .
So assume that n ≥ 2 and let f be a configuration of t. Since S is not empty by
Lemma 3.9, min(n, n − |S| + 1) = n − |S| + 1, implying that n − |S| + 1 ≤ m ≤ n.
Assume without loss of generality that
S = {n−m′ + 1, n−m′ + 2, . . . , n} (3.4)
for some m′ ∈ [n]. If m = 1, then n − |S| + 1 ≤ 1, implying that n = |S|. Hence, as
|F| = n and S = [n], every element of [n] is contained in exactly one element of F , that is
F = {{k} : k ∈ [n]}. So in this case, t({k}) = k for all k ∈ [n], the only configuration f
of t is the map defined by f(k) = 1 for all k ∈ [n], and the surjective map σ : [n] → [m],
defined by σ(k) = 1 for all k ∈ [n], satisfies f . So assume without loss of generality that
m ≥ 2.
Since n− |S|+ 1 ≤ m ≤ n, m satisfies the inequality n−m′ + 1 ≤ m ≤ n. As F is
flagged, there is an ordering F ′1, F
′
2, . . . , F
′
n of the members of F such that∣∣∣∣ k⋃
i=1
F ′i
∣∣∣∣ = k (3.5)
for all 1 ≤ k ≤ n. Define the following subfamilies of F ,
F0 = {F ∈ F : t(F ) ≤ m− 1}
B. T. Chan: A generalization of balanced tableaux and marriage problems with . . . 207
and
F1 = {F ∈ F : t(F ) ≥ m}.
We first prove that F0 is flagged. Because S is the set of elements k ∈ [n] such that
k ∈ F for exactly one member F of F , Equation (3.4) and the fact that n−m′+1 ≤ m ≤ n
implies that for all m ≤ k ≤ n, k is contained in exactly one member of F and that for all
F ∈ F1,
|F ∩ {m,m+ 1, . . . , n}| = 1.
In particular, F0 is an (m− 1)-member family of subsets of [m− 1].
Assume that there exists an integer 1 ≤ j ≤ n− 1 such that F ′j ∈ F1 and F ′j+1 ∈ F0.
Write
Xj =
j−1⋃
i=1
F ′i ,
where we assume that Xj = ∅ if j = 1. Since F ′j ∈ F1, t(F ′j) ∈ {m,m + 1, . . . , n} and
no member of F other than F ′j contains t(F ′j). Moreover, by Equation (3.5), |F ′j ∪Xj | =
|Xj | + 1. So as t(F ′j) ∈ F ′j , it follows that [m − 1] ∩ (F ′j ∪Xj) = [m − 1] ∩Xj . Since
F ′j+1 ∈ F0, F ′j+1 ⊆ [m−1]. Moreover, by Equation (3.5), |Xj∪F ′j∪F ′j+1| = |Xj∪F ′j |+1.
It follows that F ′j+1\Xj = F ′j+1\(Xj ∪ F ′j) = {k} for some k ∈ [m− 1], implying that
|F ′j+1 ∪Xj | = |Xj |+ 1. (3.6)
So the ordering F = {F ′′1 , F ′′2 , . . . , F ′′n } of the members of F , such that F ′′j = F ′j+1,
F ′′j+1 = F
′
j , and F
′′
i = F
′
i for all i ∈ [n]\{j, j+1}, satisfies the following by Equation (3.5)
and Equation (3.6). For all 1 ≤ k ≤ n,∣∣∣∣ k⋃
i=1
F ′′i
∣∣∣∣ = k. (3.7)
Furthermore, F ′′j ∈ F0 and F ′′j+1 ∈ F1. If there exists an integer 1 ≤ j′ ≤ n− 1 such that
F ′′j′ ∈ F1 and F ′′j′+1 ∈ F0, then argue again as above. Repeating this argument at most a
finite number of times, we obtain an ordering F = {F1, F2, . . . , Fn} of the members of F
where ∣∣∣∣ k⋃
i=1
Fi
∣∣∣∣ = k (3.8)
for all 1 ≤ k ≤ n, F0 = {Fk : 1 ≤ k ≤ m − 1}, and F1 = {Fk : m ≤ k ≤ n}.
In particular, Equation (3.8) holds for all 1 ≤ k ≤ m − 1, implying that F0 satisfies
Equation (3.1) of Definition 3.6. It follows, by Definition 3.6, that F0 is a flagged family
of subsets of [m− 1].
So consider the ordering F1, F2, . . . , Fn of the members of F as above and assume
without loss of generality that for all m ≤ i ≤ n, t(Fi) = i. Let t′ be the transversal of
F0 defined by t′(F ) = t(F ) for all F ∈ F0. Moreover, let f ′ = f |[m−1], where f |[m−1]
denotes the restriction of f to [m− 1]. In particular, f ′ is a configuration of t′.
Since it is assumed in the theorem that min(n, n − |S| + 1) ≤ m ≤ n, and since a
surjective map σ : [n] → [m] is a permutation if m = n, the following holds. Because F0
is flagged, because |F0| = m − 1, and because |[m − 1]| < n, the induction hypothesis
implies that there exists a permutation σ′ : [m− 1] → [m− 1] such that σ′ satisfies f ′.
208 Ars Math. Contemp. 21 (2021) #P2.03 / 201–217
If there exists an integer m ≤ j ≤ n such that f(j) = |Fj |, then there exists a surjective
map κ′ : [n] → [m] such that κ′(i) = i for all 1 ≤ i ≤ m − 1 and the following two
properties hold for all m ≤ i ≤ n.
• If f(i) = |Fi|, then κ′(i) = m.
• If f(i) < |Fi|, then κ′(i) is equal to the f(i)th smallest element of σ′(Fi\{i}).
Otherwise, if f(i) < |Fi| for all m ≤ i ≤ n, the following holds. Write σ′(Fn\{n}) =
{r1, r2, · · · , rt} where t = |Fn| − 1 and r1 < r2 < · · · < rt. Since f(n) < |Fn|, there
exists a map κ∗ : [m − 1] → [m] such that κ∗ is injective and order-preserving and such
that, with x ∈ [m]\κ∗([m−1]), x < κ∗(r1) if f(n) = 1 and κ∗(rf(n)−1) < x < κ∗(rf(n))
if f(n) ≥ 2. So there exists a surjective map κ′′ : [n] → [m] such that κ′′|[m−1] = κ∗ and
such that the following two properties hold.
• If m ≤ i < n, then κ′′(i) is equal to the f(i)th smallest element of κ′′(σ′(Fi\{i})).
• If i = n, then κ′′(i) /∈ κ′′([m− 1]) and κ′′(i) is equal to the f(i)th smallest element
of κ′′(i) ∪ κ′′(σ′(Fi\{i})).
We note that κ′′|[m−1] is injective and order-preserving because κ∗ is injective and
order-preserving.
So define a surjective map κ : [n] → [m] as follows. If there exists an integer m ≤ j ≤
n such that f(j) = |Fj |, then define κ = κ′. Otherwise, if f(i) < |Fi| for all m ≤ i ≤ n,
define κ = κ′′. Now, define the map σ : [n] → [m] by
σ(i) =
{
κ(σ′(i)) if 1 ≤ i ≤ m− 1
κ(i) if m ≤ i ≤ n.
Because σ′ : [m−1] → [m−1] is a bijection, the definition of κ implies that σ is surjective.
Moreover, because σ′ satisfies f ′ and because, for all integers m ≤ i ≤ n, i is contained in
exactly one member of F and Fi ∩ {m,m + 1, . . . , n} = {i}, the definition of κ and the
definition of σ imply that σ satisfies f . From this, the theorem follows.
A natural consequence of the above is the following which, combined with Theo-
rem 3.10, gives another characterization of flagged families of sets.
Corollary 3.11. Let F be a family of subsets of [n] such that |F| = n, assume that F
satisfies the marriage condition, and let t be a transversal of F . Moreover, let S be as
in Theorem 3.10. Lastly, let f0 be the configuration of t defined by f0(t(F )) = 1 for all
F ∈ F . Then f0 is satisfied by some permutation σ : [n] → [n] if and only if for all
integers n − |S| + 1 ≤ m ≤ n and for all configurations f of t, there exists a surjective
map σ : [n] → [m] that satisfies f .
Proof. Let f1 be the configuration of t defined by f1(t(F )) = |F | for all F ∈ F . Then
a permutation σ : [n] → [n] satisfies f0 if and only if the permutation σ′ : [n] → [n]
defined by σ′(i) = n− σ(i) + 1 for all i ∈ [n] satisfies f1. In particular, f0 is satisfied by
some permutation if and only if f1 is. The first half of the proof of Theorem 3.10 implies
that if f1 is satisfied by some permutation σ : [n] → [n], then F is flagged. Furthermore,
by Theorem 3.10, if F is flagged, then for all integers n − |S| + 1 ≤ m ≤ n and for all
configurations f of t, there exists a surjective map σ : [n] → [m] that satisfies f . From
this, the corollary follows.
B. T. Chan: A generalization of balanced tableaux and marriage problems with . . . 209
Remark 3.12. A family F of subsets of [n] such that |
⋃
F∈F F | = |F| = n is called
a critical block in [10] by Hall Jr.. He used this notion as a very important ingredient in
extending Hall’s Marriage Theorem to infinite families of finite sets.
3.2 The case of skew shapes
To explain how the results in the previous subsection relate to skew shapes, we will need
the following definitions.
Definition 3.13. Let λ/µ be a skew shape with n cells, and let 1 ≤ m ≤ n be an integer.
Then a surjective tableau of shape λ/µ is a surjective function T : λ/µ → [m] and ele-
ments in the range of T are called the entries in T . In the case m = n a surjective tableau is
a bijective tableau. Moreover, a standard skew tableau of shape λ/µ is a bijective tableau
of shape λ/µ such that the entries along every row increase from left to right and the entries
along every column increase from top to bottom.
Example 3.14. The tableaux
T1 =
3 5
6 1 2
4
, T2 =
2 3
1 5 6
4
, and T3 = 2 3
3 1 2
2
have shape (4, 3, 1)/(2). Here, T1 and T2 are bijective and T2 is standard. All three are
surjective. Moreover, for T1 and T2, m = 6 and for T3, m = 3.
In order to fully relate Definition 3.13 to Definition 3.4, we will use the following
standard definitions.
Definition 3.15. Let λ/µ be a skew shape. For any cell (i, j) ∈ λ/µ, define the corre-
sponding hook H(i,j) to be
H(i,j) = {(i, j)} ∪A(i,j) ∪ L(i,j),
where
A(i,j) = {(i, j′) ∈ λ/µ : j′ > j}
is the arm of H(i,j) and where
L(i,j) = {(i′, j) ∈ λ/µ : i′ > i}
is the leg of H(i,j). Define the corresponding hook-length h(i,j) to be
h(i,j) = |H(i,j)|.
Example 3.16. Consider the following skew shape λ/µ, where λ = (5, 4, 3, 3) and µ =
(2, 2, 1). Then r = (2, 3) is the cell of λ/µ depicted below that is filled with a bullet. The
entries of Hr are filled with asterisks, bullets or circles, so hr = 4. Moreover, the entry of
Ar is filled with an asterisk and the entries of Lr are filled with circles.
• ∗
◦
◦
210 Ars Math. Contemp. 21 (2021) #P2.03 / 201–217
Definition 3.17. Let λ/µ be a skew shape. Then define Fλ/µ to be the set
{Hr : r ∈ λ/µ}.
Example 3.18. If λ/µ is the skew shape depicted below
,
then, as λ/µ = {(1, 2), (1, 3), (2, 2), (2, 3)},
Fλ/µ = {{(1, 2), (2, 2), (1, 3)}, {(1, 3)}, {(2, 1), (2, 2)}, {(2, 2)}}.
Many families of sets that satisfy the marriage condition are not flagged. For instance,
the family F = {F1, F2}, where F1 = F2 = {1, 2}, satisfies the marriage condition but is
not flagged. However, Definition 3.6 is a very broad definition. Let λ be a Young diagram.
Then an inner corner of λ is a cell r ∈ λ such that deleting r from λ results in another
Young diagram. For instance, if λ = (4, 2, 2), then the inner corners of λ are the cells filled
with bullets.
•
•
With this definition in mind, let λ/µ be a skew shape with n cells, and consider the family
F of sets defined by F = Fλ/µ. Let r1, r2, . . . , rn be a sequence of cells in λ/µ such that:
• The cell r1 is an inner corner of λ.
• For all k ∈ [n− 1], the cell rk+1 is an inner corner of λ\{r1, r2, · · · , rk}.
Define σF : [n] → F by letting σF (k) = Hrk for all k ∈ [n]. It can be checked that
the bijection σF satisfies Equation (3.1). Now, because the skew shape λ/µ is a finite
set, regard λ/µ as being the set [n], where n is the number of cells in λ/µ. In particular,
regard Fλ/µ as a family of subsets of [n]. Then, by the above and by Definition 3.6,
F is flagged. In particular, by Proposition 3.7, F has a unique transversal. The unique
transversal tλ/µ : F → λ/µ of F is given by tλ/µ(Hr) = r for all r ∈ λ/µ.
As we are regarding the cells of a skew shape with n cells as being the elements of
[n], we can regard any surjective tableau T of shape λ/µ as being a surjective function
T : [n] → [m] in which T (i) = j if j is the entry of T in the cell of T corresponding
to i. Taking m = n, we can also regard any bijective tableau of shape λ/µ as being a
permutation T : [n] → [n]. Lastly, as we are regarding the skew shape λ/µ as being the
set [n] and as we are regarding Fλ/µ as a family of subsets of [n], we define configurations
f : λ/µ → N of tλ/µ, where tλ/µ is the unique transversal of Fλ/µ, and surjective maps
σ : λ/µ → [m] that satisfy f analogously to Definition 3.4.
Next, let λ/µ be a skew shape with n cells, consider the flagged family of sets Fλ/µ,
and let tλ/µ be the unique transversal of F . We define the configuration f0 of tλ/µ by
f0(r) = 1 for all r ∈ λ/µ. It can be seen that the standard skew tableaux of shape λ/µ
are the bijective tableaux of shape λ/µ that satisfy f0. Since we regard λ/µ as being the
set [n], we can regard f0 as being the function f0 : [n] → N defined by f0(k) = 1 for
B. T. Chan: A generalization of balanced tableaux and marriage problems with . . . 211
all k ∈ [n]. So as we regard Fλ/µ as being a family of subsets of [n], the standard skew
tableaux of shape λ/µ can be regarded as being the permutations σ : [n] → [n] that satisfy
f0.
Example 3.19. Consider the following surjective tableau of shape λ = (4, 3, 2).
1 2 3 3
1 2 3
3 3
Next, consider Fλ. Let tλ be the unique transversal of Fλ, and let f : λ → N be the
configuration of tλ defined by f(r) = 1 for all r ∈ λ/µ. It can be checked that the above
tableau satisfies f .
Edelman and Greene introduced the following class of bijective tableaux, which we
re-formulate in terms of the configurations we defined in this paper.
Definition 3.20 (Edelman and Greene [5]). Let λ be a Young diagram containing n cells.
Moreover, let tλ be the transversal of Fλ defined by tλ(Hr) = r for all r ∈ λ and let f be
the configuration defined by
f(r) = |Lr|+ 1
for all r ∈ λ. Then a balanced tableau of shape λ is a bijective tableau of shape λ that
satisfies f .
Example 3.21. Let λ = (4, 3, 2), and let tλ and f be defined from Fλ as described in
Definition 3.20. Then
T = 4 5 8 3
6 7 9
1 2
is balanced because T satisfies f . For instance, f((2, 1)) = 2 since L(2,1) = {(3, 1)} and
|Lr| + 1 = 2. So as T ((2, 1)) = 6, H(2,1) = {(2, 1), (2, 2), (2, 3), (3, 1)}, and the set of
entries in T that are contained in a cell of H(2,1) is {1, 6, 7, 9}, it follows that T ((2, 1)) is
the f((2, 1))th-smallest element of {1, 6, 7, 9}.
Remark 3.22. The surjective tableaux from Definition 3.13 that satisfy the configuration
f : [n] → N defined by f(k) = 1 for all k ∈ [n] do not correspond to semistandard
tableaux, nor do they correspond to the semistandard balanced labelings in [7].
Balanced tableaux can be regarded as permutations σ : [n] → [n] that satisfy f(r) =
|Lr| + 1. The function f(r) = |Lr| + 1 was called the hook rank of r by Edelman and
Greene and they used it to define balanced tableaux [5].
Lastly, we give examples illustrating Theorem 3.10 and Corollary 3.11.
Example 3.23. We give an example in which the lower bound min(n, n − |S| + 1) from
Theorem 3.10 cannot be improved on. Consider λ = (3, 2, 1). Next, let F = Fλ and let
t be the unique transversal of F . As discussed earlier, F is flagged. Now, let f be the
configuration of Fλ defined by f((1, 1)) = 5, f((1, 2)) = 3, f((1, 3)) = 1, f((2, 1)) = 3,
f((2, 2)) = 1, and f((3, 1)) = 1. We depict the configuration f with the below diagram.
212 Ars Math. Contemp. 21 (2021) #P2.03 / 201–217
5 3 1
3 1
1
There is exactly one cell in the Young diagram λ, the cell (1, 1), that is contained in
exactly one member of F = {Hr : r ∈ λ}. Hence, S = {(1, 1)} and min(n, n−|S|+1) =
n− |S|+1. With this in mind, set n = 6 and, as n− |S|+1 = 6− 1+1 = 6, assume that
m is an integer satisfying 1 ≤ m ≤ 5. Suppose that there exists a surjective tableau T of
shape λ, and with entries from [m], such that T satisfies the configuration f defined above.
The cells (1, 1), (1, 2) and (2, 1) are cells r ∈ λ that satisfy f(r) = hr. Moreover, because
T satisfies f , f(r) = hr implies that no two entries of T in Hr are the same and that the
entry of T in cell r is the hthr smallest element of the set of entries of T that are contained
in Hr.
So consider the cell (2, 2) of λ. Since m ≤ 5, some two entries of T in H(1,1) are the
same, or the entry of T in cell (2, 2) equals to the entry of T in some other cell, (i1, j1), in
λ. Since f((1, 1)) = 5 = h(1,1), no two entries of T in H(1,1) are the same. So the entry of
T in cell (2, 2) equals to the entry of T in some other cell, (i1, j1), in λ. If (i1, j1) = (1, 1),
then the entry of T in cell (2, 2) of λ is larger than the entries of T in cells (1, 2) and (2, 1)
of λ. But that is impossible by the above. If (i1, j1) = (2, 1) or if (i1, j1) = (3, 1), then
the entry of T in cell (2, 1) of λ is the kth smallest element of the set of entries of T that
are contained in H(2,1) for some k ≤ 2. But that is impossible by the above. By symmetry,
it is impossible for (i1, j1) = (1, 2) or for (i1, j1) = (1, 3). Hence, we have reached a
contradiction. It follows that there is no such surjective tableau T .
Example 3.24. Consider a skew shape λ/µ with n cells, and let S denote the set of cells
of λ/µ that are contained in exactly one member of {Hr : r ∈ λ/µ}. The elements of S
are also known as the outer corners of µ. Clearly, there exists a standard skew tableau of
shape λ/µ. Corollary 3.11 implies that such a tableau exists if and only if for all integers
n− |S|+ 1 ≤ m ≤ n and for all configurations f of λ/µ, there exists a surjective tableau
T of shape λ/µ, with [m] as the set of entries of T , such that T satisfies f .
3.3 The average number of generalized tableaux
Let S(n,m) denote the Stirling number of the second kind, namely the number of set
partitions of [n] into m parts. Let F be a family of subsets of [n] that satisfies the marriage
condition, let m ∈ [n], and let t be a transversal of F . If f is a configuration of t, then let
An,m(f) denote the number of surjective maps σ : [n] → [m] that satisfy f . Moreover,
let X be the set of configurations f of t such that An,m(f) ≥ 1. Then define the average
value of An,m(f) over all configurations f of t satisfying An,m(f) ≥ 1 to be
1
|X|
∑
f∈X
An,m(f)
if |X| > 0, and 0 otherwise.
Theorem 3.25. Let F be a flagged family of subsets of [n] such that |F| = n and let t be
the transversal of F . Moreover, let S ⊆ [n] be the set of elements k ∈ [n] such that k ∈ F
for exactly one member F of F , and let m be an integer satisfying
n− |S|+ 1 ≤ m ≤ n.
B. T. Chan: A generalization of balanced tableaux and marriage problems with . . . 213
Then the average value of An,m(f) over all configurations f of t satisfying
An,m(f) ≥ 1
is
m!S(n,m)∏
F∈F |F |
. (3.9)
Remark 3.26. Consider the sequence (pk(x))k=0,1,2,... of polynomials in Q[x] such that
p0(x) = 1 and, for all k,
pk+1(x)− pk+1(x− 1) = x pk(x).
If k = n−m, then S(n,m) = pk(m) (see [2, 22]). So if k is fixed, then we can compute
closed-form expressions for S(n,m). For instance, Expression 3.9 becomes
m!∏
F∈F |F |
if n = m, (
m+ 1
2
)
m!∏
F∈F |F |
if n = m+ 1, and
1
2
(
m+ 1
2
)((
m+ 1
2
)
+
2m+ 1
3
)
m!∏
F∈F |F |
if n = m+ 2.
In order to prove Theorem 3.25, we prove the following.
Lemma 3.27. Let m,n ∈ N such that m ≤ n, and let F be a family of subsets of [n]
that has a transversal t : F → [n] such that t is surjective. Then every surjective function
σ : [n] → [m] satisfies exactly one configuration f of t.
Proof. Let σ : [n] → [m] be a surjective map. Then σ satisfies the configuration f of t
defined by letting, for all F ∈ F , f(t(F )) = k where σ(t(F )) is the kth smallest element
of the set σ(F ). Now, suppose that σ satisfies more than one configuration of t. Then, let
f1 and f2 be two distinct configurations of t. Because f1 ̸= f2 and because t is surjective,
there is an element F ∈ F such that f1(t(F )) ̸= f2(t(F )). So write k1 = f1(t(F ))
and write k2 = f2(t(F )). Since σ satisfies f1, Definition 3.4 implies that σ(t(F )) is
the kth1 smallest element of σ(F ). Moreover, since σ satisfies f2, Definition 3.4 implies
that σ(t(F )) is the kth2 smallest element of σ(F ). However, this is impossible because
k1 = f1(t(F )) ̸= f2(t(F )) = k2.
Now, we prove Theorem 3.25.
Proof. By Definition 3.4, the total number of configurations of F equals to
∏
F∈F |F |.
Moreover, it is well-known that the number of surjective maps from [n] to [m] is given
by m!S(n,m). By Lemma 3.27, every surjective map satisfies exactly one configuration.
Moreover, by Theorem 3.10, every configuration of F is satisfied by some surjective map
from [n] to [m]. From this, the theorem follows.
214 Ars Math. Contemp. 21 (2021) #P2.03 / 201–217
Theorem 3.25 implies the following consequence relating to how the values An,m(f)
are distributed. By Theorem 3.10, every configuration f of t is satisfied by at least one
surjective map σ : [n] → [m]. Hence, by Theorem 3.25 and the fact that An,m(f) ≥ 1
always holds, it follows that for all constants k ≥ 1 the number of configurations f of t
that satisfy
An,m(f) ≤ k ·
m! S(n,m)∏
F∈F |F |
is at least (
1− 1
k
) ∏
F∈F
|F |.
We now illustrate Theorem 3.25 with some examples and in the process describe its
relationship with the hook-length formula.
Example 3.28. Let λ = (6, 5, 4, 3, 2, 1) and µ = (1). The skew shape λ/µ is depicted
below.
Since λ/µ has eighteen cells, let n = 18. The cells of λ/µ that are contained in exactly one
member of the family Fλ/µ are (1, 3), (2, 2), and (3, 1). Hence, S = {(1, 3), (2, 2), (3, 1)}
and n−|S|+1 = n−2. So let m = n−2 = 16. Then by Theorem 3.25 and Remark 3.26,
the average value of An,m(f) over all configurations f of λ/µ satisfying An,m(f) ≥ 1 is
given by
1
2
(
m+ 1
2
)((
m+ 1
2
)
+
2m+ 1
3
)
m!∏
F∈Fλ/µ |F |
=
=
1
2
(
16 + 1
2
)((
16 + 1
2
)
+
2 · 16 + 1
3
)
16!∏
r∈λ/µ hr
=
1
2
(
17
2
)((
17
2
)
+ 11
)
16!
(7 · 5 · 3 · 1)3 · 5 · 3 · 1 · 3 · 1 · 1
= 4014814003 +
1
5
.
The hook-length formula, first proved by Frame, Robinson, and Thrall [8], is well-
known. It is as follows. A skew shape λ/µ is a straight shape if µ = ∅. Given a Young
diagram λ, call a standard skew tableau of straight shape λ a standard Young tableaux of
shape λ. If λ is a Young diagram with n cells, then the number of standard Young tableaux
of shape λ equals
n!∏
r∈λ hr
.
B. T. Chan: A generalization of balanced tableaux and marriage problems with . . . 215
Moreover, the above formula was also proved by Edelman and Greene to equal the number
of balanced tableaux of shape λ [5]. Furthermore, the hook-length formula does not hold
for skew shapes. Taking m = n in Theorem 3.25, setting F = Fλ, and letting t be the
unique transversal of F , we see that the average value of An,m(f) over all configurations
f of t satisfying An,m(f) ≥ 1 equals to the number of standard Young tableau of shape λ.
Example 3.29. Let λ = (6, 5, 4, 3, 2, 1). The Young diagram λ is depicted below.
Since λ has twenty-one cells, let n = 21. The cell of λ that is contained in exactly one
member of the family Fλ is (1, 1). Hence, S = {(1, 1)} and n − |S| + 1 = n. So let
m = n = 21. Then by Theorem 3.25 and Remark 3.26, the average value of An,m(f) over
all configurations f of λ satisfying An,m(f) ≥ 1 is given by the hook-length formula
m!∏
F∈Fλ |F |
=
21!∏
r∈λ hr
=
21!
16 · 35 · 54 · 73 · 92 · 11
= 1100742656
and is, by the hook-length formula, equal to the number of standard Young tableaux of
shape λ.
Remark 3.30. Theorem 3.25 is versatile. For instance, possible applications of the special
case of Theorem 3.25 in the case of permutations are as follows. There is a formula for the
number of standard skew tableaux of shape λ/µ, known as Naruse’s formula. Asymptotic
properties of Naruse’s formula were analysed by Morales, Pak, and Panova in [17]. In
particular, it turns out that in general, the number of standard skew tableaux of shape λ/µ
divided by
n!∏
r∈λ/µ hr
,
where n is the number of cells of λ/µ, can be arbitrarily large. Hence, we can apply
Theorem 3.25 to Naruse’s formula and, using the work of Morales, Pak, and Panova in [17],
analyse lower bounds on the number of configurations f of λ/µ such that An,n(f) ≥ 1
and An,n(f) is strictly less than
n!∏
r∈λ/µ hr
.
Remark 3.31. Regarding Remark 3.30, there are variants and generalizations of Naruse’s
formula, the formula mentioned in Remark 3.30, for skew shifted shapes [9, 19]. What
we observe about these shapes is that the “hook-sets” for skew shifted shapes as defined
216 Ars Math. Contemp. 21 (2021) #P2.03 / 201–217
in [9, 19] also form examples of flagged families. Hence, the results in this section can
be replicated verbatim to include skew shifted shapes. Moreover, it is claimed by Morales,
Pak, and Panova in [17] that their analysis of Naruse’s formula can be extended to skew
shifted shapes. It also appears that we can even extend the above to involve posets known
as d-complete posets [19], as there is a generalization of Naruse’s formula for such posets
and the “hook-sets” in these formulas are a generalization of the “hook-sets” for the skew
shifted shapes [19].
We conclude this subsection by asking some natural enumerative questions related to
the quantity An,m(f) in Theorem 3.25.
1. Which configurations f as specified in Theorem 3.25 are such that An,m(f) is given
by Equation (3.9)?
2. Which flagged families F with transversal t are such that An,m(f), with m fixed, is
independent of the configuration f of t?
3. If the configuration f as specified in Theorem 3.25 is such that f(F ) = 1 for all
F ∈ F , when is An,m(f) maximal, and can An,m(f) be less than or equal to Equa-
tion (3.9)?
4. Let F be a flagged family and let t be a transversal of F . For m fixed, which config-
urations f of t maximize or minimize the value of An,m(f)?
5. Does the value of m in comparison to n affect answers to any of the above questions?
ORCID iDs
Brian Tianyao Chan https://orcid.org/0000-0003-0525-1362
References
[1] O. Angel, A. E. Holroyd, D. Romik and B. Virág, Random sorting networks, Adv. Math. 215
(2007), 839–868, doi:10.1016/j.aim.2007.05.019.
[2] M. Bóna, A walk through combinatorics, World Scientific Publishing Co., Inc., River Edge, NJ,
2002, doi:10.1142/4918, an introduction to enumeration and graph theory, With a foreword by
Richard Stanley.
[3] G. J. Chang, On the number of SDR of a (t, n)-family, European J. Combin. 10 (1989), 231–
234, doi:10.1016/s0195-6698(89)80056-3.
[4] P. Edelman and C. Greene, Combinatorial correspondences for Young tableaux, balanced
tableaux, and maximal chains in the weak Bruhat order of Sn, in: Combinatorics and alge-
bra (Boulder, Colo., 1983), Amer. Math. Soc., Providence, RI, volume 34 of Contemp. Math.,
pp. 155–162, 1984, doi:10.1090/conm/034/777699.
[5] P. Edelman and C. Greene, Balanced tableaux, Adv. in Math. 63 (1987), 42–99, doi:10.1016/
0001-8708(87)90063-6.
[6] N. J. Y. Fan, Standard Rothe tableaux, Discrete Math. 342 (2019), 3182–3193, doi:10.1016/j.
disc.2019.06.027.
[7] S. Fomin, C. Greene, V. Reiner and M. Shimozono, Balanced labellings and Schubert polyno-
mials, European J. Combin. 18 (1997), 373–389, doi:10.1006/eujc.1996.0109.
[8] J. S. Frame, G. d. B. Robinson and R. M. Thrall, The hook graphs of the symmetric groups,
Canad. J. Math. 6 (1954), 316–324, doi:10.4153/cjm-1954-030-1.
B. T. Chan: A generalization of balanced tableaux and marriage problems with . . . 217
[9] W. Graham and V. Kreiman, Excited Young diagrams, equivariant K-theory, and
Schubert varieties, Trans. Amer. Math. Soc. 367 (2015), 6597–6645, doi:10.1090/
s0002-9947-2015-06288-6.
[10] M. Hall, Jr., Distinct representatives of subsets, Bull. Amer. Math. Soc. 54 (1948), 922–926,
doi:10.1090/s0002-9904-1948-09098-x.
[11] P. Hall, On representatives of subsets, J. London Math. Soc 10 (1935), 26–30, doi:10.1112/
jlms/s1-10.37.26.
[12] J. L. Hirst and N. A. Hughes, Reverse mathematics and marriage problems with unique solu-
tions, Arch. Math. Logic 54 (2015), 49–57, doi:10.1007/s00153-014-0401-z.
[13] J. L. Hirst and N. A. Hughes, Reverse mathematics and marriage problems with finitely many
solutions, Arch. Math. Logic 55 (2016), 1015–1024, doi:10.1007/s00153-016-0509-4.
[14] L. Hurwicz and S. Reiter, Transversals, systems of distinct representatives, mechanism design,
and matching, in: Markets, Games, and Organizations, Springer, Berlin, Heidelberg, 2003,
doi:10.1007/978-3-540-24784-5 10.
[15] M. Konvalinka, A bijective proof of the hook-length formula for skew shapes, European J.
Combin. 88 (2020), 103104, 14, doi:10.1016/j.ejc.2020.103104.
[16] D. P. Little, Combinatorial aspects of the Lascoux-Schützenberger tree, Adv. Math. 174 (2003),
236–253, doi:10.1016/s0001-8708(02)00038-5.
[17] A. H. Morales, I. Pak and G. Panova, Asymptotics of the number of standard Young tableaux
of skew shape, European J. Combin. 70 (2018), 26–49, doi:10.1016/j.ejc.2017.11.007.
[18] A. H. Morales, I. Pak and G. Panova, Hook formulas for skew shapes I. q-analogues and bijec-
tions, J. Combin. Theory Ser. A 154 (2018), 350–405, doi:10.1016/j.jcta.2017.09.002.
[19] H. Naruse and S. Okada, Skew hook formula for d-complete posets via equivariant K-theory,
Algebr. Comb. 2 (2019), 541–571, doi:10.5802/alco.54.
[20] P. F. Reichmeider, The equivalence of some combinatorial matching theorems, Polygonal Publ.
House, Washington, NJ, 1984, doi:10.2307/3615712.
[21] R. P. Stanley, Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies
in Advanced Mathematics, Cambridge University Press, Cambridge, 1999, doi:10.1017/
cbo9780511609589, with a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.
[22] R. P. Stanley, Enumerative Combinatorics. Volume 1, volume 49 of Cambridge Studies in
Advanced Mathematics, Cambridge University Press, Cambridge, 2nd edition, 2012, doi:
10.1017/cbo9781139058520.
[23] F. Viard, A new family of posets generalizing the weak order on some coxeter groups, 2015,
arXiv:1508.06141 [math.CO].
[24] F. Viard, A natural generalization of balanced tableaux, 2016, arXiv:1407.6217v2
[math.CO].

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P2.04 / 219–241
https://doi.org/10.26493/1855-3974.2478.d1b
(Also available at http://amc-journal.eu)
Enumerating symmetric peaks in non-decreasing
Dyck paths*
Sergi Elizalde
Department of Mathematics, Dartmouth College, Hanover, NH, U.S.A.
Rigoberto Flórez †
Department of Mathematical Sciences, The Citadel, Charleston, SC, U.S.A.
José Luis Ramı́rez ‡
Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
Received 7 November 2020, accepted 25 February 2021, published online 30 September 2021
Abstract
Local maxima and minima of a Dyck path are called peaks and valleys, respectively.
A Dyck path is non-decreasing if the heights (y-coordinates) of its valleys increase from
left to right. A peak is symmetric if it is surrounded by two valleys (or endpoints of the
path) at the same height. In this paper we give multivariate generating functions, recurrence
relations, and closed formulas to count the number of symmetric and asymmetric peaks in
non-decreasing Dyck paths. Finally, we use Riordan arrays to study weakly symmetric
peaks, namely those for which the valley preceding the peak is at least as high as the valley
following it.
Keywords: Non-decreasing Dyck path, symmetric peak, generating function, Riordan array, Fibonacci
number.
Math. Subj. Class. (2020): 05A15, 05A19
*The authors are grateful to an anonymous referee for helpful comments.
†Corresponding author. The author was partially supported by the Citadel Foundation, Charleston, SC.
‡The author was partially supported by Universidad Nacional de Colombia, Project No. 46240.
E-mail addresses: sergi.elizalde@dartmouth.edu (Sergi Elizalde), rigo.florez@citadel.edu (Rigoberto
Flórez), jlramirezr@unal.edu.co (José Luis Ramı́rez)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
220 Ars Math. Contemp. 21 (2021) #P2.04 / 219–241
1 Introduction
A Dyck path is a lattice path in the first quadrant of the xy-plane that starts at the origin,
ends on the x-axis, and consists of (the same number of) up-steps X = (1, 1) and down-
steps Y = (1,−1). A peak is a subpath of the form XY , and a valley is a subpath of the
form Y X . The height of a valley is the y-coordinate of its lowest point. A Dyck path is
called non-decreasing if the heights of its valleys form a non-decreasing sequence from left
to right (see Figure 1 for an example). Non-decreasing Dyck paths have been extensively
studied in the literature, see [2, 5, 6, 8, 13, 15, 17, 20]. All the Dyck paths considered in
this paper will be non-decreasing. Following the notation from [5, 6, 13, 14], we denote by
D the set of all non-decreasing Dyck paths, and by Dn the set of all non-decreasing Dyck
paths of length 2n, where the length is defined as the number of steps. For P ∈ Dn, we
write |P |= n to denote its semilength.
A pyramid of semilength h ≥ 1 is a subpath of the form XhY h; it is maximal if it can
not be extended to a pyramid Xh+1Y h+1.
Flórez and Ramı́rez [16] introduced the concept of symmetric and asymmetric peaks
in Dyck paths, see also recent follow-up work by Elizalde [11] and Flórez et al. [14]. This
concept was motivated in part by Asakly’s [1] study of symmetric and asymmetric peaks
in k-ary words. The concept of symmetric peaks is different from the notion of degree of
symmetry, which has been considered by Elizalde [9, 10] as a measure of how symmetric a
Dyck path is.
In this paper we study symmetric peaks and asymmetric peaks in non-decreasing Dyck
paths. A peak is symmetric if the maximal pyramid containing the peak is not preceded
by an X and is not followed by a Y . A peak is weakly symmetric if the maximal pyramid
containing the peak is not preceded by an X . A peak is asymmetric if the maximal pyramid
containing the peak is either preceded by an X or followed by a Y . Geometrically, a peak is
symmetric if the maximal pyramid containing the peak is either at ground level or bounded
by two valleys at the same height, and it is asymmetric otherwise. For example, in the non-
decreasing Dyck path in Figure 1, the first, third, fourth, and sixth peaks are symmetric.
The weakly symmetric peaks are the symmetric ones along with the seventh peak. Finally,
the second, fifth, and the seventh peaks are asymmetric.
We are also interested in the size of the maximal pyramid containing a peak. We define
the weight of a pyramid XhY h to be equal to h. In [5, 7], the authors refer to this parameter
as the height, but we will use the term weight to suggest that it is not affected by the
location of the pyramid. We define the weight of a peak to be the weight of the maximal
pyramid that contains it. The symmetric weight of a path is the sum of the weights of its
symmetric peaks. Similarly, the asymmetric weight of a path is the sum of the weights of its
asymmetric peaks. For example, the weights of the symmetric peaks in the path depicted
in Figure 1 are 4, 3, 3, 2 from left to right, and so the symmetric weight of the path is 12.
The weights of asymmetric peaks are 1, 3, and 1, and the asymmetric weight of the path
is 5.
Figure 1: A non-decreasing Dyck path of length 38.
S. Elizalde et al.: Enumerating symmetric peaks in non-decreasing Dyck paths 221
The generating functions that we present throughout the paper, are given using the
symbolic method (cf. [12]). In Section 2, we give generating functions, recurrence rela-
tions, and closed formulas enumerating symmetric peaks and asymmetric peaks in non-
decreasing Dyck paths. In Section 3, we focus on the enumeration of peaks with respect
to their weight, and we give a connection to directed column-convex polyominoes. In Sec-
tion 4, we study weakly symmetric peaks, and we synthesize the results using Riordan
arrays. A summary of notation used throughout the paper appears in Tables 1 and 2 in the
appendix.
2 Counting symmetric peaks
In this section we study the distribution of the number of symmetric peaks in Dn. We give
recurrences, generating functions and closed formulas (in terms of Fibonacci numbers) that
enumerate these statistics in non-decreasing Dyck paths. Throughout the paper we will use
Fn and Ln to denote the nth Fibonacci number and the nth Lucas number, respectively.
The set Dn can be partitioned into two disjoint sets An and Bn, where An consists of
the paths that have at least one valley of ground level (height 0), and Bn = Dn \ An. Note
that
Dn = An ∪· Bn and An =
n−1⋃
·
i=1
Cn,i, (2.1)
where Cn,i consists of those paths whose first valley touches the x-axis at (2i, 0), and ∪·
denotes disjoint union. There is a natural bijection
Cn,i → Dn−i
P 7→ P \∆i,
(2.2)
obtained by removing the first pyramid ∆i = XiY i of each P ∈ Cn,i. Similarly, there is a
bijection from Bn to Dn−1 obtained by removing the first up-step and last down-step from
each path.
From (2.1), a path Q ∈ D is either empty or has one of these two forms: Q = XPY
or Q = XkY kP , where k ≥ 1 and P ∈ D is non-empty. This decomposition gives rise to
the following equation for the generating function D(x) =
∑
P∈D x
|P | =
∑
n≥0|Dn|xn:
D(x) = 1 + xD(x) +
x
1− x
(D(x)− 1). (2.3)
Solving this equation and removing the empty path, we obtain the generating function for
non-decreasing Dyck paths with respect to their semilength:
D(x) =
x(1− x)
1− 3x+ x2
=
∞∑
n=1
F2n−1x
n.
Therefore,
|Dn|= F2n−1. (2.4)
Other derivations of this generating function appear in [2, 13].
222 Ars Math. Contemp. 21 (2021) #P2.04 / 219–241
2.1 A generating function for the number of symmetric and asymmetric peaks
In this section we give a multivariate generating function enumerating symmetric peaks and
the number of asymmetric peaks in non-decreasing Dyck paths. We start by introducing
some terminology. We define the insertion vertices of a path to be the lowest point of each
valley Y X , the initial point of the path, and, if the path contains no valleys at positive
height, the final point of the path. For a path P ∈ D, we use τ(P ), σ(P ), σ(P ), ν(P ),
and ι(P ) to denote the number of peaks, the number of symmetric peaks, the number
of asymmetric peaks, the number of valleys, and the number of insertion points of P ,
respectively. We are interested in the generating function
Dσ,σ(t, r, x) =
∑
P∈D
tσ(P )rσ(P )x|P |.
The coefficient of tirjxn in Dσ,σ(t, r, x) is the number of paths of length 2n with i sym-
metric peaks and j asymmetric peaks.
Theorem 2.1. The generating function for non-decreasing Dyck paths with respect to the
number of symmetric peaks and the number of asymmetric peaks is
Dσ,σ(t, r, x) =
1− (3 + t)x+ (3 + 2t− r)x2 − (1 + t− r − r2)x3
(1− (1 + t)x)(1− (t+ 2)x+ (1 + t− r)x2)
.
Proof. In order to obtain an expression for Dσ,σ(t, r, x), we show that non-decreasing
Dyck paths where some of their symmetric peaks have been marked can be constructed
by inserting marked symmetric peaks in certain positions of smaller non-decreasing Dyck
paths.
First, we refine Equation (2.3) by introducing a variable v that keeps track of the number
of valleys in the path. Letting Dν(v, x) =
∑
P∈D v
ν(P )x|P |, the same decomposition gives
Dν(v, x) = 1 + xDν(v, x) +
vx
1− x
(Dν(v, x)− 1),
from where
Dν(v, x) =
1− (1 + v)x
1− (2 + v)x+ x2
.
Next we introduce another refinement. Let D∆ ⊆ D denote the set of paths that con-
sist of a non-empty sequence of pyramids, that is, paths of the form Xk1Y k1 · · ·XkjY kj ,
where ki ≥ 1 for 1 ≤ i ≤ j, for some j ≥ 1. Let Dτ,ι(p, q, x) =
∑
P∈D p
τ(P )qι(P )x|P |
be the generating function with respect to the number of peaks and the number of insertion
vertices. Recall that insertion vertices of P are the bottoms of the valleys, the initial point
of P , and, in the case that P ∈ D∆, the final point of P . Thus, ι(P ) = ν(P ) + 2 if
P ∈ D∆, and ι(P ) = ν(P ) + 1 otherwise. On the other hand, τ(P ) = ν(P ) + 1 unless P
is empty, in which case τ(P ) = 0. Using that
D∆ν (v, x) =
∑
P∈D∆
vν(P )x|P | =
x/(1− x)
1− vx/(1− x)
=
x
1− x− vx
,
it follows that
Dτ,ι(p, q, x) = q + pq(Dν(pq, x)−D∆ν (pq, x)− 1) + pq2D∆ν (pq, x)
= q +
pq2x(1− (2 + pq)x+ (1 + p)x2)
(1− x+ pqx)(1− (2 + pq)x+ x2)
. (2.5)
S. Elizalde et al.: Enumerating symmetric peaks in non-decreasing Dyck paths 223
By construction, the insertion vertices of P are those vertices where the insertion of a
pyramid XkY k creates a symmetric peak and results in another non-decreasing Dyck path.
Our next step is to enumerate non-decreasing Dyck paths where some of its symmetric
peaks have been marked. Formally, we are enumerating pairs (P,M) where P ∈ D and
M is a subset of the symmetric peaks of P . Let D∗ be the set of such pairs (P,M),
which we refer to as non-decreasing Dyck paths with marked symmetric peaks, and let
D∗τ (p, u, x) =
∑
(P,M)∈D∗ p
τ(P )u|M |x|P |. The key observation is that elements of D∗ can
be uniquely obtained from paths in D by inserting a possibly empty sequence of marked
pyramids (that is, pyramids whose symmetric peak is marked) at each insertion vertex.
Since replacing each insertion vertex with a sequence of marked pyramids corresponds to
the substitution
q =
1
1− upx/(1− x)
,
we get
D∗τ (p, u, x) = Dτ,ι
(
p,
1
1− upx/(1− x)
, x
)
.
In order to have a variable t that keeps track of the total number of symmetric peaks,
as opposed to marked symmetric peaks, we make the substitution u = t − 1. Note that, if
Σ(P ) is the set of symmetric peaks of a path P ∈ D, then∑
M⊆Σ(P )
(t− 1)|M | = ((t− 1) + 1)|Σ(P )| = tσ(P ). (2.6)
It follows that
Dτ,σ(p, t, x) =
∑
P∈D
pτ(P )tσ(P )x|P | =
∑
P∈D
∑
M⊆Σ(P )
pτ(P )(t−1)|M |x|P | = D∗τ (p, t−1, x).
Finally, since σ(P ) = τ(P )− σ(P ), we have
Dσ,σ(t, r, x) = Dτ,σ(r, t/r, x) = Dτ,ι
(
r,
1
1− (t− r)x/(1− x)
, x
)
,
and the formula in the statement follows now from Equation (2.5).
Corollary 2.2. The generating functions for the total number of symmetric peaks and the
total number of asymmetric peaks in non-decreasing Dyck paths are, respectively,
S(x) :=
∑
P∈D
σ(P )x|P | =
∂
∂t
Dσ,σ(t, 1, x)
∣∣∣∣
t=1
=
x(1− 5x+ 7x2 − x3 − x4)
(1− 2x)(1− 3x+ x2)2
, (2.7)
∑
P∈D
σ(P )x|P | =
∂
∂r
Dσ,σ(1, r, x)
∣∣∣∣
r=1
=
x3(2− 6x+ 3x2)
(1− 2x)(1− 3x+ x2)2
.
2.2 Recurrence relations and Fibonacci numbers
Let sn =
∑
P∈Dn σ(P ), that is, the total number of symmetric peaks in all non-decreasing
Dyck paths of semilength n. Note that S(x) =
∑
n≥1 snx
n is the generating function
in Equation (2.7). Next we give a recurrence for sn that involves the Fibonacci numbers.
Define the level of a pyramid to be the height of the base of the pyramid.
224 Ars Math. Contemp. 21 (2021) #P2.04 / 219–241
Theorem 2.3. The sequence sn satisfies the recurrence relation
sn = 3sn−1 − sn−2 + F2(n−2) − 2n−3 for n ≥ 3,
with initial values s1 = 1 and s2 = 3.
Proof. Recall the decomposition given in (2.1). It is clear from the definition of non-
decreasing Dyck paths that the first pyramid in every path in Cn,i has a symmetric peak.
Applying the bijection Cn,i → Dn−i from Equation (2.2) to all paths in Cn,i removes a
total of |Dn−i|= F2(n−i)−1 pyramids (using Equation (2.4)), each having a symmetric
peak. This implies that the number of symmetric peaks in Cn,i equals F2(n−i)−1 plus the
number of symmetric peaks in Dn−i. So, the total number of symmetric peaks in An is
given by
n−1∑
i=1
sn−i +
n−1∑
i=1
F2(n−i)−1 =
n−1∑
i=1
si + F2(n−1). (2.8)
We now count the total number of symmetric peaks in Bn, using the fact that Bn maps
bijectively into Dn−1 by deleting the first X and the last Y . Note, however, that the first
and the last peak of paths in Bn are not symmetric (unless the path is a pyramid), but they
may become symmetric after the first X and the last Y are deleted. This happens when
the associated path in Dn−1 starts or ends with a pyramid at ground level, without the path
being itself the pyramid ∆n−1 = Xn−1Y n−1, resulting in more symmetric peaks in Dn−1
than in Bn. Therefore, to count the number of symmetric peaks in Bn, we take the number
of symmetric peaks in Dn−1, which is sn−1, and subtract the total number of first and last
pyramids at ground level of paths in Dn−1 \ {∆n−1}.
First of all, we want to know the total number of pyramids at ground level that occur at
the end of the paths in Dn−1 \ {∆n−1}. Note that if the last pyramid of a non-decreasing
Dyck path is at ground level, then the path consists of a sequence of pyramids at ground
level. From [13, Corollary 6.3], we deduce that the number of paths in Dn−1 ending with
a pyramid ∆i = XiY i at ground level, for 1 ≤ i ≤ n− 2, is 2(n−1−i)−1. This implies that
the total number of last pyramids at ground level in Dn−1\{∆n−1} is
∑n−3
i=0 2
i = 2n−2−1.
From a similar analysis as in the first paragraph of this proof, we have that the total number
of first pyramids at ground level in Dn−1 \ {∆n−1} is
∑n−2
i=1 F2i−1 = F2(n−2). So, the
total number of symmetric peaks in Bn is given by sn−1 − F2(n−2) − 2n−2 + 1. Adding
this to (2.8), we get
sn =
(
n−1∑
i=1
si + F2(n−1)
)
+
(
sn−1 − F2(n−2) − 2n−2 + 1
)
,
with s1 = 1, and s2 = 3. We can simplify the recurrence by computing sn+1 − sn =
2sn − sn−1 + F2(n−1) − 2n−2. Therefore,
sn+1 = 3sn−1 − sn−2 + F2(n−2) − 2n−3.
The first few values of the sequence sn for n ≥ 1 are
1, 3, 8, 22, 62, 177, 508, 1459, 4182, 11946, . . . .
For example, Figure 2 shows the non-decreasing Dyck paths of length 6, where the total
number of symmetric peaks is s3 = 8.
S. Elizalde et al.: Enumerating symmetric peaks in non-decreasing Dyck paths 225
Figure 2: Non-decreasing Dyck paths of length 6.
Next we give another expression for sn in terms of the Fibonacci and the Lucas num-
bers.
Theorem 2.4. The sequence sn satisfies
sn = F2n +
n∑
ℓ=3
(F2ℓ−2 − 2ℓ−2)F2(n−ℓ) =
2F2n−2 + (n− 1)L2n−2
5
+ 2n−1.
Proof. We first consider the generating function of the bisection of the Fibonacci sequence
F (x) =
∑
n≥0
F2nx
n =
x
1− 3x+ x2
.
By Equation (2.7), the generating function S(x) can be decomposed as
S(x) = F (x)
1− 5x+ 7x2 − x3 − x4
(1− 2x)(1− 3x+ x2)
= F (x)
(
1 +
x2
1− 3x+ x2
− x
2
1− 2x
)
= F (x)
(
1 + xF (x)− x
2
1− 2x
)
.
Using the Cauchy product of series we obtain the desired result. The second equality
follows from the recurrence relation given in Theorem 2.3.
In [6, Theorem 2], the authors prove that the total number of peaks in Dn is
tn =
(2n− 1)F2n − (n− 5)F2n−1
5
. (2.9)
The next corollary is a direct consequence of Theorem 2.4 and Equation (2.9).
Corollary 2.5. Let sn be the total number of asymmetric peaks in Dn. Then, for n ≥ 2,
sn =
2F2n+1 + (n− 2)L2n−3
5
− 2n−1.
The first few values of the sequence sn for n ≥ 1 are
0, 0, 2, 10, 37, 122, 379, 1136, 3326, 9580, . . . .
From the identities in Theorem 2.4 and Corollary 2.5, we obtain some asymptotic re-
sults about the proportion of peaks in non-decreasing Dyck paths that are symmetric.
Theorem 2.6. Among all peaks of non-decreasing Dyck paths, the proportion of those that
are symmetric is asymptotically
lim
n→∞
sn
tn
=
−1 +
√
5
2
≈ 0.618034.
226 Ars Math. Contemp. 21 (2021) #P2.04 / 219–241
Proof. From the well-known limits
lim
n→∞
Fn+1
Fn
= ϕ =
1 +
√
5
2
and lim
n→∞
Ln
Fn
=
√
5,
we have
lim
n→∞
sn
tn
= lim
n→∞
(2F2n−2 + (n− 1)L2n−2)/5 + 2n−1
((2n− 1)F2n − (n− 5)F2n−1)/5
= lim
n→∞
2 + (n− 1)L2n−2/F2n−2 + 5 · 2n−1/F2n−2
(2n− 1)F2n/F2n−2 − (n− 5)F2n−1/F2n−2
=
√
5
2ϕ2 − ϕ
=
−1 +
√
5
2
.
Corollary 2.7. Among all peaks of non-decreasing Dyck paths, the proportion of those that
are asymmetric is asymptotically
lim
n→∞
sn
tn
=
3−
√
5
2
≈ 0.381966.
We say that a symmetric peak is low if the y-coordinate of its top vertex is one, and that
it is high if this coordinate is greater than 1. Note that every low peak is symmetric. By [6,
Corollary 6], the total number of high peaks in Dn is ((2n−1)F2n−nF2n−1)/5. Together
with Corollary 2.5, this implies the following.
Corollary 2.8. The total number of high symmetric peaks in Dn is
1
5
(F2n−3 + (n− 4)L2n−2) + 2n−1.
3 Symmetric weight and symmetric height
Recall that the weight of a pyramid XhY h is equal to h and that the weight of a peak is
the weight of the maximal pyramid that contains it. In this section we give a multivariate
generating function for non-decreasing Dyck paths with respect to the weight of their sym-
metric peaks, as well a recurrence relation for the total symmetric weight over Dn. We also
give a recurrence relation for the total sum of the heights of symmetric peaks over Dn. At
the end of the section we describe a connection with polyominoes.
3.1 A generating function for symmetric weight
We introduce an infinite family of variables t = (t1, t2, . . . ) in order to keep track of
symmetric peaks of a given weight. For P ∈ D and i ≥ 1, let ωi(P ) be the number of
symmetric peaks of weight i in P . Let ω(P ) = (ω1(P ), ω2(P ), . . . ), and let tω(P ) =∏
i≥1 t
ωi(P )
i . We are interested in the generating function
Dω(t, x) =
∑
P∈D
tω(P )x|P |.
S. Elizalde et al.: Enumerating symmetric peaks in non-decreasing Dyck paths 227
Theorem 3.1. Let P (t, x) =
∑
i≥1 tix
i. The generating function for non-decreasing Dyck
paths with respect to the weights of their symmetric peaks is
Dω(t, x) =
1− 3x+ 2x2 + x3 − (1− x)3P (t, x)
(1− x)(1− P (t, x))(1− 2x− (1− x)2P (t, x))
.
Proof. We modify the proof of Theorem 2.1 in order to keep track of the weight of the in-
serted marked symmetric peaks. Replacing insertion vertices in non-decreasing Dyck paths
with sequences of marked pyramids, with variable ui keeping track of marked pyramids of
the form XiY i for each i ≥ 1, corresponds to the substitution
q =
1
1−
∑
i≥1
uix
i
in Dτ,ι(1, q, x). A variant of Equation (2.6), where we replace Σ(P ) with the set of sym-
metric peaks of weight i, shows that the substitutions ui = ti − 1 yield the generating
function where ti keeps track of the total number of symmetric peaks of weight i in non-
decreasing Dyck paths. It follows that
Dω(t, x) = Dτ,ι
1, 1
1−
∑
i≥1
(ti − 1)xi
, x
 = Dτ,ι
(
1,
1
1
1−x − P (t, x)
, x
)
,
and the formula is now obtained from Equation (2.5).
The symmetric weight of a path P ∈ D is defined as the sum of the weights of its
symmetric peaks, and it is denoted by ω(P ) =
∑
i≥1 ωi(P ). From Theorem 3.1, one can
easily obtain a generating function for this statistic. Let
Dσ,ω(t, w, x) =
∑
P∈D
tσ(P )wω(P )x|P |
be the generating function for non-decreasing Dyck paths with respect to the number of
symmetric peaks and the symmetric weight of the path.
Corollary 3.2. The generating function Dσ,ω(t, w, x) is equal to
(1− wx)
(
1− (3 + w + tw)x+ (2 + 3w + 3tw)x2 + (1− 2w − 3tw)x3 − (1− t)wx4
)
(1− x) (1− (t+ 1)wx) (1− (2 + w + tw)x+ 2(t+ 1)wx2 − twx3)
.
Proof. By definition, Dσ,ω(t, w, x) is obtained from Dω(t, x) by making the substitution
ti = tw
i for all i ≥ 1. When applied to P (t, x), this substitution yields
∑
i≥1 tw
ixi =
twx/(1− wx), and so the formula follows immediately from Theorem 3.1.
Corollary 3.3. The generating function for the total symmetric weight in non-decreasing
Dyck paths is
W (x) :=
∑
P∈D
ω(P )x|P | =
∂
∂w
Dσ,ω(1, w, x)
∣∣∣∣
w=1
=
x(1− 5x+ 7x2 − x3 − x4)
(1− x)(1− 2x)(1− 3x+ x2)2
.
228 Ars Math. Contemp. 21 (2021) #P2.04 / 219–241
Comparing this formula with Equation (2.7), we see that
W (x) =
S(x)
1− x
.
Taking the coefficients of xn on both sides, and letting wn =
∑
P∈Dn ω(P ) denote the
total symmetric weight of Dn, we get
wn =
n∑
k=1
sk, (3.1)
that is, the total symmetric weight of paths in Dn equals the total number of symmetric
peaks of paths in
⋃n
k=1 Dk. Next we give a bijective proof of this equality.
The right-hand side of (3.1) can be interpreted as counting paths in
⋃n
k=1 Dk with a
distinguished symmetric peak. Indeed, for each k, the number of ways to choose path in
Dk and select a symmetric peak of such path equals the total number of symmetric peaks
of paths in Dk, namely sk. Similarly, the left-hand side of (3.1) can be interpreted as
counting pairs (P̂ , i), where P̂ is a path in Dn with a distinguished symmetric peak, and
i is an integer between 1 and the weight of the distinguished peak of P̂ . This is because,
for a given path P ∈ Dn, the number of ways to choose a symmetric peak of P and then
an integer i between 1 and the weight of that peak equals the sum of the weights of the
symmetric peaks of P , which is ω(P ).
Let us describe a bijection between the sets counted by both sides of (3.1). Given
a path in Dk (for some k ≤ n) with a distinguished symmetric peak, insert a pyramid
Xn−kY n−k at the top of the distinguished peak to obtain a pair (P̂ , i), where P̂ is a path in
Dn with a distinguished symmetric peak (the same distinguished peak where the pyramid
was inserted), and i = n−k. Conversely, given such a pair (P̂ , i), delete the pyramid XiY i
around the distinguished peak, to obtain a path in Dn−i with a distinguished symmetric
peak (the same distinguished peak from where the pyramid was removed).
3.2 Recurrence relations and Fibonacci numbers
Recall that wn denotes the sum of the symmetric weights of all paths in Dn. Similarly, let
wn denote the sum of the asymmetric weights of all paths in Dn. For example, the paths
in Figure 2 give w3 = 3 + 0 + 3 + 3 + 3 = 12 and w3 = 2. The next theorem follows
immediately by applying Equation (3.1) to Theorem 2.3.
Theorem 3.4. The sequence wn satisfies the recurrence relation
wn = 3wn−1 − wn−2 + F2n−3 − 2n−2 + 1 for n ≥ 3,
with initial values w1 = 1 and w2 = 4.
The first few values of the sequence wn for n ≥ 1 are
1, 4, 12, 34, 96, 273, 781, 2240, 6422, 18368, . . . .
From the expression for W (x) in Corollary 3.3, we obtain the following corollary.
S. Elizalde et al.: Enumerating symmetric peaks in non-decreasing Dyck paths 229
Corollary 3.5. We have
wn = F2n +
n∑
ℓ=1
(F2ℓ−1 − 2ℓ−1 + 1)F2(n−ℓ)
and
wn =
1
5
(nL2n−1 − F2n) + 2n − 1.
In [5, Theorem 8] the authors prove that the sum of the weights of all peaks in Dn is
2nF2n+1 + (2− n)F2n
5
.
As a direct application of Corollary 3.5, we obtain the following formula for the sum of the
asymmetric weights of all paths in Dn.
Corollary 3.6. We have
wn =
1
5
(3F2n + nL2n−2)− 2n + 1.
3.3 Symmetric height
The height of a peak is the y-coordinate of the vertex at the top of the peak. Denote by hn
the total sum of the heights of all symmetric peaks of paths in Dn. For example, from the
paths in Figure 2, we see that h3 = 12.
Theorem 3.7. The sequence hn satisfies the recurrence relation
hn = 3hn−1 − hn−2 +
nL2n−5 + 7F2n−5
5
− 2n−2 + 1 for n ≥ 3,
with initial values h1 = 1 and h2 = 4.
Proof. We will find the total sum of the heights of all symmetric peaks of paths in Dn =
An ∪· Bn by adding the total sum of the heights of all symmetric peaks in An and the
total sum of the heights of all symmetric peaks in Bn. Recall that An = ∪· n−1i=1 Cn,i, and
that the first peak of every path in Cn,i is symmetric. From (2.2) we know that every path
P ∈ Cn,i is a concatenation of the pyramid ∆i = XiY i with a path Q ∈ Dn−i. So, the
total sum of the heights of all symmetric peaks in P is given by the hight of ∆i (which is
equal to i) plus the total sum of the heights of all symmetric peaks in Q. Summing over
all paths P ∈ Cn,i, we deduce that the total sum of the heights of all symmetric peaks of
Cn,i is i|Dn−i|+hn−i = iF2(n−i)−1 + hn−i (using that |Dn−i|= F2(n−i)−1, see (2.4)).
Therefore, the total sum of the heights of all symmetric peaks in An is given by
n−1∑
i=1
hn−i +
n−1∑
i=1
iF2(n−i)−1 =
n−1∑
i=1
hi + F2n−1 − 1. (3.2)
We now count the sum of the heights of all symmetric peaks in Bn, using the fact that
Bn is in bijection with Dn−1, for which the sum of the heights of all symmetric peaks
is hn−1. The bijection is given by removing the first and the last step of the path. Let
230 Ars Math. Contemp. 21 (2021) #P2.04 / 219–241
us carefully analyze how the sum of the heights of the symmetric peaks is changed by
this bijection. On the one hand, removing the first and last step of the path decreases the
heights of the peaks by one. On the other hand, for paths in Dn−1 that begin or end with
a pyramid at ground level, those pyramids contain a symmetric peak that does not give a
symmetric peak in the corresponding path in Bn. To account for these cases, we subtract,
from the total sum of heights of symmetric peaks in Dn−1, the heights of the first and last
peaks belonging to pyramids at ground level, and then we add one for each symmetric peak
whose height has increased.
We recall that the paths in Dn−1 \ {∆n−1}, whose first pyramid is at ground level have
the form ∆iPn−1−i, where Pn−1−i ∈ Dn−1−i and 1 ≤ i ≤ n− 2. For fixed i, the height
of all first pyramids in all such paths is given by i |Dn−1−i|= i F2(n−1−i)−1. So, the total
height of all first pyramids at ground level of paths in Dn−1 \ {∆n−1} is given by
n−2∑
i=1
(n− 1− i)F2i−1 = F2n−3 − 1. (3.3)
We count the total height of pyramids at ground level that occur at the end of the paths in
Dn−1 \ {∆n−1}. If the last pyramid of a non-decreasing Dyck path is at ground level, then
the whole path consists of a sequence of pyramids at ground level. From [13, Corollary 6.3],
we deduce that the number of paths in Dn−1 ending with a pyramid ∆i at ground level, for
1 ≤ i ≤ n − 2, is 2(n−1−i)−1. So, the total height of all last pyramids at ground level of
paths in Dn−1 \ {∆n−1} is given by
n−2∑
i=1
i 2n−i−2 = 2n−1 − n. (3.4)
Now, —to account for the increase by one of peak heights caused by the addition of the
initial X and the final Y to paths in Dn−1— we add the total number of symmetric peaks
in Dn−1, which equals sn−1 (see Theorem 2.4). But this results in some over-counting due
to the first and last pyramids at ground level of the paths in Dn−1, so we have to subtract
F2n−4 and 2n−2 − 1. All in all, the term that needs to be added to account for the increase
in peak heights is(
2F2n−4 + (n− 2)L2n−4
5
+ 2n−2
)
− F2n−4 − 2n−2 + 1. (3.5)
Adding (3.2), hn−1, and (3.5), and subtracting (3.3) and (3.4), we get the recurrence
relation
hn =
n−1∑
i=1
hi + F2n−1 − 1 + hn−1+(
2F2n−4 + (n− 2)L2n−4
5
+ 2n−2 − F2n−4 − 2n−2 + 1
)
−
(
F2n−3 − 1 + 2n−1 − n
)
.
Simplifying, we have that
hn =
n−1∑
i=1
hi + hn−1 +
F2n−1 + nL2n−4 + L2n−5
5
− 2n−1 + n+ 1,
S. Elizalde et al.: Enumerating symmetric peaks in non-decreasing Dyck paths 231
where h1 = 1, and h2 = 4. Now it is easy to see that
hn+1 − hn = 2hn − hn−1 +
F2n + nL2n−3 + 5F2n−3
5
− 2n−1 + 1.
Therefore,
hn = 3hn−1 − hn−2 +
nL2n−5 + 7F2n−5
5
− 2n−2 + 1.
The first few values of the sequence hn for n ≥ 1 are
1, 4, 12, 35, 104, 315, 964, 2957, 9044, 27502, . . . .
3.4 Connections with dccp-polyominoes
Non-decreasing Dyck paths are in bijection with a family of polyominoes called directed
column-convex polyominoes (dccp). A polyomino is directed if each of its cells can be
reached from its bottom left-hand corner by a path which is contained in the polyomino
and uses only north and east steps. A dccp polyomino is a directed polyomino such that
every column consists of contiguous cells [3]. Deutsch and Prodinger [8] give a bijection
between the set of non-decreasing Dyck paths of length 2n and the set of dccp of area n,
where the area of a polyomino is defined as its number of cells. Figure 3 shows a dccp of
area 19. The numbers in the first (second) row represent the final (initial) altitude of each
column.
4
0
2
0
4
1
4
1
5
1
4
2
3
2
Figure 3: A direct column-convex polyomino (dccp).
The bijection from [8] can be described as follows. Given a dccp whose columns have
initial altitudes A = (0, a2, . . . , ak) and final altitudes B = (b1, b2, . . . , bk), from left to
right, its corresponding non-decreasing Dyck path has valleys at heights (a2, . . . , ak), and
peaks at heights (b1, b2, . . . , bk), from left to right. For example, the dccp in Figure 3 is
mapped to the path in Figure 1.
We say that two consecutive columns in a dccp polyomino are at the same level if
their initial altitudes are the same. For example, the polyomino in Figure 3 has 4 pairs of
consecutive columns at the same level; columns 1 and 2, columns 3 and 4, columns 4 and 5,
and columns 6 and 7. Thus, the sequence sn that we introduced in Section 2.2 also counts
the total number of pairs of consecutive columns at the same level in all dccp polyominoes
232 Ars Math. Contemp. 21 (2021) #P2.04 / 219–241
of area n. Moreover, if we define the weight of a pair of consecutive columns at the same
level as the number of cells in the first of these two columns, then the total weight over all
dccp polyominoes of area n is given by wn.
4 Weakly symmetric peaks
In this section we consider a variation of symmetric peaks. We recall from Section 1 that
a peak is weakly symmetric if the maximal pyramid containing the peak is not preceded
by an X . Figure 4 shows different possibilities for the steps preceding and following the
maximal pyramid of a weakly symmetric peak. Note that the last configuration in Figure 4
can only occur in the last peak of a path.
In Section 2, we gave generating functions to count symmetric and asymmetric peaks
in non-decreasing Dyck paths, in this section we also give generating functions to count the
number of weakly symmetric peaks. Surprisingly, the generating functions in this section
have a simpler construction.
We will find formulas, involving Fibonacci numbers, for the total number of weakly
symmetric peaks, as well as the sum of their weights, using generating functions and recur-
rence relations. The results in this section are synthesized using Riordan arrays.
Figure 4: Weakly symmetric peaks.
4.1 A generating function for the number of weakly symmetric peaks
Let s̃n be the total number of weakly symmetric peaks in Dn. For example, we see from
the paths in Figure 2 that s̃3 = 9. The first few values of s̃n for n ≥ 1 are
1, 3, 9, 27, 80, 234, 677, 1941, 5523, 15615, . . . ,
which correspond to sequence A059502 in [23].
Given a non-decreasing Dyck path P , we denote by σ̃(P ) the number of weakly sym-
metric peaks of P , and recall that |P | denotes the semilength of P . We introduce the
generating function
Dσ̃(x, y) =
∑
P∈D
x|P |yσ̃(P ).
Theorem 4.1. The generating function Dσ̃(x, y) is given by
Dσ̃(x, y) =
(1− x)xy
1− (2 + y)x+ yx2
.
Proof. Recall the decomposition in (2.1). Non-empty paths in Bn can be written as XY or
XT ′Y , where T ′ is a non-decreasing Dyck paths. Paths in An are of the form X∆Y T ′′,
where ∆ is a pyramid and T ′′ is a non-decreasing Dyck paths. Figure 5 illustrates the three
cases.
S. Elizalde et al.: Enumerating symmetric peaks in non-decreasing Dyck paths 233
Figure 5: Decomposition of a non-decreasing Dyck path.
Using the symbolic method, we obtain the relation
Dσ̃(x, y) = xy + x(Dσ̃(x, y)−
xy
1− x
Dσ̃(x, y) +
x
1− x
Dσ̃(x, y)︸ ︷︷ ︸
(a)
) +
xy
1− x
Dσ̃(x, y).
The term (a) corresponds to the case where T ′ starts with a pyramid, which was symmetric
in T ′ but is no longer weakly symmetric in the big path. This completes the proof.
Corollary 4.2. The total number of weakly symmetric peaks in Dn satisfies these
(i) The generating function for s̃n is given by
∞∑
n=1
s̃nx
n =
(1− x)(1− 2x)x
(1− 3x+ x2)2
.
(ii) The sequence s̃n satisfies the recurrence relation
s̃n = 6s̃n−1 − 11s̃n−2 + 6s̃n−3 − s̃n−4 for n ≥ 5,
with initial values s̃1 = 1, s̃2 = 3, s̃3 = 9 and s̃4 = 27.
(iii) The sequence s̃n satisfies the recurrence relation
s̃n = 3s̃n−1 − s̃n−2 + F2(n−2) for n ≥ 3,
with initial values s̃1 = 1 and s̃2 = 3.
(iv) For n ≥ 1, we have the convolution
s̃n =
n−1∑
ℓ=0
F2ℓ−1F2(n−ℓ)−1.
(v) The sequence s̃n satisfies that s̃n = (3F2n + nL2n−2) /5.
Proof. By Theorem 4.1,
∞∑
n=0
s̃nx
n =
∂Dσ̃(x, y)
∂y
∣∣∣∣
y=1
=
(1− x)(1− 2x)x
(1− 3x+ x2)2
.
This proves part (i). The recurrence in part (ii) is obtained from this rational generating
function. The proof of (iii) is similar to the proof of Theorem 2.3, but in this case we do
not subtract the last pyramid at ground level of paths in Bn.
234 Ars Math. Contemp. 21 (2021) #P2.04 / 219–241
To prove part (iv), note that
∞∑
n=1
s̃nx
n =
(
(1− x)x
1− 3x+ x2
)(
1− 2x
1− 3x+ x2
)
=
( ∞∑
n=1
F2n−1x
n
)( ∞∑
n=0
F2n−1x
n
)
=
∞∑
n=1
(
n−1∑
ℓ=0
F2ℓ−1F2(n−ℓ)−1
)
xn.
Comparing coefficients of xn yields the identity.
Finally, it is easy to verify that the right side of part (v) satisfies the same recurrence
relation as s̃n given in part (2), or alternatively in (3).
From Part (v) of Corollary 4.2 and Equation (2.9), we conclude the following.
Theorem 4.3. Among all peaks of all non-decreasing Dyck paths, the proportion of those
that are weakly symmetric is asymptotically
lim
n→∞
s̃n
tn
=
−1 +
√
5
2
≈ 0.618034.
Notice that this coincides with the asymptotic proportion of symmetric peaks given in
Theorem 2.6.
4.2 A connection with Riordan arrays
In this section we use Riordan arrays to describe the distribution of the number of weakly
symmetric peaks in non-decreasing Dyck paths. We start by giving some background on
Riordan arrays [22]. We will say that an infinite column vector (a0, a1, . . . )T has generat-
ing function f(x) if f(x) =
∑
n≥0 anx
n, and we index rows and columns starting at 0. A
Riordan array is an infinite lower triangular matrix whose kth column has generating func-
tion g(x)f(x)k for all k ≥ 0, for some formal power series g(x) and f(x) with g(0) ̸= 0,
f(0) = 0, and f ′(0) ̸= 0. Such a Riordan array is denoted by (g(x), f(x)). If we multiply
this matrix by a column vector (c0, c1, . . . )T having generating function h(x), then the re-
sulting column vector has generating function g(x)h(f(x)). This property is known as the
fundamental theorem of Riordan arrays, or as the summation property.
The product of two Riordan arrays (g(x), f(x)) and (h(x), l(x)) is defined by
(g(x), f(x)) ∗ (h(x), l(x)) = (g(x)h(f(x)), l(f(x))) . (4.1)
Under this operation, the set of all Riordan arrays is a group [22]. The identity element is
I = (1, x), and the inverse of (g(x), f(x)) is
(g(x), f(x))−1 =
(
1/
(
g ◦ f<−1>
)
(x), f<−1>(x)
)
, (4.2)
where f<−1>(x) denotes the compositional inverse of f(x).
Let rn,k be the number of paths in Dn with exactly k weakly symmetric peaks, that is,
Dσ̃(x, y) =
∑
n,k≥1
rn,kx
nyk.
S. Elizalde et al.: Enumerating symmetric peaks in non-decreasing Dyck paths 235
By definition,
∑n
k=1 k rn,k = s̃n.
Consider the matrix R = [rn,k]n,k≥1. The first few rows of R are
R = [rn,k]n,k≥1 =

1 0 0 0 0 0 0 0 · · ·
1 1 0 0 0 0 0 0
2 2 1 0 0 0 0 0
4 5 3 1 0 0 0 0
8 12 9 4 1 0 0 0 · · ·
16 28 25 14 5 1 0 0
32 64 66 44 20 6 1 0
64 144 168 129 70 27 7 1
...
...
. . .

,
which correspond to array A105306 in [23]. Even though rows and columns of Riordan
arrays are indexed starting at 0, the elements of R are shifted so that the entry in row 0 and
column 0 is in fact r1,1. The goal of this shift is to simplify some of our formulas.
Theorem 4.4. The matrix R is a Riordan array given by
R =
(
1− x
1− 2x
,
x(1− x)
1− 2x
)
.
Proof. Multiplying the right-hand side of the equality by the vector (1, y, y2, . . . )T , which
has generating function 11−xy , and using the summation property, the resulting vector has
bivariate generating function(
1− x
1− 2x
,
x(1− x)
1− 2x
)
1
1− xy
=
1− x
1− 2x
1
1− x(1− x)
1− 2x
y
=
1− x
1− (2 + y)x+ yx2
=
Dσ̃(x, y)
xy
,
by Theorem 4.1.
Theorem 4.5. For n, k ≥ 0,
rn+1,k+1 =
n∑
ℓ=0
(
k + 1
ℓ
)(
n− ℓ
k
)
(−1)ℓ2n−k−ℓ.
Proof. From the definition of the Riordan array R, we have
rn+1,k+1 = [x
n]
1− x
1− 2x
(
x(1− x)
1− 2x
)k
=
[
xn−k
]( 1− x
1− 2x
)k+1
=
[
xn−k
]∑
n≥0
n∑
ℓ=0
(
k + 1
ℓ
)(
k + n− ℓ
n− ℓ
)
(−1)ℓ2n−ℓxn.
236 Ars Math. Contemp. 21 (2021) #P2.04 / 219–241
Let P =
[(
n
k
)]
n,k≥0, often called Pascal’s matrix, and let P = [pi,j ] be the matrix
defined by
pi,j =
{(
(i+j)/2
j
)
, if i+ j ≡ 0 (mod 2);
0, otherwise.
It is easy to show that P and P are Riordan arrays given by
P =
(
1
1− x
,
x
1− x
)
and P =
(
1
1− x2
,
x
1− x2
)
.
Theorem 4.6. The matrix R factors as R = PP .
Proof. By Equation (4.1),
PP =
(
1
1− x
,
x
1− x
)(
1
1− x2
,
x
1− x2
)
=
 1
1− x
 1
1−
(
x
1−x
)2
 , x1−x
1−
(
x
1−x
)2
 .
Simplifying,
PP =
(
1− x
1− 2x
,
x(1− x)
1− 2x
)
= R.
From above theorem and the product of matrices we obtain the following combinatorial
identities.
Theorem 4.7. For n, k ≥ 0,
rn+1,2k+1 =
⌊n2 ⌋∑
ℓ=0
(
n
2ℓ
)(
ℓ+ k
2k
)
,
rn+1,2k+2 =
⌊n2 ⌋∑
ℓ=0
(
n
2ℓ+ 1
)(
ℓ+ k + 1
2k + 1
)
.
Rogers [21], observed that every element not belonging to row 0 or column 0 in a Rior-
dan array can be expressed as a fixed linear combination of the elements in the preceding
row. The A-sequence is defined to be the sequence coefficients of this linear combination.
Similarly, Merlini et al. [19] introduced the Z-sequence, that characterizes the elements
in column 0, except for the top one. Therefore, the A-sequence, the Z-sequence and the
upper-left element completely characterize a Riordan array. We summarize this character-
ization in the following two theorems.
Theorem 4.8 ([19]). An infinite lower triangular array F = [dn,k]n,k≥0 is a Riordan array
if and only if d0,0 ̸= 0 and there exist two sequences (a0, a1, a2, . . . ), with a0 ̸= 0, and
(z0, z1, z2, . . . ) (called the A-sequence and the Z-sequence, respectively), such that
dn+1,k+1 = a0dn,k + a1dn,k+1 + a2dn,k+2 + · · · for n, k ≥ 0,
dn+1,0 = z0dn,0 + z1dn,1 + z2dn,2 + · · · for n ≥ 0.
S. Elizalde et al.: Enumerating symmetric peaks in non-decreasing Dyck paths 237
Theorem 4.9 ([18, 19]). Let F = (g(x), f(x)) be a Riordan array with inverse F−1 =
(d(x), h(x)). Then the A-sequence and the Z-sequence of F have generating functions
A(x) =
x
h(x)
, Z(x) =
1
h(x)
(1− d0,0d(x)) ,
respectively.
Next we describe the A-sequence and Z-sequence for the Riordan array R.
Theorem 4.10. If Cn denotes the n-th Catalan number, then for n, k ≥ 2,
rn,k =
n∑
ℓ=0
rn−1,k−1+ℓ cℓ,
where
cn =

1, if n = 0, 1;
(−1)n+22 Cn−2
2
, if n ≥ 2 is even;
0, otherwise.
Moreover, for n ≥ 2
rn,1 =
n∑
ℓ=0
rn−1,k−1+ℓ cℓ+1,
with initial value r1,1 = 1.
Proof. By Equation (4.2), the inverse of the matrix R is given by
R−1 =
(
1 + 2x−
√
1 + 4x2
2x
,
1 + 2x−
√
1 + 4x2
2
)
.
Therefore, by Theorem 4.9, the A-sequence and Z-sequence of the Riordan array R have
generating functions given by
A(x) =
1 + 2x+
√
1 + 4x2
2
and Z(x) =
−1 + 2x+
√
1 + 4x2
2x
.
We recall that the generating function of the Catalan numbers is given by
C(x) =
∑
n≥0
Cnx
n =
1−
√
1− 4x
2x
.
Therefore, A(x) = 1 + x + x2C(−x2) =
∑
n≥0 cnx
n, where cn is as in the statement of
the theorem. Similarly, Z(x) = 1 + xC(−x2). The recurrences from Theorem 4.8 now
give the desired result.
The first few values of the sequence cn for n ≥ 0 are
1, 1, 1, 0, −1, 0, 2, 0, −5, 0, 14, 0, −42, 0, 132, . . . .
238 Ars Math. Contemp. 21 (2021) #P2.04 / 219–241
So, the recurrence for rn,k starts as
rn−1,k−1 + rn−1,k + rn−1,k+1 − rn−1,k+3 + 2rn−1,k+5 − 5rn−1,k+7 + · · · .
Next we analyze the central diagonal of the matrix R, that is, the sequence un =
r2n+1,n+1 for n ≥ 0 (recall that the entry in row i and column j of R is ri+1,j+1). The
first few values of un are
1, 2, 9, 44, 225, 1182, 6321, 34232, 187137, 1030490, 5707449, . . . ,
which correspond to the sequence A176479 in [23].
Barry [4] proved that for any Riordan array (g(x), f(x)) = [dn,k]n,k≥0 the generating
function of its central diagonal is given by∑
n≥0
d2n,nx
n =
v(x)g(v(x))
f(v(x))
v′(x),
where
v(x) =
(
x2
f(x)
)<−1>
.
Therefore, by Theorem 4.4,∑
n≥0
unx
n =
3− x+
√
1− 6x+ x2
4
√
1− 6x+ x2
.
Other combinatorial interpretations of the sequence un are given in [23]. For example, it
counts the number of Dyck paths having exactly n peaks at height 1, n peaks at height 2,
and no other peaks. It is also equal to n+1 times the nth little Schröder number. The little
Schröder numbers have several combinatorial interpretations in terms of leaves in plane
trees, parenthesizations, and dissections of convex polygons [24].
4.3 A generating function for total weight
Let ω̃(P ) be the sum of the weights of the weakly symmetric peaks of a path P . Define the
generating function
Dω̃(x, y) =
∑
P∈D
x|P |yω̃(P ).
Theorem 4.11. The generating function Dω̃(x, y) is given by
Dω̃(x, y) =
(1− x)2xy
1− 2(1 + y)x+ 4yx2 − yx3
.
Proof. We again use the refinement of the decomposition (2.1) illustrated in Figure 5: every
non-empty non-decreasing Dyck path can be written as either XY , XT ′Y , or X∆Y T ′′,
where T ′ and T ′′ are non-decreasing Dyck paths and ∆ is a pyramid. It follows that
Dω̃(x, y) = xy + x(Dω̃(x, y)−
xy
1− xy
Dω̃(x, y) +
x
1− x
Dω̃(x, y)︸ ︷︷ ︸
(a)
− xy
1− xy
+
xy2
1− xy︸ ︷︷ ︸
(b)
))
+
xy
1− xy
Dω̃(x, y).
S. Elizalde et al.: Enumerating symmetric peaks in non-decreasing Dyck paths 239
The correction term (a) corresponds to the case where T ′ consists of a pyramid followed by
a non-empty path, whereas the term (b) corresponds to the case where T ′ is a pyramid.
From Theorem 4.11 we obtain the following corollary, whose proof is similar to that of
Corollary 4.2. Let w̃n be the sum of the weights of all weakly symmetric peaks of paths
in Dn.
Corollary 4.12. The sum of the weights of all weakly symmetric peaks in Dn satisfies the
following:
(i) The generating function for w̃n is given by
∞∑
n=1
w̃nx
n =
(1− 2x)x
(1− 3x+ x2)2
.
(ii) The sequence w̃n satisfies the recurrence relation
w̃n = 6w̃n−1 − 11w̃n−2 + 6w̃n−3 − w̃n−4 for n ≥ 5,
with initial values w̃1 = 1, w̃2 = 4, w̃3 = 13 and w̃4 = 40.
(iii) For n ≥ 1, we have the convolution
w̃n =
n∑
ℓ=0
F2ℓ−1F2(n−ℓ) =
4F2n + nL2n−1
5
.
The first few values of w̃n for n ≥ 1 are
1, 4, 13, 40, 120, 354, 1031, 2972, 8495, 24110, . . . ,
which correspond to the sequence A238846 in [23].
Let qn,k be the number of paths in Dn which have weakly symmetric weight k, that is,
Dω̃(x, y) =
∑
n,k≥1
qn,kx
nyk.
Notice that
∑n
k=1 k qn,k = w̃n. Consider the matrix defined by Q = [qn,k]n,k≥1. The first
few rows of Q are
Q = [qn,k]n,k≥1 =

1 0 0 0 0 0 0 0 · · ·
0 2 0 0 0 0 0 0
1 0 4 0 0 0 0 0
2 3 0 8 0 0 0 0
4 6 8 0 16 0 0 0 · · ·
8 13 16 20 0 32 0 0
16 28 37 40 48 0 64 0
32 60 84 98 96 112 0 128
...
...
. . .

.
Again, as in the matrix R, the elements of Q are shifted so that the entry in row 0 and
column 0 is q1,1. The proof of our last result is similar to that of Theorem 4.4.
Theorem 4.13. The matrix Q is a Riordan array given by
Q =
(
1− 2x+ x2
1− 2x
,
2x− 4x2 + x3
1− 2x
)
.
240 Ars Math. Contemp. 21 (2021) #P2.04 / 219–241
ORCID iDs
Sergi Elizalde https://orcid.org/0000-0003-4116-2455
Rigoberto Flórez https://orcid.org/0000-0002-3644-9358
José Luis Ramı́rez https://orcid.org/0000-0002-8028-9312
References
[1] W. Asakly, Enumerating symmetric and non-symmetric peaks in words, Online J. Anal. Comb.
(2018), 7, https://hosted.math.rochester.edu/ojac/articles.html.
[2] E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci
numbers, Discrete Math. 170 (1997), doi:10.1016/s0012-365x(97)82778-1.
[3] E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recur-
rence relations, in: TAPSOFT ’93: theory and practice of software development (Orsay,
1993), Springer, Berlin, volume 668 of Lecture Notes in Comput. Sci., pp. 282–298, 1993,
doi:10.1007/3-540-56610-4 71.
[4] P. Barry, On the central coefficients of Riordan matrices, J. Integer Seq. 16 (2013), Ar-
ticle 13.5.1, 12pp, https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry1/
barry242.html.
[5] E. Czabarka, R. Flórez and L. Junes, Some enumerations on non-decreasing Dyck paths, Elec-
tron. J. Combin. 22 (2015), Paper 1.3, 22, doi:10.37236/3941.
[6] E. Czabarka, R. Flórez, L. Junes and J. L. Ramı́rez, Enumerations of peaks and valleys on
non-decreasing Dyck paths, Discrete Math. 341 (2018), 2789–2807, doi:10.1016/j.disc.2018.
06.032.
[7] A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math. 137
(1995), 155–176, doi:10.1016/0012-365x(93)e0147-v.
[8] E. Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes
and ordered trees of height at most three, Theoret. Comput. Sci. 307 (2003), 319–325, doi:
10.1016/s0304-3975(03)00222-6, random generation of combinatorial objects and bijective
combinatorics.
[9] S. Elizalde, Measuring symmetry in lattice paths and partitions, Sém. Lothar. Combin. 84B
(2020), Art. 26, 12pp, https://www.mat.univie.ac.at/˜slc/.
[10] S. Elizalde, The degree of symmetry of lattice paths, Ann. Comb. (2021), doi:10.1007/
s00026-021-00551-6.
[11] S. Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, Discrete Math. 344 (2021),
112364, doi:10.1016/j.disc.2021.112364.
[12] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cam-
bridge, 2009.
[13] R. Flórez, L. Junes and J. L. Ramı́rez, Enumerating several aspects of non-decreasing Dyck
paths, Discrete Math. 342 (2019), 3079–3097, doi:10.1016/j.disc.2019.06.018.
[14] R. Flórez, L. Junes and J. L. Ramı́rez, Counting asymmetric weighted pyramids in non-
decreasing Dyck paths, Australas. J. Combin. 79 (2021), 123–140, https://ajc.maths.
uq.edu.au/?page=get_volumes&volume=79.
[15] R. Flórez and J. L. Ramı́rez, Some enumerations on non-decreasing Motzkin paths, Aus-
tralas. J. Combin. 72 (2018), 138–154, https://ajc.maths.uq.edu.au/?page=
get_volumes&volume=72.
S. Elizalde et al.: Enumerating symmetric peaks in non-decreasing Dyck paths 241
[16] R. Flórez and J. L. Ramı́rez, Enumerating symmetric and asymmetric peaks in Dyck paths,
Discrete Math. 343 (2020), 112118, doi:10.1016/j.disc.2020.112118.
[17] R. Flórez and J. L. Ramı́rez, Enumerations of rational non-decreasing Dyck paths with integer
slope, Graphs and Combinatorics (2021), doi:10.1007/s00373-021-02392-9.
[18] T.-X. He and R. Sprugnoli, Sequence characterization of Riordan arrays, Discrete Math. 309
(2009), 3962–3974, doi:10.1016/j.disc.2008.11.021.
[19] D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations
of Riordan arrays, Canad. J. Math. 49 (1997), 301–320, doi:10.4153/cjm-1997-015-x.
[20] H. Prodinger, Words, Dyck paths, trees, and bijections, in: Words, semigroups, & transductions,
World Sci. Publ., River Edge, NJ, pp. 369–379, 2001, doi:10.1142/9789812810908 0028.
[21] D. G. Rogers, Pascal triangles, Catalan numbers and renewal arrays, Discrete Math. 22 (1978),
301–310, doi:10.1016/0012-365x(78)90063-8.
[22] L. W. Shapiro, S. Getu, W. Woan and L. Woodson, The Riordan group, Discrete Appl. Math.
34 (1991), 229–239, doi:10.1016/0166-218x(91)90088-e.
[23] N. J. A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org/.
[24] R. P. Stanley, Hipparchus, Plutarch, Schröder, and Hough, Amer. Math. Monthly 104 (1997),
344–350, doi:10.1080/00029890.1997.11990645.
A Appendix. Notation tables
type of peaks
symmetric asymmetric weakly symmetric all
number of such peaks in P σ(P ) σ(P ) σ̃(P ) τ(P )
total number over Dn sn sn s̃n tn
vector of peak weights of P ω(P ) = (ω1(P ), . . . )
sum of peak weights of P ω(P ) ω̃(P )
total sum of weights over Dn wn w̃n
total sum of heights over Dn hn
Table 1: Summary of notation for peak statistics.
Notation Page Notation Page Notation Page
Dn, D 220 ι(P ), ν(P ) 222 rn,k 234
An, Bn, Cn,i 221 S(x) 223 qn,k 239
Table 2: Other notation, along with the page where it is first introduced.

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P2.05 / 243–257
https://doi.org/10.26493/1855-3974.2351.07b
(Also available at http://amc-journal.eu)
Density results for Graovac-Pisanski’s
distance number
Lowell Abrams
University Writing Program and Department of Mathematics,
The George Washington University, Washington, DC 20052, USA
Lindsey-Kay Lauderdale *
Department of Mathematics, Towson University, Towson, MD 21252, USA
Received 2 June 2020, accepted 13 May 2021, published online 30 October 2021
Abstract
The sum of distances between every pair of vertices in a graph G is called the Wiener
index of G. This graph invariant was initially utilized to predict certain physico-chemical
properties of organic compounds. However, the Wiener index of G does not account for
any of its symmetries, which are also known to effect these physico-chemical properties.
Graovac and Pisanski modified the Wiener index of G to measure the average distance
each vertex is displaced under the elements of the symmetry group of G; we call this the
Graovac-Pisanski (GP) distance number of G. In this article, we prove that the set of all
GP distance numbers of graphs with isomorphic symmetry groups is dense in a half-line.
Moreover, for each finite group Γ and each rational number q within this half-line, we
present a construction for a graph whose GP distance number is q and whose symmetry
group is isomorphic to Γ. This construction results in graphs whose vertex orbits are not
connected; we also consider an analogous construction which ensures that all vertex orbits
are connected.
Keywords: Wiener index, distance number, Graovac-Pisanski index, graph automorphism group,
chemical graph theory.
Math. Subj. Class. (2020): 05C12, 05C25, 05C35, 05C92
*Corresponding author. The author is the Jess and Mildred Fisher Endowed Professor of Mathematics in the
Fisher College of Science and Mathematics at Towson University and is partially supported by this endowment.
E-mail addresses: labrams@gwu.edu (Lowell Abrams), llauderdale@towson.edu (Lindsey-Kay Lauderdale)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
244 Ars Math. Contemp. 21 (2021) #P2.05 / 243–257
1 Introduction
Throughout this article, all graphs considered are simple and finite, and all groups consid-
ered are finite. We let V (G) and E(G) denote the vertex set and edge set of a graph G,
respectively. The Wiener index of G is the sum of all distances between pairs of vertices in
G, namely
W (G) :=
1
2
∑
u∈V (G)
∑
v∈V (G)
d(u, v),
where d(u, v) is the length of a shortest path between u and v in G. This graph invariant
was original defined by Wiener [14], where he considered graphical representations of
molecules. In particular, each vertex in V (G) represents an atom of a molecule and each
edge in E(G) represents a bond between atoms. Wiener [14] used this graph invariant to
establish an equation that predicts the boiling points of paraffin molecules.
Other physico-chemical properties of organic molecules, including refractive index,
heat of isomerization, heat of vaporization, density, surface tension, viscosity, and chro-
matographic retention time, were later linked to the Wiener index [5]. Consequently, the
Wiener index of classes of compounds, including benzenoids [6], chains [13], and trees [2],
were calculated; Mohar and Pisanski [11] described numerous algorithms that compute the
Wiener index of a graph in general. An interested reader can see [10] and the references
within for more results on this graph invariant.
The symmetries of molecules are known to effect certain physico-chemical properties
of organic compounds [12]. In this article, we are interested in a modification ofW (G) that
accounts for these symmetries of G. Recall the set of adjacency-preserving permutations
of V (G) is called the automorphism group of G and is denoted by AutG. Graovac and
Pisanski [4] defined the distance number of G to be the average
δ(G) :=
1
|AutG||V (G)|
∑
u∈V (G)
∑
σ∈AutG
d
(
u, σ(u)
)
.
We call this invariant the Graovac and Pisanski (GP) distance number. Graovac and Pisan-
ski [4] established some basic properties of δ(G) and computed δ(G) provided G is a path,
cube, cycle graph, complete bipartite graph, or lattice graph. Note that the results in this
article only hold for the GP distance number and not what is currently referred to in the
literature as the Graovac-Pisanski index, namely Ŵ (G) := 12 |V (G)|
2δ(G).
The GP distance number and the GP index were the subject of prior research by a num-
ber of authors. For example, Ashrafi and Shabani [1] computed the GP index of graphs that
resulted via standard graph operations on trees. The GP index of truncation graphs, Thorn
graphs, and caterpillars were calculated by Iranmanesh and Shabani [7]. Additionally, Knor
et al. [8] considered the maximum GP index among all graphs of a fixed order. Note that
these results on the GP index have direct implications for the GP distance number.
In this article, we consider a dual problem to that of computing the maximum GP dis-
tance number among all graphs of a fixed order; this approach better represents how the GP
distance numbers of classes of compounds can predict their physico-chemical properties.
Specifically, for a given group Γ, we establish the possible values of δ(G) among all graphs
G with AutG ∼= Γ. When AutG ∼= Γ, we call G a Γ -graph. Our main result is stated
below.
L. Abrams and L.-K. Lauderdale: Density results for Graovac-Pisanski’s distance number 245
Theorem 1.1. Given a group Γ, define
DΓ := {δ(G) : G is a Γ-graph}.
The setDΓ is dense in (inf(DΓ),∞). Moreover, for each rational number q ∈(inf(DΓ),∞),
there exists a Γ-graph G with δ(G) = q.
Our results will establish the exact value of inf(DΓ), as well as give two infinite families
of Γ-graphs whose GP distance numbers equal this infimum.
We prove Theorem 1.1 by constructing a family of Γ-graphs whose vertex orbits under
the Γ-action are not necessarily connected. Consideration of Γ-graphs whose vertex orbits
are all connected yields a more restricted result, Theorem 6.3, in which the interval of
potential GP distance numbers is finite and, moreover, not every rational number in the
interval can be obtained as a GP distance number of a graph in the constructed family.
This article is organized as follows. In Section 2, we describe an alternative formula
to compute δ(G) for a given graph G, and then use it to state bounds on this invariant in
terms of W (G). Next, for a given group Γ, we construct an infinite family of Γ-graphs
in Section 3. The results of Section 4 establish their associated GP distance numbers, and
in Section 5, we present a proof of our main result, Theorem 1.1. Finally, we conclude in
Section 6 with a discussion leading to Theorem 6.3.
2 Preliminaries
The definition of δ(G) for a graphG can be reformulated by considering the orbits of V (G)
under the action of AutG. For ease of notation, define
d(v, V ) :=
∑
u∈V
d(v, u),
where v ∈ V ⊆ V (G). Graovac and Pisanski connected this alternative expression for
δ(G) to the Wiener index of the vertex orbits of G; we state their results below.
Theorem 2.1 (Graovac and Pisanski [4]). If V0, V1, . . . , Vp−1 are the orbits of V (G) de-
termined by AutG and vi ∈ Vi for each i ∈ {0, 1, . . . , p− 1}, then
δ(G) =
1
|V (G)|
p−1∑
i=0
d(vi, Vi) =
2
|V (G)|
p−1∑
i=0
W (Vi)
|Vi|
. (2.1)
For the remainder of this article, we will use Equation (2.1) to compute the GP distance
number of a given graph. As simple examples, we calculate the GP distance numbers of
both complete graphs and paths below.
Example 2.2. Let Kn denote the complete graph with n vertices. If v ∈ V (Kn), then
δ(Kn) =
1
n
d
(
v, V (Kn)
)
=
n− 1
n
,
where the first equality holds because Kn is vertex-transitive (i.e., p = 1) and the second
equality holds because v is adjacent to all vertices in V (Kn) except itself.
246 Ars Math. Contemp. 21 (2021) #P2.05 / 243–257
Example 2.3. Let Pn denote the path of order n ≥ 2, and label this graph so that uiui+1 ∈
E(Pn) for each i ∈ {0, 1, . . . , n−2}. Since Pn is a Z2-graph, there are
⌊
n+1
2
⌋
vertex orbits
under the action of Aut(Pn). Set p =
⌊
n+1
2
⌋
and label these orbits by V0, V1, . . . , Vp−1
so that ui ∈ Vi for each i ∈ {0, 1, . . . , p − 1}. Under these assumptions, ui and un−1−i
comprise the orbit Vi and
d(ui, Vi) = d(ui, ui) + d(ui, un−1−i) = 0 + (n− 1− 2i) = n− 1− 2i
for all i ∈ {0, 1, . . . , p− 1}. Therefore,
δ(Pn) =
1
n
p−1∑
i=0
(n− 1− 2i︸ ︷︷ ︸
d(ui,Vi)
) =
1
n
[
p(n− 1)− 2
(
1
2
(p− 1)p
)]
=
{
n
4 if n is even
n2−1
4n if n is odd,
where the first equality holds by Equation (2.1) and the last equality holds because p =⌊
n+1
2
⌋
.
Paths and complete graphs represent important families of graphs in the context of
the Wiener index. In particular, Knor, Škrekovski, and Tepeh [9] observed that if G is a
connected graph of order n, then(
n
2
)
=W (Kn) ≤W (G) ≤W (Pn) =
(
n+ 1
3
)
. (2.2)
For a given graph G, this observation allows us to place simple bounds on δ(G) in terms of
W (G).
Lemma 2.4. Let G be a graph. If the induced subgraph on each vertex orbit of G under
the action of AutG is connected with order k, then
k − 1
k
≤ δ(G) ≤ k
2 − 1
3k
.
Proof. Let V0, V1, . . . , Vp−1 denote the vertex orbits of G under the action of AutG. Be-
cause each orbit has size k and |V (G)| = kp, Equation (2.1) implies
δ(G) =
2
k2p
p−1∑
i=0
W (Vi).
Combining the equation above with Equation (2.2), we obtain
k − 1
k
=
2
k2p
· p
(
k
2
)
≤ δ(G) ≤ 2
k2p
· p
(
k + 1
3
)
=
k2 − 1
3k
,
as desired.
The lower bound stated in Lemma 2.4 is realized by G = Kn (see Example 2.2). As
demonstrated by Example 2.3, the upper bound in Lemma 2.4 is not realized by G = Pn.
Moreover, we conjecture this upper bound is not sharp under the stated assumptions.
For a given group Γ, Theorem 1.1 implies that there is no maximum value of δ(G)
among all Γ-graphs. In fact, the values of GP distance numbers of graphs in general are not
bounded; Lemma 2.4 foreshadows how these graphs must be built. To construct a family of
graphs with arbitrarily large GP distance numbers, the induced subgraphs on some of the
vertex orbits must be disconnected. We continue by constructing such graphs in the next
section.
L. Abrams and L.-K. Lauderdale: Density results for Graovac-Pisanski’s distance number 247
3 Graph construction
To investigate the set DΓ, we will construct an infinite family of Γ-graphs, parameterized
by non-negative integers a and c, from a given Γ-graph G. Specifically, each graph φac (G)
in this family will be constructed by appending to G, in a special way, a anti-cliques of
order |V (G)| and c cliques of order |V (G)| (see Definition 3.1 below). Every vertex in
φac (G) will have two labels; the superscript of a vertex indicates its distance to G and the
subscript label represents the vertex in G it is closest to. The parameters a and c are used
in Section 5 to increase and decrease the value of δ
(
φac (G)
)
, respectively.
Definition 3.1. Let Γ be a group, and suppose G is a Γ-graph with V (G) = {u00, u01, . . . ,
u0n−1}. Given a, c ∈ N, construct a new graph from G, denoted φac (G), with n(1 + a+ c)
vertices and
E(G) + an+ c
(
n+
1
2
n(n− 1)
)
edges as follows:
1. For each i ∈ {0, 1, . . . , n−1}, attach a path of length a to vertex u0i and sequentially
label the vertices on that path by u0i , u
1
i , u
2
i , . . . , u
a
i .
2. For each i ∈ {0, 1, . . . , n − 1}, attach a path of length c to u0i and sequentially
label the vertices w0i , w
1
i , w
2
i , . . . , w
c
i , where w
0
i := u
0
i ; thereupon, provided c ̸= 0,
include the edgeswki w
k
j for all k ∈ {1, 2, . . . , c} and distinct i, j ∈ {0, 1, . . . , n−1}.
Observe that G and φac (G) are equal when a = 0 = c. The graph φ
a
0(Cn) is depicted
in Figure 1, where Cn denotes the cycle graph of order n. We discuss the structure of the
vertex orbits of φac (G) under the action of Aut
(
φac (G)
)
in the following remark.
Remark 3.2. Let Γ be a group. IfG and φac (G) are both Γ-graphs, then the vertex orbits of
φac (G) under its Γ-action depend on the vertex orbits of G under its Γ-action. In particular,
let V0, V1, . . . , Vp−1 denote the vertex orbits of G under its Γ-action. By construction, we
obtain a + c vertex orbits of φac (G) under its Γ-action for each Vi, so, in total, φ
a
c (G) has
(1 + a+ c)p vertex orbits under its Γ-action.
We continue with an example in which we compute the value of δ
(
φa0(Cn)
)
for all
a, n ∈ N with n ≥ 3.
Example 3.3. Let us compute the GP distance number of the graph φa0(Cn), which is
illustrated in Figure 1. Recall that Cn is vertex-transitive. If Aj is the orbit of u
j
0 under
the dihedral action of Aut
(
φa0(Cn)
) ∼= D2n for all j ∈ {0, 1, . . . , a}, then A0, A1, . . . , Aa
form a partition of V
(
φa0(Cn)
)
. We claim the value of d(uj0, A
j) depends on the parity of
n.
Consider the vertices uj0, u
j
i ∈ Aj , where i ∈ {1, 2, . . . , n− 1} and j ∈ {0, 1, . . . , a}.
A shortest path between these vertices is constructed by concatenating the uj0, u
0
0-path of
length j, a u00, u
0
i -path of minimum length inCn, and the u
0
i , u
j
i -path of length j. Therefore,
if n = 2ℓ+ 1 is odd, then
d(uj0, A
j) =
n−1∑
i=1
d(uj0, u
j
i ) = 2
ℓ∑
k=1
(2j + k) = 4jℓ+ ℓ(ℓ+ 1) = 2(n− 1)j + n
2 − 1
4
,
248 Ars Math. Contemp. 21 (2021) #P2.05 / 243–257
u03
u02
u01
u00 u0n−1
u13
u12
u11
u10
u1n−1
ua−13
ua−12
ua−11
ua−10
ua−1n−1
ua3
ua2
ua1
ua0
uan−1
Figure 1: Depiction of the graph φa0(Cn).
and, if n = 2ℓ is even, then
d(uj0, A
j) =
n−1∑
i=1
d(uj0, u
j
i ) = (2j+ ℓ)+2
ℓ−1∑
k=1
(2j+k) = 4jℓ−2j+ ℓ2 = 2(n−1)j+ n
2
4
.
Since |V
(
φa0(Cn)
)
| = n(1 + a), we have that
δ
(
φa0(Cn)
)
=
1
n(1 + a)
a∑
j=0
d(uj0, A
j) =

4(n− 1)a+ n2 − 1
4n
if n = 2ℓ+ 1
4(n− 1)a+ n2
4n
if n = 2ℓ.
The statements in Remark 3.2 are based on the assumption that G and φac (G) have
isomorphic automorphism groups. The following proposition proves that this is almost
always the case.
Proposition 3.4. Let Γ be a group. IfG is a nontrivial connected Γ-graph and either a ̸= 0
or G is not a complete graph, then φac (G) is also a Γ-graph.
Proof. To prove that Γ is isomorphic to a subgroup of Aut
(
φac (G)
)
, we note that each
element of AutG induces a (subscript) label-preserving automorphism of φac (G). In
particular, if σ ∈ AutG, then σ induces a permutation on {0, 1, . . . , n − 1}, denoted
ρσ , such that ρσ(i) is the subscript of σ(u0i ) for all i ∈ {0, 1, . . . , n − 1}. Define the
map πσ : V
(
φac (G)
)
→ V
(
φac (G)
)
by πσ(u
j
i ) = u
j
ρσ(i)
and πσ(wki ) = w
k
ρσ(i)
for all
j ∈ {0, 1, . . . , a} and k ∈ {0, 1, . . . , c}. Since πσ preserves the adjacency relations in
φac (G) and Γ ∼= {πσ : σ ∈ AutG}, Γ is isomorphic to a subgroup of Aut
(
φac (G)
)
.
L. Abrams and L.-K. Lauderdale: Density results for Graovac-Pisanski’s distance number 249
It remains to prove that any element of Aut
(
φac (G)
)
is equal to πσ for some σ ∈
AutG. Clearly if a = 0 = c, then φac (G) = G and the proposition holds. Thus, in what
follows we assume that at least one of a or c is nonzero.
Suppose a ̸= 0, and consider the image of the degree-1 vertex uai under ψ ∈
Aut
(
φac (G)
)
, where i ∈ {0, 1, . . . , n − 1}. Since the only vertices in φac (G) that have
degree 1 are of the form uaℓ , it follows that ψ(u
a
i ) = u
a
ℓ for some ℓ ∈ {0, 1, . . . , n − 1}.
In turn, ψ
(
ua−1i
)
= ua−1ℓ because u
a−1
i and u
a−1
ℓ are the only neighbors of u
a
i and
uaℓ in φ
a
c (G), respectively. Proceeding by induction, assume that ψ
(
uj
′
i
)
= uj
′
ℓ for all
j′ ∈ {j, j+1, . . . , a}. If j ≥ 1, then uji has exactly two neighbors, namely u
j+1
i and u
j−1
i ,
while uj+1ℓ and u
j−1
ℓ are the only neighbors of vertex u
j
ℓ . In this case, ψ
(
uj−1i
)
= uj−1ℓ as
ψ
(
uj+1i
)
= uj+1ℓ by induction. Therefore, ψ
(
uji
)
= ujℓ for all j ∈ {0, 1, . . . , a}.
Now define W k := {wk0 , wk1 , . . . , wkn−1} for each k ∈ {0, 1, . . . , c}. If c ̸= 0, then
each vertex in W c has degree n, and thus ψ(wci ) is also a vertex of degree n in φ
a
c (G).
The only vertices in φac (G) that have degree n are in W
0 ∪W c. However, each element
in W c is adjacent to at least n − 1 vertices of degree n, and because G is not a complete
graph or a ̸= 0, each vertex in W 0 = V (G) is adjacent to at most n − 2 vertices of
degree n. Consequently, W c is ψ-invariant; assume that ψ(wci ) = w
c
m for some m ∈
{0, 1, . . . , n − 1}. Both wci and wcm have exactly one neighbor that is not an element of
W c; hence, ψ
(
wc−1i
)
= wc−1m and we claim that ψ
(
wki
)
= wkm for all k ∈ {0, 1, . . . , c}.
Since this claim holds for k ∈ {c − 1, c}, we again proceed by induction. Assume that
ψ
(
wk
′
i
)
= wk
′
m for all k
′ ∈ {k, k + 1, . . . , c}. When k ≥ 1, the only neighbors of wki not
in W k are wk+1i and w
k−1
i ; moreover, w
k+1
m and w
k−1
m are the only neighbors of w
k
m not
in W k. Since ψ
(
wk+1i
)
= wk+1m by induction, it follows that ψ
(
wk−1i
)
= wk−1m and the
claim holds.
Our work above proves that ψ
(
uji
)
= ujℓ for all j ∈ {0, 1, . . . , a} and that ψ
(
wki
)
=
wkm for all k ∈ {0, 1, . . . , c}. Since u0i = w0i by definition of φac (G), we have ℓ =
m. Consequently, there exists σ ∈ Aut(G) such that ψ = πσ , and φac (G) is also a Γ-
graph.
We are now ready to compute the GP distance number of φac (G) when the graphs G
and φac (G) have isomorphic automorphism groups.
4 GP distance number of φac(G)
If G and φac (G) have isomorphic automorphism groups, then the value of δ
(
φac (G)
)
nat-
urally depends on the value δ(G); however, it also depends on the value of c in a special
way. In particular, if c ̸= 0, then the distance between any two vertices of G is at most 3.
Recalling that V0, V1, . . . , Vp−1 are the vertex orbits of G under the action of AutG, we
define
δ′(c,G) :=
{
δ(G) if c = 0
δ3(G) if c ̸= 0,
where
δ3(G) :=
1
|V (G)|
p−1∑
i=0
d3(ui, Vi) and d3(ui, Vi) :=
∑
u∈Vi
min{d(ui, u), 3}.
With this notation in hand, we compute the value of δ
(
φac (G)
)
below.
250 Ars Math. Contemp. 21 (2021) #P2.05 / 243–257
Proposition 4.1. Let Γ be a group, and assume that G and φac (G) are both Γ-graphs. If G
has order n and p vertex orbits under the action of AutG, then
δ
(
φac (G)
)
=
(n− p)(a2 + a+ c) + n(a+ 1)δ′(c,G)
n(1 + a+ c)
.
Proof. Let V0, V1, . . . , Vp−1 denote the p vertex orbits of G under the action of AutG.
After a possible relabelling of V (G), assume that u0i ∈ Vi for all i ∈ {0, 1, . . . , p− 1}. For
each Vi, there are a+ c associated vertex orbits of φac (G) under the action of Aut
(
φac (G)
)
by Remark 3.2; label these orbits by A1i , A
2
i , . . . , A
a
i and C
1
i , C
2
i , . . . , C
c
i , where u
j
i ∈ A
j
i
for j ∈ {1, 2, . . . , a} and wki ∈ Cki for k ∈ {1, 2, . . . , c}. Under these assumptions
δ
(
φac (G)
)
=
1∣∣V (φac (G))∣∣
 a∑
j=0
p−1∑
i=0
d(uji , A
j
i ) +
c∑
k=1
p−1∑
i=0
d(wki , C
k
i )
 , (4.1)
where A0i = Vi for i ∈ {0, 1, . . . , p − 1}. We evaluate each of these sums in one of the
following cases.
First, observe that d(wki , C
k
i ) = |Cki | − 1 for all k ∈ {1, 2, . . . , c} as the induced
subgraph on Cki is a clique. Since
c∑
k=1
|Cki | = c|Vi| and
p−1∑
i=0
|Vi| = |V (G)| = n,
it follows that
c∑
k=1
p−1∑
i=0
d(wki , C
k
i ) =
p−1∑
i=0
c∑
k=1
(|Cki | − 1) =
p−1∑
i=0
c(|Vi| − 1) = c(n− p). (4.2)
For the second case, if u0ℓ ∈ A0i , then a shortest path between vertices u
j
i ∈ A
j
i and
ujℓ ∈ A
j
i is constructed by concatenating the following three paths:
1. the uji , u
0
i -path in φ
a
c (G) of length j;
2. a u0i , u
0
ℓ -path of minimum length in G if c = 0 or in φ
0
1(G) provided c ̸= 0; and
3. the u0ℓ , u
j
ℓ-path in φ
a
c (G) of length j.
It follows that
d(uji , A
j
i ) = 2j(|A
j
i | − 1) + d
′(c, u0i , A
0
i ),
where
d′(c, u0i , A
0
i ) :=
{
d(u0i , A
0
i ) if c = 0
d3(u
0
i , A
0
i ) if c ̸= 0.
Since |Aji | = |Vi| for all j ∈ {0, 1, . . . , a}, we have
a∑
j=0
p−1∑
i=0
d(uji , A
j
i ) =
p−1∑
i=0
a∑
j=0
(
2j(|Aji | − 1) + d
′(c, u0i , A
0
i )︸ ︷︷ ︸
d(uji ,A
j
i )
)
=
p−1∑
i=0
(
2
1
2
a(a+ 1)(|Vi| − 1) + (a+ 1)d′(c, u0i , A0i )
)
= a(a+ 1)(n− p) + n(a+ 1)δ′(c,G).
(4.3)
L. Abrams and L.-K. Lauderdale: Density results for Graovac-Pisanski’s distance number 251
Since
∣∣V (φac (G))∣∣ = n(1 + a + c), combining Equations (4.2) and (4.3) with Equa-
tion (4.1) yields
δ
(
φac (G)
)
=
(n− p)(a2 + a+ c) + n(a+ 1)δ′(c,G)
n(1 + a+ c)
,
as desired.
Consider the value of δ
(
φac (G)
)
given in Proposition 4.1 for a fixed graph G. The pa-
rameters a and c can be used to increase and decrease the value of δ
(
φac (G)
)
, respectively;
that is,
lim
a→∞
δ
(
φac (G)
)
= ∞ and lim
c→∞
δ
(
φac (G)
)
=
n− p
n
,
provided c and a are fixed, respectively. There are several infinite families of order-n
graphs whose GP distance numbers are equal to n−pn , where p is the number of vertex
orbits under the action of their respective automorphism groups. These families arise when
the induced subgraph on every vertex orbit is a clique; Example 2.2 demonstrates that
the complete graphs Kn comprise one such family. The following example establishes a
second such family of graphs that, in contrast, are not vertex-transitive under the action of
their respective automorphism groups.
Example 4.2. Let Zk denote the cyclic group of order k, where k ≥ 3. In this example, we
construct an infinite family of Zk-graphs, denoted by Gn; each graph Gn has order n = 6k
and p = 6 edge orbits under the action of Aut(Gn). We will prove that δ(Gn) = n−pn .
Define the order-7 gadget graph H with edge set
E(H) =
{
h0h1, h1h2, h1h4, h2h3, h2h5, h5h6
}
,
which is depicted in Figure 2(A). Let Ck denote the cycle graph of order k, and label its
edges so that vivi+1 ∈ E(Ck) for all i ∈ {0, 1, . . . , k − 2}. Replace each edge in Ck with
a copy of H , where the vertices vi and vi+1 are identified with h0 and h3, respectively; we
call the resulting graph H(k). The graph H(4) is illustrated in Figure 2(B). Observe that
H(k) is a Zk-graph with order n = 6k, which has six size-k vertex orbits under the action
of Aut
(
H(k)
)
.
Finally, we construct the graph Gn by including the 3(k − 1)k edges necessary to turn
each vertex orbit of H(k) into a clique. By design Gn is also a Zk-graph, where each of its
six edge orbits under the action of Aut(Gn) is a clique of order k. Its GP distance number
is
δ(Gn) =
n− 6
n
=
k − 1
k
,
as desired.
5 Proof of Theorem 1.1
In this section, we will prove our main result, Theorem 1.1. To do so, we make use of the
following proposition.
Proposition 5.1. Let Γ be a group, and suppose G is a nontrivial connected Γ-graph
with order n and p vertex orbits under the action of AutG. For any rational number
q ∈ (n−pn ,∞), there exist a, c ∈ N such that δ
(
φac (G)
)
= q.
252 Ars Math. Contemp. 21 (2021) #P2.05 / 243–257
h0
h1
h2
h3
h4
h5
h6
(a) The gadget graph H (b) The Z4-graph H(4)
Figure 2: Depictions of the graphs H and H(4), which were defined in Example 4.2.
Proof. Choose r, s ∈ N such that q = rs , and define
b := 2max
{
1,
⌈
nr − nsδ3(G)
(n− p)s
⌉}
.
Let
a :=
(
nr − (n− p)s
)
b− 1, (5.1)
and notice that a ≥ 0 because n−pn < q =
r
s . Now define
c := −
(
nr − (n− p)as− nsδ3(G)
)
b. (5.2)
Since G has order n, nδ3(G) is an integer, and thus c is as well. In fact, c ∈ N because the
inequality
a =
(
nr − (n− p)s
)
b− 1 ≥ b− 1 ≥ 1
2
b ≥ nr − nsδ3(G)
(n− p)s
implies that
nr − (n− p)as− nsδ3(G)
is nonpositive. Consequently, our choices of a and c are valid when considering the graph
φac (G), and since a ̸= 0, φac (G) is also a Γ-graph by Proposition 3.4. Proposition 4.1 then
implies that the GP distance number of φac (G) is
δ
(
φac (G)
)
=
(n− p)(a2 + a+ c) + n(a+ 1)δ′(c,G)
n(1 + a+ c)
.
A tedious algebraic computation shows that combining our choices of a and c
(
stated in
Equations (5.1) and (5.2)
)
with the equation above yields δ
(
φac (G)
)
= rs , as desired.
The equation δ
(
φac (G)
)
= q that appears in Proposition 5.1 does not have a unique
solution. In fact, taking any integer value of b greater than the one specified in the proof
will also yield a choice of a and c which satisfies the theorem. We now provide an example
showing that it is also possible to obtain smaller values of a and c which work.
L. Abrams and L.-K. Lauderdale: Density results for Graovac-Pisanski’s distance number 253
Example 5.2. Let G be K4 − e for any edge e of K4, in which case AutG ∼= Z2 × Z2
and δ(G) = 34 . Applying the proof of Proposition 5.1 with q =
4
5 , we obtain b = 2 and
then a = 11 and c = 218. However, we can in fact take b = 1 and still obtain a solution to
δ
(
φac (G)
)
= q, namely a = 5 and c = 49. The solution with the smallest possible values
of both a and c, not obtainable through the construction in that proof, is a = 1 and c = 3.
We conclude this section with a proof of our main result.
Proof of Theorem 1.1. Let the group Γ be given, and recall that
DΓ := {δ(G) : G is a Γ-graph}.
Frucht [3] proved that there exists a graph whose automorphism group is isomorphic to Γ;
among all such Γ-graphs G with order nG and with pG vertex orbits under the action of
AutG, choose G so that nG−pGnG is minimal. Under these assumptions, if G has order n
and p vertex orbits under the action of AutG, then
inf(DΓ) =
n− p
n
.
For each rational number q ∈ (inf(DΓ),∞), there exists a Γ-graph with GP distance num-
ber equal to q by Proposition 5.1. Consequently, DΓ is dense in (inf(DΓ),∞), as the
rational numbers are dense in this interval. The result now follows.
6 Graphs with connected vertex orbits
For a given group Γ, Theorem 1.1 proved that there was no maximum value of δ(G) among
all Γ-graphs; such arbitrarily large values of δ(G) were obtained from graphs with discon-
nected induced subgraphs on the vertex orbits of G under the action of AutG. If we
assume that the induced subgraph on every vertex orbit of G under the action of AutG
is connected, then we obtain a bounded interval of potential GP distance numbers. While
these stricter assumptions preserve density, we no longer can produce a graph with a given
GP distance number using a similar construction. We will conclude this article with a
result analogous to that of Theorem 1.1 which makes the aforementioned connectedness
assumption.
Let the group Γ be given. If a Γ-set V has size n, let GΓ,n denote any choice of a
connected graph on the Γ-set V which has a Γ-action compatible with the Γ-action on V
and has the maximum possible GP distance number among all such graphs. Note thatGΓ,n
need not be a Γ-graph. We use δΓ(GΓ,n) to denote the GP distance number obtained by
considering the Γ-action on GΓ,n rather than the action of Aut(GΓ,n).
Suppose now that G is a Γ-graph with p orbits V0, V1, . . . , Vp−1 of sizes n0, n1, . . . ,
np−1, respectively. Each orbit itself has a Γ-action, so we consider the graphs GΓ,n0 , . . . ,
GΓ,np−1 ; let ĜΓ denote GΓ,n0 ⊔ · · · ⊔GΓ,np−1 , where ⊔ denotes disjoint union. Define
δ̂(G) :=
1
n0 + · · ·+ np−1
p−1∑
i=0
niδΓ(GΓ,ni);
δ̂(G) is the maximum possible GP distance number relative to Γ for all graphs with a
Γ-action and vertex set the Γ-set V (G). Note, however, that Aut(ĜΓ) may contain an
isomorphic copy of Γ as a proper subgroup.
254 Ars Math. Contemp. 21 (2021) #P2.05 / 243–257
Definition 6.1. Let Γ be a group, and supposeG is a Γ-graph with vertices u00, u01, . . . , u0n−1
and vertex orbits V0, V1, . . . , Vp−1. Without loss of generality, we assume that u0i ∈ Vi for
each i ∈ {0, 1, . . . , p − 1}. We define a new graph φ̂ac (G) iteratively with respect to the
natural numbers c and a as follows. Given φ̂ac (G), define φ̂
a+1
c (G) to be the graph obtained
by carrying out the following steps:
1. introduce new vertices ua+10 , u
a+1
1 , . . . , u
a+1
n−1; we refer to these vertices as being in
“level a+ 1”;
2. connect these new vertices with new edges uai u
a+1
i for each i ∈ {0, 1, . . . , n − 1};
and
3. for each orbit Vi, add new edges to build a copy of the Γ-graph GΓ,|Vi| on the orbit
of vertices in level a+ 1 corresponding to the Γ-set Vi.
Given φ̂ac (G), let w
0
i := u
0
i for each i ∈ {0, 1, . . . , n− 1}. Define φ̂ac+1(G) by connecting
an n-clique on new vertices wc+1i with new edges w
c
iw
c+1
i for each i ∈ {0, 1, . . . , n− 1}.
Note that, under the Γ-action, we have enhanced G with cp orbits whose induced sub-
graphs are cliques and with ap orbits whose induced subgraphs are disjoint unions of con-
nected GP-distance-number-maximizing graphs.
Let G be a Γ-graph for a given group Γ. The following proposition shows that φ̂ac (G)
is also a Γ-graph in most cases. We omit its proof, which is similar to the proof of Propo-
sition 3.4.
Proposition 6.2. Let Γ be a group, and suppose G is a nontrivial connected Γ-graph that
is not complete. If either c ̸= 0 or G ̸∼= ĜΓ, then φ̂ac (G) is also a Γ-graph.
We now present our result analogous to Theorem 1.1 that makes an assumption on the
connectedness of graphs.
Theorem 6.3. Let Γ be a group. If G is a connected Γ-graph of order n having p vertex
orbits, each of which induces a connected subgraph of G, then{
δ
(
φ̂ac (G)
)
| a, c ∈ N and φ̂ac (G) is a Γ-graph
}
is dense in
(
n−p
n , δ̂(G)
)
.
Proof. Given any ϵ > 0 and any q ∈
(
n−p
n , δ̂(G)
)
, it suffices to find a′, c′ ∈ N such that∣∣∣q − δ (φ̂a′c′ (G))∣∣∣ < ϵ. We first determine an expression for δ (φ̂ac (G)), and then explain
how to choose a′ and c′.
Let V0, V1, . . . , Vp−1 be the Γ-orbits in V (G). For each Vi, there are a + c associated
vertex orbits of φ̂ac (G) under the action of Aut
(
φ̂ac (G)
)
; for i ∈ {0, 1, . . . , p − 1}, label
these orbits by A1i , A
2
i , . . . , A
a
i and C
1
i , C
2
i , . . . , C
c
i , where u
j
i ∈ A
j
i for j ∈ {1, 2, . . . , a}
and wki ∈ Cki for k ∈ {1, 2, . . . , c}. For X ∈ {G, ĜΓ}, let dX denote the distance function
inX , and let dX,3 denote the function given by min(dX(u, v), 3) for vertices u, v ∈ V (X).
Write d′G = dG for c = 0 and d
′
G = dG,3 for c ≥ 1.
For each i ∈ {0, 1, . . . , p−1} and any k, any two distinct vertices in Cki are at distance
1 from each other. Choosing a representative in each orbit Ck1 , C
k
2 , . . . , C
k
p−1, we find that
the total distance over all the orbits in level k is
p−1∑
i=0
(|Cki | − 1) = n− p.
L. Abrams and L.-K. Lauderdale: Density results for Graovac-Pisanski’s distance number 255
For i ∈ {0, 1, . . . , p − 1} and any j, a shortest path between any two vertices ujℓ , ujm
in Aji is either a shortest path in layer j, or is a path obtained by concatenating a shortest
ujℓ , u
0
ℓ -path and a shortest u
0
m, u
j
m-path with a shortest u
0
ℓ , u
0
m-path in G if c = 0 and with
a shortest u0ℓ , u
0
m-path in φ̂
0
1(G) if c > 0. Thus, the length of a shortest u
j
ℓ , u
j
m-path is
min
{
dĜΓ
(
ujℓ , u
j
m
)
, 2j + d′G
(
u0ℓ , u
0
m
)}
.
Writing diam(X) for the length of a longest path in graph X , if j ≥ diam(ĜΓ)/2 then we
have
min
{
dĜΓ
(
u0ℓ , u
0
m
)
, 2j + d′G
(
u0ℓ , u
0
m
)}
= dĜΓ
(
u0ℓ , u
0
m
)
.
Note that, to prove the result, it suffices to presume that a > diam(ĜΓ)/2. Choosing a rep-
resentative in each orbit, we can calculate the total distance for levels 0 to
⌈
diam(ĜΓ)/2
⌉
;
write D for this value. Also, for each j >
⌈
diam(ĜΓ)/2
⌉
, the total distance in level j is
nδ̂(G). Thus, we have
δ
(
φ̂ac (G)
)
=
(n− p)c+D +
(
a−
⌈
diam(ĜΓ)/2
⌉)
nδ̂(G)
(1 + a+ c)n
.
In order to choose appropriate a and c, observe first that, for any positive a, c ∈ N, we
have
δ
(
φ̂ac−1(G)
)
− δ
(
φ̂ac (G)
)
=
D +
(
a−
⌈
diam(ĜΓ)/2
⌉)
nδ̂(G)− (n− p)(a+ 1)
(a+ c)(1 + a+ c)n
<
D + anδ̂(G)
(a+ c)2n
.
Let ∆(a, c) denote this upper bound, and note that ∆(a, c) has negative derivative with
respect to both a and to c.
We now choose a′ and c′. Since
lim
a→∞
δ(φ̂a0(G)) = δ̂(G) > q,
we can choose a′ ∈ N so that a′ >
⌈
diam(ĜΓ)/2
⌉
, ∆(a′, 0) < ϵ, and δ
(
φ̂a
′
0 (G)
)
> q.
Because
lim
c→∞
δ
(
φ̂a
′
c (G)
)
=
n− p
n
< q
we can then choose
c′ := min
{
c ∈ N
∣∣δ(φ̂a′c (G)) ≤ q} .
Observe that c′ > 0 because we have chosen a′ to ensure that δ
(
φ̂a
′
0 (G)
)
> q. Since
δ
(
φ̂a
′
c′ (G)
)
< q ≤ δ
(
φ̂a
′
c′−1(G)
)
,
we have
q − δ
(
φ̂a
′
c′ (G)
)
< ∆(a′, c′) < ∆(a′, 0) < ϵ,
as desired. Furthermore, since c′ > 0, Proposition 6.2 guarantees that φ̂a
′
c′ (G) is a Γ-
graph.
256 Ars Math. Contemp. 21 (2021) #P2.05 / 243–257
Let Γ be a group, and suppose G is a connected Γ-graph of order n with p vertex orbits
under the action of AutG. If the induced subgraph on each vertex orbit of G is connected,
then we claim that there exists infinitely many rational numbers in
(
n−p
n , δ̂(G)
)
that are
not the GP distance numbers of graphs of the form φ̂ac (G). We demonstrate our claim with
the following example.
Example 6.4. LetG be the graph constructed from an 8-cycle on vertices u00, u01, u02, . . . , u07
and a 4-cycle on vertices u08, u
0
9, u
0
10, u
0
11, by including edges
u00u
0
8, u
0
1u
0
8, u
0
2u
0
9, u
0
3u
0
9, u
0
4u
0
10, u
0
5u
0
10, u
0
6u
0
11, and u
0
7u
0
11.
The graph G, which is illustrated in Figure 3, is a D8-graph with two vertex orbits under
the action of AutG (here D8 denotes the dihedral group of order 8).
u00
u01u
0
2
u03
u04
u05 u
0
6
u07
u08u
0
9
u010 u
0
11
Figure 3: The D8-graph G constructed in Example 6.4.
Observe that ĜD8 is equal to C8 ⊔ C4. Moreover, δ(G) = 2012 = δ̂(G), and thus The-
orem 6.3 established that
{
δ
(
φ̂ac (G)
)
| a, c ∈ N
}
is dense in the interval
(
5
6 ,
5
3
)
. Observe
that
δ
(
φ̂ac (G)
)
=

20
12
if c = 0
19 + 20a+ 10c
12(1 + a+ c)
if c ̸= 0,
and suppose δ
(
φ̂ac (G)
)
= rs for some
r
s ∈
(
5
6 ,
5
3
)
. Solving for c in the case when c > 0 we
obtain
c =
(20s− 12r)a+ 19s− 12r
12r − 10s
.
Notice that if s is odd, then the numerator of this expression for c is odd whereas the
denominator is even, and thus this value of c is not an integer. It follows that s is even, so
no rational number in reduced form with an odd denominator is δ
(
φ̂ac (G)
)
for any values
of a and c. Finally, the reader may be entertained by the observation that both the set of GP
distance numbers and non-GP distance numbers in
(
5
6 ,
5
3
)
are dense.
ORCID iDs
Lowell Abrams https://orcid.org/0000-0002-8174-5957
L. Abrams and L.-K. Lauderdale: Density results for Graovac-Pisanski’s distance number 257
References
[1] A. R. Ashrafi and H. Shabani, The modified Wiener index of some graph operations, Ars Math.
Contemp. 11 (2016), 277–284, doi:10.26493/1855-3974.801.968.
[2] D. Bonchev and N. Trinajstić, Information theory, distance matrix, and molecular branching, J.
Chem. Phys. 67 (1977), 4517–4533, doi:10.1063/1.434593.
[3] R. Frucht, Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Math. 6
(1939), 239–250, http://www.numdam.org/item?id=CM_1939__6__239_0.
[4] A. Graovac and T. Pisanski, On the Wiener index of a graph, J. Math. Chem. 8 (1991), 53–62,
doi:10.1007/bf01166923, mathematical chemistry and computation (Dubrovnik, 1990).
[5] I. Gutman and T. Körtvélyesi, Wiener indices and molecular surfaces, Z. Naturforsch. 50
(1995), 669–671, doi:10.1515/zna-1995-0707.
[6] I. Gutman and O. E. Polansky, Wiener numbers of polyacenes and related benzenoid molecules,
Match 20 (1986), 115–123, https://match.pmf.kg.ac.rs/content20.htm.
[7] M. A. Iranmanesh and H. Shabani, The symmetry-moderated Wiener index of truncation graph,
Thorn graph and caterpillars, Discrete Appl. Math. 269 (2019), 41–51, doi:10.1016/j.dam.2018.
05.040.
[8] M. Knor, J. Komornı́k, R. Škrekovski and A. Tepeh, Unicyclic graphs with the maximal value
of Graovac-Pisanski index, Ars Math. Contemp. 17 (2019), 455–466, doi:10.26493/1855-3974.
1925.57a.
[9] M. Knor, R. Škrekovski and A. Tepeh, Chemical graphs with the minimum value of Wiener
index, Match Commun. Math. Comput. Chem. 81 (2019), 119–132, https://match.pmf.
kg.ac.rs/content81n1.htm.
[10] M. Knor, R. Škrekovski and A. Tepeh, Mathematical aspects of Wiener index, Ars Math. Con-
temp. 11 (2016), 327–352, doi:10.26493/1855-3974.795.ebf.
[11] B. Mohar and T. Pisanski, How to compute the Wiener index of a graph, J. Math. Chem. 2
(1988), 267–277, doi:10.1007/bf01167206.
[12] R. Pinal, Effect of molecular symmetry on melting temperature and solubility, Org. Biomol.
Chem. 2 (2004), 2692–2699, doi:10.1039/b407105k.
[13] H. Wiener, Correlation of heats of isomerization, and differences in heats of vaporization of
isomers, among the paraffin hydrocarbons, J. Am. Chem. Soc. 69 (1947), 2636–2638, doi:
10.1021/ja01203a022.
[14] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947),
17–20, doi:doi.org/10.1021/ja01193a005.

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P2.06 / 259–269
https://doi.org/10.26493/1855-3974.1926.6ad
(Also available at http://amc-journal.eu)
Decompositions of the automorphism groups
of edge-colored graphs into the direct product of
permutation groups
Mariusz Grech *
Wrocław University of Science and Technology,
Wybrzeże Wyspiańskiego Str. 27, Wrocław, Poland
Received 31 January 2019, accepted 1 June 2021, published online 11 October 2021
Abstract
In the paper Graphical complexity of products of permutation groups, M. Grech, A. Jeż,
and A. Kisielewicz have proved that the direct product of automorphism groups of edge-
colored graphs is itself the automorphism groups of an edge-colored graph. In this paper,
we study the direct product of two permutation groups such that at least one of them fails
to be the automorphism group of an edge-colored graph. We find necessary and sufficient
conditions for the direct product to be the automorphism group of an edge-colored graph.
The same problem is settled for the edge-colored digraphs.
Keywords: Colored graph, automorphism group, permutation group, direct product.
Math. Subj. Class. (2020): 05E18
1 Introduction
For permutation groups (A, V ), (B,W ), the direct product of A and B (with product ac-
tion) is a permutation group (A×B, V ×W ) with the action given by
(a, b)(x, y) = (a(x), b(y)).
The study of the direct product of automorphism groups of graphs was initiated by
G. Sabidussi [19] in 1960. The problem was taken up in 1971 by M. Watkins [20]. In
1972, L. Nowitz and M. Watkins [21], and independently W. Imrich [13], have described
the conditions under which the direct product of regular permutation groups that are auto-
morphism groups of graphs is itself the automorphism group of a graph. This result was
*Supported in part by Polish NCN grant UMO-2016/21/B/ST1/03079.
E-mail address: mariusz.grech@pwr.edu.pl (Mariusz Grech)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
260 Ars Math. Contemp. 21 (2021) #P2.06 / 259–269
a contribution to the description of all regular automorphism groups of graphs, which has
been completed in 1978 by C. Godsil [5] for graphs, and in 1980 by L. Babai [1] for di-
graphs. The above results in [13, 21] have been extended to arbitrary permutation groups
in [6], where the description of the conditions, under which the direct product of automor-
phism groups of graphs is itself an automorphism group of a graph, is given. In [8], the
same is done for digraphs.
All the above results are motivated more or less directly by trying to make a contribution
to the solution of the concrete version of König problem asking about a characterization
of those permutation groups that are the automorphism groups of graphs (see [14]). There
are a number of results (see e.g. [9, 10, 18] and [14]) showing that it is more natural and
effective to study the automorphism groups of (edge-)colored graphs (rather than simple
graphs), which is essentially the approach suggested by Wielandt [23].
In [14], A. Kisielewicz has introduced the notion of graphical complexity of permuta-
tion groups and suggested the study of constructions of permutation groups in this context.
By G(k), we denote the class of the automorphism groups of k-edge-colored graphs (those
using at most k colors), and by GR, the union of all the classes G(k), which in Wielandt’s
terminology [23] is the class of 2∗-closed groups. Similarly, by DG(k) we denote the class
of the automorphism groups of k-edge-colored digraphs, and by DGR the union of all the
classes DG(k) (which in Wielandt’s terminology is the class of 2-closed groups). Clearly,
GR ⊆ DGR, and G(k) ⊆ DG(k), for any k.
The main general problem is to determine which permutation groups are the automor-
phism groups of edge-colored graphs. Various aspects of this general problem are investi-
gated. For example, it leads to the concept of colored totally symmetric graphs, that was
described in [11, 12]. This coincides to a large extent with the research on homogeneous
factorization of graphs (c.f., [4, 15, 16]). One direction of research is to consider various
constructions of permutation groups and to ask the following question: is it true that if the
components of the construction belong to a particular class G(k), then the result belongs
to G(k), as well? And if not, how many colors one must add to make sure that the result of
the construction belong to G(k + r)?
For the direct product the problem has been solved in [9, Theorem 2.2].
Theorem 1.1 (Grech, Jeż, Kisielewicz). If permutation groups A,B ∈ GR, then A×B ∈
GR. Also, if A,B ∈ DGR, then A×B ∈ DGR.
Note that the second part of this theorem was also shown in [3, Theorem 5.1]
This result, with some exceptions, is also true for particular classes G(k) and DG(k)
(for details see [7]). In this paper we consider the converse of the theorem above. We
show that while for DGR the converse also holds (Theorem 3.1), for GR it is not generally
true. The main results is Theorem 3.2 characterizing the conditions under which the direct
product of two arbitrary permutation groups belongs to GR.
2 Preliminaries
We assume that the reader has basic knowledge in the areas of graphs and permutation
groups, so we omit an introduction to standard terminology. If necessary, additional details
can be found in [2, 24].
By a k-edge-colored graph G, we mean a pair G = (V,E), where V is the set of
vertices of G, and E the edge-color function from the set P2(V ) of unordered pairs of
M. Grech: Decompositions of the automorphism groups of edge-colored graphs into the . . . 261
vertices into the set of colors {0, . . . , k − 1} (E : P2(V ) → {0, . . . , k − 1}). Thus, G is a
complete simple graph with colored edges. Similarly, by a k-edge-colored digraph G, we
mean a pair (V,E) where E is a color function from the set of ordered pairs of different
elements of V to the set of colors {0, . . . , k − 1} (E : ((V × V ) \ {(v, v); v ∈ V }) →
{0, . . . , k − 1}).
An automorphism of an edge-colored graph G is a permutation a of the set V preserv-
ing the edge function: E({v, w}) = E({a(v), a(w)}), for all v, w ∈ V . The group of
automorphisms of G will be denoted by Aut(G), and considered as a permutation group
(Aut(G), V ) acting on the set of the vertices V . Edge-colored digraphs are defined simi-
larly.
All groups considered in this paper are groups of permutations. They are considered up
to permutation group isomorphism. Generally, a permutation group A acting on a set V is
denoted (A, V ) or just A, if the set V is clear from the context or not important. By Sn we
denote the symmetric group on n elements, and by In, the one element group acting on n
elements (consisting of the identity only, denoted by id).
We shall consider the natural actions of a given permutation group A = (A, V ) on the
sets of ordered and unordered pairs of V , V × V and P2(V ), respectively. Let a ∈ A and
v, w ∈ V . Then, the first action of a is given by the formula
a((v, w)) = (a(v), a(w)),
while the second action is given by
a({v, w}) = {a(v), a(w)}.
The orbits of A in the action on V × V are called orbitals of A. Since in this paper we
concider graphs (digraphs) without loops, we exclude trivial orbitals consisting of pairs of
the form (v, v). For two orbitals O1, O2 we say that O1 is paired with O2 if and only if
O2 = {(w, v) : (v, w) ∈ O1}. We call an orbital O self-paired if it is paired with itself.
Moreover, we say that a permutation a transposes O1 and O2, if a(O1) = O2.
In addition, the orbits of A in the action on P2(V ) will be called here 2∗-orbitals. Note
that we can think of a 2∗-orbital either as a self paired orbital or as a pair of paired orbitals.
Since A×I1 = I1×A = A (up to permutation isomorphism), in this paper, we consider
only the direct products A×B with both the permutation groups A,B different from I1.
Let A = (A, V ) be a permutation group, and let O∗1 , . . . O
∗
k be all the 2
∗-orbitals of A.
We define an edge-colored graph G∗(A) (called 2∗-orbital graph) as follows.
G∗(A) = (V,E), where E : P2(V ) → {0, . . . k − 1}.
E({v, w}) = i if and only if the edge {v, w} belongs to the 2∗-orbital O∗i .
Now, we define A∗ = Aut(G∗(A)). Obviously, A ⊆ A∗. It should be clear that A∗ is
the smallest permutation group on V that contains A and belongs to GR. (Indeed, if G′
is a colored graph whose automorphism group contains A, then edges in each 2∗-orbital
of A have to have the same color. Hence, each permutation in Aut(G∗(A)) belongs to
Aut(G′).) In particular, we have that A ∈ GR if and only if A = A∗.
Similarly we define the orbital digraph G(A) replacing 2∗-orbitals by orbitals. In the
same way, denoting A = Aut(G(A)), we have that A is the smallest permutation group on
X that contains A and belongs to DGR. Moreover, A ∈ DGR if and only if A = A. In
addition, A ⊆ A ⊆ A∗.
For direct products of permutation groups we have the following inclusions
262 Ars Math. Contemp. 21 (2021) #P2.06 / 259–269
Lemma 2.1.
(i) A×B ⊆ Aut(G∗(A×B)) ⊆ A∗ ×B∗,
(ii) A×B ⊆ Aut(G(A×B)) ⊆ A×B,
Proof. The first inclusion holds for all permutation groups, as it was remarked above. We
prove the second inclusion.
Consider the edges of the form {(v1, w), (v2, w)}, which we may refer as edges be-
longing to the rows. Obviously, they form a union of 2∗-orbitals, and therefore the edges
{(v1, w1), (v2, w2)} with w1 ̸= w2 in Aut(G∗(A × B)) have different colors than those
belonging to the rows. The same is true for columns, i.e. the edges of the form {(w, v1),
(w, v2)}. Thus, rows can be mapped only onto rows by automorphisms of G∗(A × B),
and columns can be mapped only onto columns. This implies that Aut(G∗(A × B)) ⊆
A1 × B1, for some A1 and B1. Now let (a, b) ∈ Aut(G∗(A × B)). Then, the edges
(a, b)({(v1, w), (v2, w)}) and {(v1, w), (v2, w)} have the same color. Therefore, there is
(a1, b1) ∈ A × B such that (a1, b1)({(v1, w), (v2, w)}) = {(v1, w), (v2, w)}. Hence,
(a−11 a, b
−1
1 b) ∈ Aut(G∗(A × B)) preserves the row with the edge {(v1, w), (v2, w)}.
Since every row in Aut(G∗(A × B)) is a copy of G∗(A) (up to recoloring), we have that
a−11 a ∈ A∗, which implies that a ∈ A∗. In a similar way, b ∈ B∗, which completes the
proof of the first part of the theorem. The second part is proved similarly.
We observe that if C = Aut(G∗(A × B)), then C∗ may be a proper subgroup of
A∗ × B∗. The smallest example is I2 × I2, where Aut(G∗(I2 × I2)) = I2 × I2, while
I2
∗ × I2∗ = S2 × S2.
We observe also that if a ∈ A∗, then it not only preserves 2∗-orbitals of A (by defini-
tion), but it also preserves orbits of A.
Lemma 2.2. Let A ̸= I2 be a permutation group. If a ∈ A∗, then a preserves the orbits of
A.
Proof. Let Qt, t ∈ {1, . . . ,m} be the orbits of A. The claim is obvious if A = It for any
t > 2, so we may assume that there is an orbit Qi that has at least two elements. Then, the
set P2(Qi) is nonempty. Moreover, it is clear that P2(Qi) is the union of 2∗-orbitals of A.
Hence, the edges of G∗(A) that belong to P2(Qi) have different colors than the remaining
edges. This implies that a preserves the orbit Qi.
Now, if there is another orbit Qt, t ̸= i, then obviously, the edges {v, w} with v ∈ Qi
and w ∈ Qt have different colors than the remaining edges. Consequently, every orbit is
preserved by a.
3 Results
We proceed to the main problem of this paper to describe conditions under which A × B
belongs to GR or DGR. The case of directed graphs is pretty easy.
Theorem 3.1. Let A and B be permutation groups. Then, A × B ∈ DGR if and only if
both A and B are in DGR.
Proof. In view of the Theorem 1.1 quoted in the introduction we need to prove merely the
“only if” part. It is enough to prove, without loss of generality, that if A /∈ DGR, then
M. Grech: Decompositions of the automorphism groups of edge-colored graphs into the . . . 263
A × B /∈ DGR. Let A = (A, V ) and B = (B,W ). We assume that A /∈ DGR. Then,
A ̸= I2 (since I2 ∈ DGR). Moreover, we may choose a ∈ A \ A. By definition, it
preserves all orbitals of A.
Let idB be the identity in the permutation group B. We show that the permutation
(a, idB) belongs to Aut(G(A × B)). To this end, we show that for every directed edge
e = ((v1, w1), (v2, w2)), where v1, v2 ∈ V , w1, w2 ∈ W , the image (a, idB)(e) has the
same color as e.
Assume first that v1 ̸= v2. Since a preserves orbitals of A, for every pair (v1, v2),
there is a permutation a2 ∈ A such that a(v1) = a2(v1) and a(v2) = a2(v2). We have
(a, idB)(e) = (a2, idB)(e), and therefore the directed edges (a, idB)(e) and e belong to
the same orbital of A × B. So, by the definition of the edge-colored digraph G(A × B),
(a, idB)(e) and e have the same color in G(A×B).
If v1 = v2, then since A ̸= I2, we may use Lemma 2.2 and find a permutation a1 ∈ A
such that a1(v1) = a(v1). We have (a, idB)(e) = (a1, idB)(e), and therefore the directed
edges (a, idB)(e) and e belong to the same orbital of A×B. So, they have the same color.
Thus, in all the cases (a, idB) ∈ Aut(G(A × B)), but (a, idB) does not belong to
A×B. Therefore, A×B /∈ DGR.
This settles the problem for the case of edge-colored digraphs. The case of edge-colored
graphs is different and more complex.
Theorem 3.2. Let A and B be permutation groups. Then, A × B ∈ GR, except for the
following cases:
(i) A×B /∈ DGR, that is, either A /∈ DGR or B /∈ DGR,
(ii) either every orbital of A ∈ GR is self-paired and B /∈ GR∪ {I2} or every orbital of
B ∈ GR is self-paired and A /∈ GR ∪ {I2},
(iii) A,B ∈ DGR \ (GR ∪ {I2}), and there exist a ∈ A∗ \A and b ∈ B∗ \B, such that
a transposes every pair of paired orbitals in A, and b transposes every pair of paired
orbitals in B.
Proof. We consider a few cases. An obvious consequence of Theorem 3.1 is the following
Corollary 3.3. Let A /∈ DGR and B be an arbitrary permutation group. Then, A × B /∈
GR.
Accordingly to this corollary, we will assume further that both the components of A×B
belongs to DGR. The next three lemmas deal with the case when one of the groups belongs
to GR or is equal to I2.
Lemma 3.4. Let A ∈ DGR \ (GR ∪ {I2}) and B ∈ GR. If every orbital of B is self-
paired, then A×B ̸∈ GR.
Proof. Denote A = (A, V ) and B = (B,W ). Let a ∈ A∗ \ A, and idB be the identity in
the permutation group B. Let e = {(v1, w1), (v2, w2)}, where v1, v2 ∈ V , w1, w2 ∈ W .
We show that the edges e and (a, idB)(e) have the same color. To this end it is enough to
prove that (a, idB)(e) belongs to the same 2∗-orbital of A×B as e.
264 Ars Math. Contemp. 21 (2021) #P2.06 / 259–269
If w1 = w2, then the statement holds by the fact that a preserves all 2∗-orbitals of A.
Assume v1 = v2. Since A ̸= I2, by Lemma 2.2, a preserves all orbits of A (in its action on
V ). Hence, there is a1 ∈ A such that a(v1) = a1(v1). We have,
(a, idB)({(v1, w1), (v1, w2)}) = {(a(v1), w1), (a(v1), w2)}
= (a1, idB)({(v1, w1), (v1, w2)}).
Thus, e and (a, idB)(e) belong to the same 2∗-orbital of A×B.
Now let v1 ̸= v2 and w1 ̸= w2. If the pair a((v1, v2)) belongs to the same orbital of
A as the pair (v1, v2), then there is a1 ∈ A such that a1(v1) = a(v1) and a1(v2) = a(v2).
Similarly as above, we have,
(a, idB)({(v1, w1), (v2, w2)}) = {(a(v1), w1), (a(v2), w2)}
= (a1, idB)({(v1, w1), (v2, w2)}).
Assume, finally, that v1 ̸= v2, w1 ̸= w2 and the pairs a((v1, v2)), (v1, v2) belong to
different orbitals of A. Since a ∈ A∗, we know that a preserves all 2∗-orbitals of A. This
implies that, the pairs a((v1, v2)) and (v2, v1) belong to the same orbital of A. Hence,
there is a1 ∈ A such that a1((v2, v1)) = a((v1, v2)). Moreover, since all orbitals of B are
self-paired, there is b ∈ B such that b((w1, w2)) = (w2, w1). Consequently,
(a, idB)(e) = {(a1(v2), b(w2)), (a1(v1), b(w1))} = (a1, b)(e).
Thus (a, idB)(e) and e belongs to the same 2∗-orbital of A×B, and consequently, (a, idB)
does not change the color of the edges.
It follows that (a, idB) ∈ Aut(G∗(A×B)) = (A×B)∗. Since a ∈ A∗ \A, (a, idB) /∈
A×B, and therefore A×B ̸= (A×B)∗, which completes the proof.
Lemma 3.5. Let A ∈ DGR\(GR∪{I2}) and let B ∈ GR have at least one not-self-paired
orbital. Then, A×B ∈ GR.
Proof. Let A = (A, V ) and B = (B,W ). We know, by Lemma 2.1(1), that Aut(G∗(A×
B)) ⊆ A∗ ×B. Therefore, every c ∈ Aut(G∗(A×B)) has the form (a, b), where a ∈ A∗
and b ∈ B. We show that, in fact, a always belongs to A. Assume, to the contrary, that
a ∈ A∗ \ A. In this case, since A ∈ DGR \ (GR ∪ {I2}), there is an (ordered) pair
(v1, v2), v1, v2 ∈ V such that a((v1, v2)) ̸= a1((v1, v2)), for every a1 ∈ A. Since B
has an orbital which is not-self-paired, there are w1, w2 ∈ W such that b((w1, w2)) ̸=
(w2, w1) for every b ∈ B. Now, observe that the edges (a, b)({(v1, w1), (v2, w2)}) and
{(v1, w1), (v2, w2)} belong to different 2∗-orbitals of A×B. Indeed, if the edges (a, b)({
(v1, w1), (v2, w2)}) and {(v1, w1), (v2, w2)} belong to the same 2∗-orbital of A×B, then
either there are a1 ∈ A and b1 ∈ B such that a((v1, v2)) = a1((v1, v2)) and b((w1, w2)) =
b1((w1, w2)) or there are a2 ∈ A and b2 ∈ B such that a((v1, v2)) = a2((v2, v1)) and
b((w1, w2)) = b2((w2, w1)). The first case is impossible by the assumption on a. In the
second case, we get b−12 b((w1, w2)) = (w2, w1), which contradicts the assumption. This
implies that E((a, b)({(v1, w1), (v2, w2)})) ̸= E({(v1, w1), (v2, w2)}), which contradicts
the fact that (a, b) ∈ Aut(G∗(A×B)). Consequently, we have Aut(G∗(A×B)) ⊆ A×B,
which completes the proof.
We summarize Lemma 3.4 and Lemma 3.5.
M. Grech: Decompositions of the automorphism groups of edge-colored graphs into the . . . 265
Corollary 3.6. Let A ∈ DGR \ (GR ∪ {I2}) and B ∈ GR. Then, A × B ∈ GR if and
only if there exists a non-self-paired orbital of B.
The following special case must be considered separately.
Lemma 3.7. Let B ∈ GR. Then, B × I2 ∈ GR.
Proof. By Lemma 2.1(1), Aut(G∗(B× I2)) is equal either to B× I2 or to B×S2. By our
general assumption B ̸= I1, hence, in G∗(B × I2), there is at least one edge of the form
{(v, 0), (w, 0)}, and being in different orbitals, it has a different color than {(v, 1), (w, 1)}.
Thus, Aut(G∗(B × I2)) = B × I2. Therefore, B × I2 ∈ GR.
This completes the description in all the cases where at least one of the components
belongs to GR.
The remaining case occurs where A,B ∈ (DGR \GR). We start with the following.
Lemma 3.8. Let A,B ∈ (DGR \ GR). If for every b ∈ B∗ there exists a pair of paired
orbitals O1 ̸= O2 of B such that b does not transpose O1 and O2, then A×B ∈ GR.
Proof. Let A = (A, V ) and B = (B,W ). Assume to the contrary that there exists (a, b) ∈
Aut(G∗(A×B))\(A×B).
First, assume that a ∈ A; then, b /∈ B. Since A ∈ (DGR \GR), there is an (ordered)
pair (v1, v2), where v1, v2 ∈ V , which belongs to a non-self paired orbital of A. Since
B ∈ DGR, there is an (ordered) pair (w1, w2) where w1, w2 ∈ W , for which there is no
b1 ∈ B such that b1((w1, w2)) = b((w1, w2)). We prove that the edge {(v1, w1), (v2, w2)}
belongs to a different 2∗-orbital than the edge (a, b)({(v1, w1), (v2, w2)}). Indeed, if the
edges (a, b)({(v1, w1), (v2, w2)}) and {(v1, w1), (v2, w2)} belong to the same 2∗-orbital,
then either there are a1 ∈ A and b1 ∈ B such that a((v1, v2)) = a1((v1, v2)) and
b((w1, w2)) = b1((w1, w2)) or there are a2 ∈ A and b2 ∈ B such that a((v1, v2)) =
a2((v2, v1)) and b((w1, w2)) = b2((w2, w1)). In the former, by assumption on b and
w1, w2, this is impossible. In the latter, since a ∈ A it is also impossible. Hence, the edges
(a, b)({(v1, w1), (v2, w2)}) and {(v1, w1), (v2, w2)} have different colors in G∗(A × B).
This contradicts the assumption that (a, b) ∈ Aut(G∗(A×B)).
Next, consider the case where a /∈ A. Since A ∈ DGR, there is an ordered pair
(v1, v2), where v1, v2 ∈ V , for which there is no permutation a1 ∈ A such that a1((v1, v2))
= a((v1, v2)). Let O1, O2 be orbital from the statement of the lemma. By assump-
tion, there are w1, w2 ∈ W such that {w1, w2} ∈ O1 and b((w1, w2)) ∈ O1. Thus,
b((w1, w2)) = b1((w1, w2)) for some b1 ∈ B. A similar proof as above shows that the
edge
(a, b)({(v1, w1), (v2, w2)}) = (a, b1)({(v1, w1), (v2, w2)})
belongs to a different 2∗-orbital than the edge {(v1, w1), (v2, w2)}. Again, this contradicts
the assumption that (a, b) ∈ Aut(G∗(A×B)).
Now, we consider the case where one of the groups is equal to I2.
Lemma 3.9. Let A ∈ (DGR \GR). Then, A× I2 ∈ GR.
Proof. Let A = (A, V ) and I2 = (I2, {w1, w2}). Assume to the contrary that there is
(a, b) ∈ Aut(G∗(A × I2)) \ (A × I2). Since, for any v1, v2, v3, v4 ∈ V , the edges
{(v1, w1), (v2, w1)} and {(v3, w2), (v4, w2)} have different colors, b = id. In the same
way as in the second case of the proof of the Lemma 3.8, we get a contradiction.
266 Ars Math. Contemp. 21 (2021) #P2.06 / 259–269
Now, we consider the last case.
Lemma 3.10. Let A,B ∈ DGR \ (GR∪ I2). If there exists a ∈ A∗ \A which transposes
all the pairs of the paired orbitals of A and there exists b ∈ B∗ \B which transposes all the
pairs of the paired orbitals of B, then A×B ̸∈ GR. Moreover, A×B is transitive.
Proof. Let A = (A, V ) and B = (B,W ). Since A ̸= I2 and B ̸= I2, by Lemma 2.2,
every permutation a ∈ A∗ \ A preserves the orbits of A (in its action on V ) and every
permutation b ∈ B∗ \ B preserves the orbits of B (in its action on W ). Hence, we ob-
tain immediately, under the assumptions on A and B, that the permutation groups A and
B have to be transitive. Consequently, for every a ∈ A∗, b ∈ B∗, v, v1, v2 ∈ V , and
w,w1, w2 ∈ W , the edge (a, b)({(v, w1), (v, w2)}) has the same color in G∗(A × B) as
the edge {(v, w1), (v, w2)}, and moreover, the edge (a, b)({(v1, w), (v2, w)}) has the same
color as the edge {(v1, w), (v1, w)}.
We choose a and b as in the statement of the lemma, and fix the elements v1 ̸= v2 ∈ V
and w1 ̸= w2 ∈ W . Since a and b preserves no non-self-paired orbital, the ordered
pair a((v1, v2)) belongs to the orbital of the ordered pair (v2, v1) and the ordered pair
b((w1, w2)) belongs to the orbital of the ordered pair (w2, w1). Hence, there are a1 ∈ A and
b1 ∈ B such that a((v1, v2)) = a1((v2, v1)) and b((w1, w2)) = b1((w2, w1)). Therefore,
we have
E((a, b)({(v1, w1), (v2, w2)})) = E({(a(v1), b(w1)), (a(v2), b(w2))})
= E({(a1(v2), b1(w2)), (a1(v1), b1(w1))})
= E((a1, b1)({(v1, w1), (v2, w2)}))
= E({(v1, w1), (v2, w2)}).
The vertices v1, v2, w1, and w2 are arbitrary. Hence, the permutation (a, b) preserves all
colors. Consequently, (a, b) ∈ Aut(G∗(A×B) \ (A×B)).
This exhausts all cases and ends the proof of the theorem.
4 Corollaries and problems
First, it is worth noting that for some subclasses the result may be stated in a nice simple
form. Since all intransitive permutation groups have a non-self-paired orbital, we have the
following.
Corollary 4.1. Let A ∈ DGR, and B ∈ GR be intransitive. Then, A×B ∈ GR.
Also, it is easy to observe that the only regular groups with all self-paired orbitals are
Sn2 , n ≥ 1. This implies that:
Corollary 4.2. Let A ∈ DGR, and B ∈ GR be regular. Then, A × B ∈ GR if and only
if B ̸= Sn2 , for every n.
Next, we give an alternative proof of the known fact, that was first observed in [22,
Example 3.15]
Corollary 4.3. Every regular permutation group belongs to DGR.
M. Grech: Decompositions of the automorphism groups of edge-colored graphs into the . . . 267
Proof. Let U be an nonsolvable regular group. Then, for every regular group A, the group
A × U is nonsolvable. By [5], we have A × U ∈ G(2) ⊆ DGR. By Theorem 3.1,
A ∈ DGR.
The next fact, it seems, was not recognized so far.
Corollary 4.4. Except for the abelian groups of exponent greater than two and generalized
dicyclic groups, all the finite regular permutation groups belong to the class GR.
Proof. Let A be an abelian group of exponent greater than two or a generalized dicyclic
group. It is proved in [5], that in such a case A /∈ G(2). The proof shows, in fact, that
A /∈ GR. Assume that A is not as those groups mentioned above. Then, it is well known
(see [5]) that A × S42 ∈ G(2). Since S42 ∈ GR and it has all orbitals self-paired, then by
Theorem 3.2 (ii), A ∈ GR.
Theorem 3.2 suggests a few open problems.
Problem 4.5. Describe the permutation groups that have all orbitals self-paired.
This does not seem to be an easy problem. Examples of groups whose all orbitals are
self-paired are Sn and their transitive products (direct product, wreath product, etc.). In
particular, all groups of the form Sk2 (the direct power) belong to this class. Yet, there are
other examples, like the automorphism groups of totally symmetric graphs described in
[11]. Note that if a permutation group A having all orbitals self-paired is an automorphism
group of a colored digraph D, A = Aut(D), then D is, in fact, an undirected colored
graph, and so A ∈ GR.
It would be also desirable to have a description of permutation groups with the property
given in Theorem 3.2(iii).
Problem 4.6. Describe all transitive permutation groups A having a permutation σ ∈ A∗ \
A transposing all pairs of paired orbitals.
We note that all regular abelian group of exponent greater than two and regular general-
ized dicyclic groups have this property. However, there are also many other examples. For
instance, the group A = ⟨(0, 1, 2, 3, 4, 5, 6), (1, 2, 4)(3, 6, 5)⟩ is one of them. This group is
a subgroup of Frobenius group F7 generated by translations and multiplication by 2 (which
is a permutation of order 3). This suggest the following.
Problem 4.7. Let A be a subgroup of the permutation group AGLn(p) generated by trans-
lations and ω2k, where ω is a generator of the the multiplicative group F ∗pn , and k divides
n. Moreover, let −1 be not quadratic in Fpn . Is it true that for each such group there is an
element a ∈ A∗ \A transposing all pairs of paired orbitals?
ORCID iDs
Mariusz Grech https://orcid.org/0000-0002-7328-8792
References
[1] L. Babai, Finite digraphs with given regular automorphism groups, Period. Math. Hungar. 11
(1980), 257–270, doi:10.1007/bf02107568.
268 Ars Math. Contemp. 21 (2021) #P2.06 / 259–269
[2] L. Babai, Automorphism groups, isomorphism, reconstruction, in: Handbook of combinatorics,
Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, pp. 1447–1540, 1995.
[3] P. J. Cameron, M. Giudici, G. A. Jones, W. M. Kantor, M. H. Klin, D. Marušič and L. A.
Nowitz, Transitive permutation groups without semiregular subgroups, J. London Math. Soc.
(2) 66 (2002), 325–333, doi:10.1112/s0024610702003484.
[4] M. Giudici, C. H. Li, P. Potočnik and C. E. Praeger, Homogeneous factorisations of graphs and
digraphs, European J. Combin. 27 (2006), 11–37, doi:10.1016/j.ejc.2004.08.003.
[5] C. D. Godsil, GRR’s for non solvable groups, in: Algebraic methods in graph theory, 1978 pp.
221–239.
[6] M. Grech, Direct products of automorphism groups of graphs, J. Graph Theory 62 (2009),
26–36, doi:10.1002/jgt.20385.
[7] M. Grech, The graphical complexity of direct products of permutation groups, J. Graph Theory
66 (2011), 303–318, doi:10.1002/jgt.20504.
[8] M. Grech, W. Imrich, A. D. Krystek and L. u. J. Wojakowski, Direct product of automorphism
groups of digraphs, Ars Math. Contemp. 17 (2019), 89–101, doi:10.26493/1855-3974.1498.
77b.
[9] M. Grech, A. Jeż and A. Kisielewicz, Graphical complexity of products of permutation groups,
Discrete Math. 308 (2008), 1142–1152, doi:10.1016/j.disc.2007.04.004.
[10] M. Grech and A. Kisielewicz, Direct product of automorphism groups of colored graphs, Dis-
crete Math. 283 (2004), 81–86, doi:10.1016/j.disc.2003.11.008.
[11] M. Grech and A. Kisielewicz, Totally symmetric colored graphs, J. Graph Theory 62 (2009),
329–345, doi:10.1002/jgt.20397.
[12] M. Grech and A. Kisielewicz, All totally symmetric colored graphs, Discrete Math. Theor.
Comput. Sci. 15 (2013), 133–145, doi:10.46298/dmtcs.630.
[13] W. Imrich, On products of graphs and regular groups, Israel J. Math. 11 (1972), 258–264,
doi:10.1007/bf02789317.
[14] A. Kisielewicz, Supergraphs and graphical complexity of permutation groups, Ars Combin. 101
(2011), 193–207.
[15] C. H. Li, T. K. Lim and C. E. Praeger, Homogeneous factorisations of complete graphs
with edge-transitive factors, J. Algebraic Combin. 29 (2009), 107–132, doi:10.1007/
s10801-008-0127-2.
[16] C. H. Li and C. E. Praeger, On partitioning the orbitals of a transitive permutation group, Trans.
Amer. Math. Soc. 355 (2003), 637–653, doi:10.1090/s0002-9947-02-03110-0.
[17] L. A. Nowitz and M. E. Watkins, Graphical regular representations of non-abelian groups.
I, II, Canadian J. Math. 24 (1972), 993–1008; ibid. 24 (1972), 1009–1018, doi:10.4153/
cjm-1972-101-5.
[18] W. Peisert, Direct product and uniqueness of automorphism groups of graphs, Discrete Math.
207 (1999), 189–197, doi:10.1016/s0012-365x(99)00044-8.
[19] G. Sabidussi, raph multiplication, Math. Z. 72 (1959), 446–457, doi:10.1007/bf01162967.
[20] M. E. Watkins, On the action of non-Abelian groups on graphs, J. Combinatorial Theory Ser.
B 11 (1971), 95–104, doi:10.1016/0095-8956(71)90019-0.
[21] M. E. Watkins and L. A. Nowitz, On graphical regular representations of direct products of
groups, Monatsh. Math. 76 (1972), 168–171, doi:10.1007/bf01298284.
M. Grech: Decompositions of the automorphism groups of edge-colored graphs into the . . . 269
[22] H. Wielandt, Permutation groups through invariant relations and invariant functions [83], in:
B. Huppert and H. Schneider (eds.), Volume 1 Group Theory, De Gruyter, pp. 237–296, 1969,
doi:10.1515/9783110863383.237.
[23] H. Wielandt, Permutation groups through invariant relation and invariant functions, in: B. Hup-
pert and H. Schneider (eds.), Mathematical works, vol. 1, Group theory, Walter de Gruyter Co.,
Berlin, pp. 237–266, 1994.
[24] H. P. Yap, Some Topics in Graph Theory, volume 108 of London Mathematical Society Lecture
Note Series, Cambridge University Press, Cambridge, 1986, doi:10.1017/cbo9780511662065.

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P2.07 / 271–282
https://doi.org/10.26493/1855-3974.2454.892
(Also available at http://amc-journal.eu)
Efficient proper embedding of a daisy cube*
Aleksander Vesel
Faculty of Natural Sciences and Mathematics, University of Maribor,
SI-2000 Maribor, Slovenia
Received 4 October 2020, accepted 14 June 2021, published online 25 October 2021
Abstract
For a set X of binary words of length h the daisy cube Qh(X) is defined as the subgraph
of the hypercube Qh induced by the set of all vertices on shortest paths that connect vertices
of X with the vertex 0h. A vertex in the intersection of all of these paths is a minimal vertex
of a daisy cube. A graph G isomorphic to a daisy cube admits several isometric embeddings
into a hypercube. We show that an isometric embedding is proper if and only if the label
0h is assigned to a minimal vertex of G. This result allows us to devise an algorithm which
finds a proper embedding of a graph isomorphic to a daisy cube into a hypercube in linear
time.
Keywords: Daisy cube, partial cube, isometric embedding, proper embedding.
Math. Subj. Class. (2020): 05C12, 05C85
1 Introduction
Hypercube is one of the most important interconnection scheme for multicomputers. An
obstacle to a direct application of a hypercube is the fact that the number of different hy-
percubes is very small with respect to the wanted (maximum) number of nodes, that is to
say, the number of vertices of a hypercube is always equal to a power of two. For that
reason, several other interconnection topologies for multicomputers based on hypercubes
have been proposed. These graphs have been devised to preserve a hypercube’s most essen-
tial properties while allowing more variety of resulting specific graphs. The corresponding
families of graphs are mostly various subgraphs of a hypercube, of which its isometric sub-
graphs, i.e. its induced subgraphs that preserve distances, are of particular importance. A
crucial problem in this scope is to find an embedding of a graph of this type to a hypercube
(see for example [1, 4, 16]).
*This work was supported by the Slovenian Research Agency under the grants P1-0297, J1-2452, J1-9109 and
J1-1693.
E-mail address: aleksander.vesel@um.si (Aleksander Vesel)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
272 Ars Math. Contemp. 21 (2021) #P2.07 / 271–282
Quite recently, a new concept which led to the class of graphs called daisy cubes has
been proposed in [9]. It has been shown that daisy cubes are isometric subgraphs of a
hypercube, moreover, they include several other important classes of graphs, some well-
known examples are Fibonacci and Lucas cubes (see, for example [2, 5, 8, 11]) as well
as some other families of generalized Fibonacci cubes and generalized Lucas cubes [3, 6,
7, 15]. Daisy cubes play an essential role in showing that specific generalized Fibonacci
cubes’ cube-complements are isometric subgraphs of a hypercube [13]. It is also proven
that a class of graphs, which is of significant importance in chemical graph theory, also
belongs to daisy cubes [14].
In [12], daisy cubes are characterized in terms of an expansion procedure. For a given
graph G isomorphic to a daisy cube, but without the corresponding embedding into a hy-
percube, an algorithm which finds a proper embedding of G into a hypercube in O(mn)
time is also presented.
Several challenging open problems concerning daisy cubes have been proposed [9, 12].
In this paper, we focus our study to the following one.
Problem 1.1. Is there a faster way of finding the vertex 0h of a daisy cube Qh(X) than the
one provided in [12]?
It is also noted that a positive answer to Problem 1 would give a linear time algorithm
for finding a proper embedding of a graph isomorphic to a daisy cube.
The paper is organized as follows. In the next section some basic definitions, concepts
and results needed in the sequel are given. In Section 3, a notion of a minimal vertex of a
daisy cube is introduced. Some necessary and sufficient conditions that a minimal vertex
has to fulfill are also given. In Section 4, it is shown that an isometric embedding of a graph
isomorphic to a daisy cube, but without the corresponding embedding into a hypercube, can
be constructed in linear time even if a minimal vertex of a daisy cube is unknown. The last
section shows that an isometric embedding devised in the Section 4 can be applied in order
to find a proper embedding within the same time bound.
2 Preliminaries
Let B = {0, 1}. If b is a word of length h over B, that is, b = (b1, . . . , bh) ∈ Bh, then we
will briefly write b as b1 . . . bh. If x, y ∈ Bh, then the Hamming distance H(x, y) between
x and y is the number of positions in which x and y differ.
We will use [n] for the set {1, 2, . . . , n}.
The hypercube of order h or simply h-cube, denoted by Qh, is the graph G = (V,E)
where the vertex set V (G) is the set of all binary strings b = b1b2 . . . bh, bi ∈ {0, 1} for
all i ∈ [h], and two vertices x, y ∈ V (G) are adjacent in Qh if and only if the Hamming
distance between x and y is equal to one.
For a binary string b = b1b2 . . . bn, let bi = 1 − bi for i ∈ [h]. The weight of u ∈ Bh
is w(u) =
∑h
i=1 ui, in other words, w(u) is the number of 1s in the word u. For the
concatenation of bits the power notation will be used, for instance 0h = 0 . . . 0 ∈ Bh.
If G is a connected graph, then the distance dG(u, v) (or simply d(u, v)) between ver-
tices u and v is the length of a shortest u, v-path (that is, a shortest path between u and v)
in G. The set of vertices lying on all shortest u, v-paths is called the interval between u
and v and denoted by IG(u, v) [10]. We will also write I(u, v) when G will be clear from
the context.
A. Vesel: Efficient proper embedding of a daisy cube 273
If G is a graph and X ⊆ V (G), then G[X] denotes the subgraph of G induced by X .
If u is a vertex of a graph G, let N(u) denote the set of neighbors of u. Moreover, let
N [u] = N(u) ∪ {u}.
Let G = (V,E) be a graph. A mapping α : V (G) → V (Qh) is an isometric embedding
of G into Qh if dQh(α(u), α(v)) = dG(u, v) for every u, v ∈ V (G). If u ∈ V (G), we will
denote the i-th coordinate of α(u) as α(i)(u).
Let G be a connected graph. The isometric dimension of G is the smallest integer h
such that G admits an isometric embedding into Qh. Isometric subgraphs of hypercubes
are called partial cubes.
Let ≤ be the partial order on V (Qh) defined with u1 . . . uh ≤ v1 . . . vh if ui ≤ vi holds
for all i ∈ [h]. For X ⊆ V (Qh) the graph induced by the set {v ∈ V (Qh) | v ≤ x for some
x ∈ X} is a daisy cube of Qh generated by X and denoted by Qh(X).
Let also ∨, ∧ and ⊕ denote the bitwise OR, bitwise AND and bitwise exclusive OR
operator, respectively.
By a slight abuse of definition, we will say that a graph G is a daisy cube if it is
isomorphic to a daisy cube generated by some X ⊆ V (Qh). If G is a daisy cube Qh(X),
then G may admit more than one isometric embedding of G into the h-cube. Let XG ⊆ Bh
be the set of labels of the vertices of G assigned by an isometric embedding α, i.e. XG =
α(V (G)). We say that α is a proper embedding of G if G is isomorphic to Qh(XG).
Let G be a graph isomorphic to a daisy cube of Gh and let α denote a proper embedding.
Note that every permutation of indices of α yields basically the “same” embedding. We
say that proper embeddings α and β are equivalent if β can be obtained from α by a
permutation of its indices.
For a daisy cube Qh(X), let X̂ denote the antichain consisting of the maximal elements
of the poset (X,≤). It was shown in [9] that Qh(X) = Qh(X̂). Hence, for a given set
X ⊆ Bn it is enough to consider the antichain X̂ . The vertices of Qh(X) from X̂ are
called the maximal vertices of Qh(X). More generally, if G is a daisy cube of Qh with a
proper embedding α such that α(v) = 0h, then X ⊆ V (G) is the set of maximal vertices
of G with respect to v if G ∼= Qh(α(X)) and α̂(X) = α(X). Moreover, v is the minimal
vertex of G with respect to α. We also say that v is a minimal vertex of G if there exists a
proper embedding α such that α(v) = 0h.
The following result shows that a daisy cube is a subgraph of Qh induced by the union
of intervals between 0h and the vertices from X̂ [9].
Lemma 2.1. Let X ⊆ Bh. Then Qh(X) = Qh[∪x∈X̂I(0
h, x)].
3 Minimal vertices of a daisy cube
If u ∈ V (Qh(X)), then I(0n, u) induces a w(u)-cube in Qh(X). Note that if x ∈ X̂ , then
the cube induced by I(0n, x) is maximal in Qh(X), i.e., it is not contained in any other
cube that belongs to Qh(X).
If x ∈ Bh, let Sx denote the set of indices of v with xi = 1, i.e., Sx = {i |xi =
1 and i ∈ [h]}.
Let v ∈ Bh and let vβ : Bh → Bh be the function defined as
vβ(i)(u) =
{
ui, vi = 0
ūi, vi = 1
274 Ars Math. Contemp. 21 (2021) #P2.07 / 271–282
Lemma 3.1. Let G be a graph isomorphic to a daisy cube of Qh with a proper em-
bedding α such that α(v0) = 0h and X̂ ⊆ V (G) is its corresponding maximal set. If
v ∈ ∩x∈X̂I(v
0, x), then
(i) vβ restricted to α(V (G)) is a bijection that maps to α(V (G)),
(ii) vβ ◦ α is a proper embedding of G with the minimal vertex v and the maximal vertex
set Y = {y | vβ(α(y)) = α(x) and x ∈ X̂}.
Proof. (i) We have to show that if v ∈ ∩x∈X̂I(v
0, x), then for every u ∈ α(V (G))) there
is exactly one vβ(u) ∈ α(V (G)). Note that α−1(u) ∈ I(v0, x) and v ∈ I(v0, x) for some
x ∈ X̂ . Thus, Su ⊆ Sα(x) and Sα(v) ⊆ Sα(x). It follows that Svβ(u) ⊆ Sα(x). Since α is
proper, α(V (G)) = ∪x∈X̂I(0
h, α(x)) by Lemma 2.1 and we obtain vβ(u) ∈ V (α(G)).
In order to see that vβ is injective, note that vβ(vβ(u)) = u for every u ∈ α(V (G)).
Suppose to the contrary that there exist u, z ∈ α(V (G)), u ̸= z, such that vβ(u) =v β(z).
It follows that vβ(vβ(u)) =vβ(vβ(z)) and thus u = z, which yields a contradiction.
(ii) By (i), vβ maps from α(V (G)) to α(V (G)). Let x ∈ X̂ and recall that vβ(vβ(α(x))) =
α(x). Thus, if y ∈ V (G) such that α(y) = vβ(α(x)), we have vβ(α(y)) = α(x). More-
over, vβ(v) = 0h. It follows that Y = {y | vβ(α(y)) = α(x) and x ∈ X̂} is the maximal
vertex set of G with respect to vβ ◦ α, while v is the corresponding minimal vertex.
00000
00011
00100 00111
01000 01011
00001
0001000010 0000110000 10011
10010 10001
10011 1000001011 0100000111 00100
10001 1001000011 0000001001 0101001010 0100100101 0011000110 00101
x
x’
v
v
3
0
y
y’ z’
z
Figure 1: Two proper embeddings of a daisy cube.
Figure 1 shows two proper embeddings of a daisy cube G. The embedding on the left
hand side, say α, admits the set of maximal vertices X̂ = {x, y, z} with labels α(x) =
10011, α(y) = 01011 and α(z) = 00111. Let v0 ∈ V (G) such that v0 = α−1(00000).
Then I(v0, x) ∩ I(v0, y) ∩ I(v0, z) = {v0, v1, v2, v3}, where α(v3) = 00011. The em-
bedding on the right hand side of Figure 1 is v
3
β ◦ α with the set of maximal vertices
Y = {x′, y′, z′}, where the corresponding labels are α(x′) = 10000, α(y′) = 01000
and α(z′) = 00100. Note also that v
3
β(α(x′)) = 10011, v
3
β(α(y′)) = 01011 and
v3β(α(z′)) = 00111.
Let u ∈ V (G) where G = Qh(X) and let Xu be the maximal subset of X̂ with the
property u ∈ ∩x∈XuI(0h, x). Let Gu be the graph induced by the set ∪x∈XuI(0h, x), i.e.
Gu = G[∪x∈XuI(0h, x)]. Note that by Lemma 3.1 and Lemma 2.1, Gu is a daisy cube of
A. Vesel: Efficient proper embedding of a daisy cube 275
Qh and u is its minimal vertex. Observe for example the graph Q4(0111, 1011, 1101, 1110)
on the right hand side of Figure 2: if u = 1100, then Xu = {1110, 1101}.
As noted in [12], an efficient way of finding a minimal vertex of a daisy cube G would
give a linear time algorithm for finding a proper embedding of G. It was also shown that
if G is a daisy cube of Qh, then a minimal vertex of G is of degree h. It is not difficult to
see that a vertex of degree h need not to be a minimal vertex of G. Note for example that
Q−h (that is a vertex deleted Qh) admits 2
h − h − 1 vertices of degree h and exactly one
minimal vertex (see also Figure 2, where Q−4 is depicted).
Proposition 3.2. Let u ∈ V (G), where G = Qh(X) and d(u) = h . Moreover, let Xu be
the maximal subset of X̂ such that u ∈ ∩x∈XuI(0h, x). Then for every proper embedding
α, the minimal vertex of G with respect to α belongs to ∩x∈XuI(0h, x).
Proof. Let v be the minimal vertex of G with respect to some proper embedding. Note that
for every x ∈ X̂ and every u ∈ I(0h, x) we have d(v, u) ≤ |Sx|. Suppose to the contrary
that v ̸∈ ∩x∈XuI(0h, x). It follows that there exists x ∈ Xu such that v ̸∈ I(0h, x). Since
u ∈ I(0h, x), it follows that Su ⊆ Sx. Moreover, since v ̸∈ I(0h, x), there exists an index
j ̸∈ Sx such that vj = 1. It follows that the string u defined by
ui =
{
v̄i, i ∈ Sx
0, otherwise
is a vertex of I(0h, x) with d(v, u) > |Sx| and we obtain a contradiction.
Theorem 3.3. If G = Qh(X) and x̂ = ∧x∈X̂x, then for every proper embedding α, v is
the minimal vertex of G with respect to α if and only if v ∈ ∩x∈X̂I(0
h, x) = I(0h, x̂).
Proof. By Lemma 3.1 and Proposition 3.2, v is a minimal vertex of G, if and only if
v ∈ ∩x∈X̂I(0
h, x). Note that v ∈ ∩x∈X̂I(0
h, x) if and only if Sv ⊆ ∩x∈XSx. Since
Sx̂ = ∩x∈XSx, for every v ∈ V (G) we have v ∈ ∩x∈X̂I(0
h, x) if and only if v ≤ x̂. It
follows that ∩x∈XI(0h, x) = I(0h, x̂) and the assertion follows.
4 Isometric embedding
If v is a vertex of a partial cube G, then NvG(u) (or simply N
v(u) ) is the set of neighbors
of u which are closer to v than u, more formally NvG(u) := {z | z ∈ N(u) and d(v, z) =
d(v, u)− 1},
If G is a graph isomorphic to a hypercube (but without an embedding), then its isometric
embedding is easy to obtain as shown in the next result.
Proposition 4.1. Let G be a graph isomorphic to a h-cube, v an arbitrary vertex of G and
α : V (G) → V (Qh) a function such that α(v) = 0d, the vertices of N(v) obtain pairwise
different labels of the form 0i−110h−i, i ∈ [h], while for the other vertices u ∈ V (G)
ordered by an increasing distance from v, we set α(u) = ∨z∈Nv(u)α(z). Then α is an
isometric embedding of G into Qh. Moreover, when a labeling of vertices in N [v] is chosen,
α is unique.
Proof. Since a hypercube is vertex-transitive, we may choose an arbitrary vertex v of G
and set α(v) = 0h. Moreover, for every u ∈ V (G) with d(v, u) = s, s ≥ 1, we must have
Nv(u) = {z | α(i)(z) = α(i)(u) = 1 for exactly one i ∈ [h] and α(j)(z) = α(j)(u) for
276 Ars Math. Contemp. 21 (2021) #P2.07 / 271–282
every j ∈ [h] \ {i}}. Thus, α(u) = ∨z∈Nv(u)α(z). It follows that for chosen labeling of
vertices in N [v], α is unique.
Lemma 4.2. Let G be partial cube of isometric dimension h, u a vertex of degree h in
G and let for every v ∈ V (G) \ N [u] it holds that |Nu(v)| ≥ 2. Define the function
α : V (G) → V (Qh) such that α(u) = 0h, the vertices of N(u) obtain pairwise different
labels of the form 0i−110h−i, i ∈ [h], while for the other vertices v ∈ V (G) ordered by an
increasing distance from u, we set α(v) = ∨z∈Nu(v)α(z). Moreover,
(i) α is an isometric embedding of G into Qh,
(ii) when a fixed embedding of vertices in N [v] is chosen, α is unique.
Proof. Since G is a partial cube of dimension h, we may assume that G is an isometric
subgraph of an (unlabeled) h-cube H . Let β be an embedding of H with respect to v as
defined in Proposition 4.1 and let α be an embedding of G such that for every z ∈ N [u] we
set α(z) = β(z). Since |NuG(v)| ≥ 2 and NuG(v) ⊆ NuH(v) for every v ∈ V (G) \N [u], it
follows that α(v) = β(v) for every vertex v ∈ V (G). By Proposition 4.1, β is an isometric
embedding of H into Qh. Thus, α is an isometric embedding of H into Qh. Moreover, by
Proposition 4.1, α is unique for a fixed embedding of vertices in N [v].
Corollary 4.3. Let G be a graph isomorphic to a daisy cube of order h. If v is a minimal
vertex of G and α an isometric embedding with α(v) = 0h, then α is proper.
Proof. Since v is a minimal vertex of G, there exist a proper embedding, say β, such that
β(v) = 0h. We may also assume w.l.o.g. that for every u ∈ N(v) we have β(u) = α(u).
From Lemma 4.2 then it follows that β(u) = α(u) for every v ∈ V (G).
Remark 4.4. If G is isomorphic to a daisy cube and α a proper embedding of G, then
different selections of labels for vertices of N(u) yield different but equivalent proper em-
beddings.
If G is a partial cube and α its isometric embedding to Qh, let Wi(G) denote the set of
vertices of G with weight i, i.e. Wi(G) = {v |w(α(v)) = i}.
We will also need the following result.
Proposition 4.5. If G is a partial cube, α its isometric embedding to Qh and v ∈ V (G)
such that w(α(v)) = i, then |N(v) ∩Wi−1(G)| ≤ i.
Proof. Since α is isometric embedding of G to Qh, for every v ∈ V (G) with w(α(v)) = i,
we have NG(v) ⊆ NQh(v). Moreover, |N(v) ∩ Wi−1(Qh)| = i and therefore |N(v) ∩
Wi−1(G)| ≤ i.
Proposition 4.6. Let G = Qh(X), x, y ∈ X̂ and x ̸= y. If u ∈ I(0h, x) and v ∈ I(0n, y)
such that u, v ̸∈ I(0n, x) ∩ I(0h, y) then uv ̸∈ E(G).
Proof. Suppose to the contrary that there exist u ∈ I(0h, x) and v ∈ I(0h, y) such that
u, v ̸∈ I(0h, x) ∩ I(0h, y) and d(u, v) = 1. Since X̂ is maximal, there exist at least two
indices i, j ∈ [h], such that xi ̸= yi and xj ̸= yj (otherwise we have either x ≤ y or
y ≤ x). Suppose w.l.o.g. xi = 1, yj = 1 and uk = vk for every k ∈ [h] \ {i, j}. If ui = 0
(resp. vj = 0), then u ∈ I(0h, y) (resp. v ∈ I(0h, x)). It follows that ui = vj = 1. But
then u = v and we obtain a contradiction.
A. Vesel: Efficient proper embedding of a daisy cube 277
Proposition 4.7. Let G = Qh(X), Xu be the maximal subset of X̂ such that u ∈
∩x∈XuI(0h, x) and Gu = G[∪x∈XuI(0h, x)]. If u ∈ V (G) and d(u) = h, then N(u) ⊆
V (Gu).
Proof. Suppose to the contrary that there exists v ∈ N(u) such that v ̸∈ ∪x∈XuI(0h, x).
It follows that there exists y ∈ X̂ − Xu such that v ∈ I(0h, y). Since u ∈ I(0h, x) for
some x ∈ X̂ and x ̸= y, Proposition 4.6 yields a contradiction.
Proposition 4.8. Let G = Qh(X), u ∈ V (G) and Xu be the maximal subset of X̂ such
that u ∈ ∩x∈XuI(0h, x). If d(u) = h, then | ∪x∈Xu Sx| = h.
Proof. Suppose | ∪x∈Xu Sx| < h. It follows that there exist j ∈ [h] such that for all
v ∈ ∪x∈XuI(0h, x) we have vj = 0. Since d(u) = h , there exists z ∈ N(u) such that
zj = 1. It follows that z ̸∈ ∪x∈XuI(0h, x). Thus, there exists y ∈ X̂ − Xu such that
v ∈ I(0h, y). Proposition 4.7 yields a contradiction.
Lemma 4.9. Let G = Qh(X) and u ∈ V (G) such that d(u) = h. Then |Nu(v)| ≥ 2 for
every v ∈ V (G) \N [u].
Proof. Let Xu be the maximal subset of X̂ with the property u ∈ ∩x∈XuI(0h, x) and
Gu = G[∪x∈XuI(0h, x)]. By Lemma 3.1 and Lemma 2.1, Gu is a daisy cube and u its
minimal vertex. It follows that the lemma holds for every v ∈ V (Gu). Suppose then that
v ̸∈ ∪x∈XuI(0h, x). Thus, there exists y ∈ X̂ − Xu, such that v ∈ I(0h, y). Note that
Su ⊆ ∩x∈XuSx.
Let Su+ = {i |ui = 1 and vi = 0} and Su− = {i | vi = 1 and ui = 0}.
We first show that |Su−| ≠ 1. Suppose to the contrary that there exists exactly one
index i ∈ [h] \ Su+, such that vi = 1 and ui = 0. Since d(u) = h, by Proposition 4.8,
there exists x ∈ Xu such that xi = 1. Note also that Su ⊆ Sx and since xi = 1, we have
Sv ⊆ Sx. It follows that v ≤ x and we obtain a contradiction.
If |Su+| = 0, then vertices of I(u, v) induce a |Su−|-cube in G. Thus, v admits |Su−|
neighbors at distance d(u, v)−1 from u. Clearly, |Su+| = 0 implies |Su−| > 0. Moreover,
since we show above that |Su−| ≠ 1, we have |Su−| ≥ 2 and the case is settled.
If |Su+| > 0, we may find i, j ∈ Su− such that i ̸= j. Let z and z′ be vertices
obtained from v by setting the i-th and j-th coordinate to zero, respectively. Obviously,
z, z′ ∈ Nu(v).
Since we show that we obtain |Nu(v)| ≥ 2 for every value of |Su+|, the lemma holds
for every v ∈ V (G) \N [u]. This assertion concludes the proof.
Lemma 4.9 is the basis for the next algorithm which finds an isometric embedding for
an unlabeled graph isomorphic to a daisy cube of dimension h.
Procedure Embedding(G, h, β, u);
1. u is a vertex of degree h in G;
2. β(u) := 0h;
3. i := 1;
4. Q := ∅; {Q is an empty queue}
5. for all v ∈ V (G) do p(v) := 0;
6. for all v ∈ N(u) do begin
β(v) := 0i−110h−i;
278 Ars Math. Contemp. 21 (2021) #P2.07 / 271–282
0000
1100
0001
1101
1101
0001
1001
0101
0100
1000
1011
0111
0101
1001
1000
0100
0111
1011
1111
0011
1100 0000
0010
1110
1110
0010
1010
0110
0110 1010
x
u
y
z
Figure 2: An isometric (left) and proper (right) embedding of a daisy cube isomorphic to
Q−4 .
i := i+ 1;
p(v) := u;
Insert v in the end of Q;
end;
7. while Q ̸= ∅ do begin
7.1 Remove the first vertex v from Q;
7.2. for all z ∈ N(v) do
if p(z) = 0 then begin
p(z) := v;
Append z to the end of Q;
end
else β(z) := β(v) ∨ β(p(z));
end.
Theorem 4.10. If G is a daisy cube, then an isometric embedding of G can be found in
linear time.
Proof. Note first that Lemma 4.2 defines the procedure to construct an isometric embed-
ding of G into Qh. Let α and β be isometric embeddings as defined in Lemma 4.2 and
algorithm Embedding, respectively. Suppose that u is the vertex being labeled 0h both by
the algorithm and by the construction of Lemma 4.2. Clearly, for every v in N [u] we could
have α(v) = β(v). Note also that in the essence the algorithm performs a BFS search
in G (see for example [4, Section 17.3]). Thus, for every z ∈ N(v) of Step 7.2 we have
d(u, z) = d(u, p(z)) + 1 = d(u, v) + 1. It follows that v, p(z) ∈ Nu(z). By Lemma 4.9,
since d(u) = h, for every v ∈ V (G)\N [u] we have |NuG(v)| ≥ 2. Therefore, α(z) = β(z)
for every z ∈ V (G) \N [u].
For the time complexity of the algorithm, note that the number of the executions of the
body of the loop in Step 7.2 is bounded by the number of edges of a graph. Since the time
complexity of the body of the loop is constant, the overall number of step of the algorithm
is linear in the number of the edges of the graph.
A. Vesel: Efficient proper embedding of a daisy cube 279
5 Proper embedding
Lemma 5.1. Let G be a daisy cube of Qh, v a minimal vertex of G and u a vertex of degree
h of G. If β is an isometric embedding of G such that β(u) = 0h, then vβ ◦ β is a proper
embedding of G.
Proof. Note that vβ(β(v)) = 0h. Since β is isometric, it is easy to see that vβ ◦ β is also
isometric. Corollary 4.3 now yields the assertion.
Let u be a vertex of degree h of G = Qh(X). Let Xu be the maximal subset of X̂
with the property u ∈ ∩x∈XuI(0h, x) and Gu = G[∪x∈XuI(0h, x)]. Recall that Gu is a
daisy cube of Qh and u its minimal vertex. If β is an isometric embedding of G such that
β(u) = 0h, let Y u be the set of maximal vertices of Gu with respect to u and let Zu be the
set of vertices z of V (G) \ V (Gu) with the property Nu(z) = N(z).
Proposition 5.2. Let u be a vertex of degree h of G = Qh(X). If β is an isometric
embedding of G such that β(u) = 0h, then Y u = {y |β(y) = x and x ∈ Xu}.
Proof. As noted above, Gu is a daisy cube of Qh and u its minimal vertex. Since u is
of degree h and β(u) = 0h, the restriction of β to V (Gu) is a proper embedding of Gu.
Moreover, since every permutation of indices of a proper embedding yields an equivalent
embedding, we may assume w.l.o.g. that for every z ∈ N(u) we have β(z) = 0i−110h−i
if and only if ui ̸= zi. It follows that for every w ∈ N(0h) we have uβ(β(w)) = w.
By Lemma 3.1, uβ ◦ β is proper. Moreover, by Lemma 4.2, uβ(β(v)) = v for every
v ∈ V (Gu). From Lemma 3.1 then follows that Y u = {y |β(y) = x and x ∈ Xu}.
Proposition 5.3. Let u be a vertex of degree h of G = Qh(X) and z ∈ Zu. If β is
an isometric embedding of G and β(u) = 0h, then there exists y ∈ X̂ − Xu such that
z ∈ I(0h, y). Moreover,
β(i)(z) =
{
0, i ∈ Su
yi, i ̸∈ Su
Proof. Let Xu be the maximal subset of X̂ with the property u ∈ ∩x∈XuI(0h, x). By
Lemma 2.1, since z ̸∈ ∪x∈XuI(0h, x), there must be y ∈ X̂ −Xu such that z ∈ I(0h, y).
By Nu(z) = N(z), we have d(u, z) ≥ d(u, v) for every v ∈ I(0h, y). If vi = 1 for
some i ∈ Su, then let v′ be the vertex of G such that v′j = vj for every j ̸= i and
v′i = 0. Obviously, v
′ ≤ y, thus v′ ∈ I(0h, y). Moreover, since β(i)(v′) = 1, we have
d(u, v′) > d(u, v) and we obtain a contradiction. It follows that the assertion holds for
every i ∈ Su. If i ̸∈ Su, then β(i)(v) = vi for every v ∈ I(0h, y). Since y is maximal in
I(0h, y), the assertion follows.
Theorem 5.4. Let u be a vertex of degree h of G = Qh(X). If β is an isometric embedding
of G such that β(u) = 0h, ŷ = ∧y∈Y uβ(y), ẑ = ∧z∈Zuβ(z)(∧1h) and v = β−1(ŷ ∧ ẑ),
then v is the minimal vertex of G with respect to vβ ◦ β.
Proof. Note first that β = β−1, thus, for every b ∈ Bh and every i ∈ [h] it holds
β(i)(b) = β
−1
(i) (b) =
{
b̄i, i ∈ Su
bi, i ̸∈ Su
(5.1)
280 Ars Math. Contemp. 21 (2021) #P2.07 / 271–282
Let x̂ = ∧x∈Xux. By Proposition 5.2, we have Y u = {y |β(y) = x and x ∈ Xu}.
Thus, x̂ = ŷ. Note that by Proposition 3.2, every minimal vertex of G belongs to I(0h, x̂).
If Xu = X̂ , then Zu = ∅ and we get β−1(ŷ ∧ ẑ) = β−1(ŷ) = β−1(x̂). By equation
(5.1), we have β−1(x̂) ≤ x. It follows that β−1(x̂) ∈ I(0h, x̂) and we are done.
Otherwise, let z ∈ Zu be such that z ∈ I(0h, y) for some y ∈ X̂ − Xu. We
have to show that β−1(x̂ ∧ β(z)) is a minimal vertex of ∪x∈XuI(0h, x) ∪ I(0h, y), i.e.
Sβ
−1(x̂∧β(z)) ⊆ Sx̂∧y .
By Proposition 5.3, we have
β(i)(z) =
{
0, i ∈ Su
yi, i ̸∈ Su
Since Su ⊆ Sx̂, we have
(x̂ ∧ β(z))i =
{
yi, i ̸∈ Sx̂ \ Su
0, otherwise
By equation (5.1), we have β−1i (x̂ ∧ β(z)) = 0 for every i ∈ [h] \ Sx̂∧y . Since we can
repeat the above discussion for every z ∈ Zu, we showed that β−1(x̂ ∧ ẑ) = β−1(ŷ ∧ ẑ)
is a minimal vertex of G. Moreover, since by Lemma 5.1 it follows that β
−1(ŷ∧ẑ)β ◦ β is a
proper embedding of G, the proof is complete.
Figure 2 shows two embeddings of a daisy cube G isomorphic to Q−4 . The embedding
β on the left hand side is determined such that β(u) = 0000 (note that d(u) = 4). Since u is
not minimal in G, the embedding β is isometric but not proper. From Xu = Y u = {x, y}
and Zu = {z} we get ŷ = 1110∧1101 = 1100, ẑ = 1111 and ŷ∧ẑ = 1100∧1111 = 1100.
Moreover, the minimal vertex of G is v = β−1(1100) and vβ ◦ β is the proper embedding
of G as described in Lemma 5.1. That is to say, we obtain the proper embedding of G by
assigning β(w)⊕ 1100 to every w ∈ V (G).
Theorem 5.4 is the basis for the next algorithm, which finds a proper embedding of a
graph isomorphic to a daisy cube of dimension h.
Procedure Proper(G, h, α);
1. Embedding(G, h, β, u);
2. for i := 1 to h+ 1 do Wi := ∅;
3. for all v ∈ V (G) do Ww(β(v)) := Ww(β(v)) ∪ {v};
4. for all v ∈ V (G) do q(v) := 0;
5. for i := 1 to h do begin
5.1. for all x ∈ Wi do
5.1.1 if
∑
y∈N(x)∩Wi−1 q(y) = i(i− 1) then begin
q(x) := i;
for all y ∈ N(x) ∩Wi−1 do q(y) := 0;
end
5.1.2 else if N(x) ∩Wi+1 = ∅ then q(x) := i
6. s := 1h;
7. for all v ∈ V (G) do
7.1. if q(v) ̸= 0 then s := s ∧ β(v);
8. for all v ∈ V (G) do α(v) := s⊕ β(v);
end.
A. Vesel: Efficient proper embedding of a daisy cube 281
Theorem 5.5. A proper embedding of an unlabeled graph isomorphic to a daisy cube can
be found in linear time.
Proof. We first show that the algorithm Proper finds a proper embedding of G. As shown
in Theorem 4.10, embedding β provided by the algorithm Embedding is isometric. With
respect to Theorem 5.4 and Step 7, we have to show that if q(v) ̸= 0, then either v ∈ Y u
or v ∈ Zu. Clearly, in Step 3, all vertices at distance i from u are inserted in Wi, while in
Step 4, q(v) is set to 0 for every v ∈ V (G). The value of q(v) is altered either in Step 5.1.1
or in Step 5.1.2.
Let w(x) = i. We show that q(x) = i in the i-th iteration of for loop if and only if
either I(u, x) induces an i-cube or x ∈ Zu. Note that I(u, x) induces an i-cube, if and only
|N(x)∩Wi−1| = i and for every y ∈ N(x)∩Wi−1 the set I(u, y) induces a (i− 1)-cube.
Moreover, if x ∈ Y u, then I(u, x) induces a maximal i-cube in Gu.
In the first iteration of Step 5, for every vertex of W1 the value of q is set to 1. In the
next iteration, when a vertex x of W2 is considered, these values for two vertices of W1,
say y and y′, are set to zero if {u, y, y′, x} induce a 2-cube. Thus, for every x, y ∈ W1∪W2
we have
- q(y) = 1 if and only if x ∈ N(u) and there is no vertex y ∈ W2 such that I(u, y) ⊆
I(u, x) and I(u, x) induces Q2.
- q(x) = 2 if and only I(u, x) induces Q2.
Suppose now that for i ≥ 3 and y ∈ Wi−1 it holds that q(y) = i−1 if and only if I(u, y)
induces a maximal cube in G[W1∪W2 . . .∪Wi−1] or Nu(y) = N(y); otherwise, q(y) = 0.
Let w(x) = i. Note that |N(x) ∩ Wi−1| ≤ i by Proposition 4.5. Thus, the condition of
the if statement in Step 5.1.1 is fulfilled if and only if for every y ∈ N(x)∩Wi−1 we have
q(y) = i − 1, i.e. for every y ∈ N(x) ∩Wi−1 the set I(u, y) induces an (i − 1)-cube. If
the condition of the if statement returns true, then q(x) obtains the value i while for every
y ∈ N(x) ∩Wi−1 the value of q(y) is set to 0. If the condition of the if statement returns
false, then q(x) is set to i if and only if N(x) ∩Wi+1 = ∅, i.e. x ∈ Zu. Thus, we showed
that in the i-th iteration of the for loop q(x) = i if and only if either I(u, x) induces an i-
cube or x ∈ Zu. Since the claim holds for every i, we showed that if q(v) ̸= 0, v ∈ V (G),
then either v ∈ Y u or v ∈ Zu. From Theorem 5.4 then it follows that the string s computed
in Step 7 is equal to ŷ ∧ ẑ, where ŷ = ∧y∈Y uβ(y) and ẑ = ∧z∈Zuβ(z). By Theorem 5.4,
β−1(s) = v is a minimal vertex of G while the embedding α obtained in Step 8 is equal to
vβ ◦ β. Moreover, α is proper by Lemma 5.1.
In order to consider the time complexity of the algorithm, note first that all steps of the
algorithm except Step 5 can be executed in O(m) time, where m is the number of edges
of G. For the time complexity of Step 5 it is convenient to store the weights of vertices in
a vector, which allows that the weight of a vertex and therefore its inclusion in a set Wi
can be determined in constant time. Thus, the time complexity of Steps 5.1.1 and 5.1.2
is linear in the number of edges incident with the vertex x. Since Step 5 is performed for
every vertex of the graph, the total number of steps is bounded by the number of edges of
G. This assertion concludes the proof.
ORCID iDs
Aleksander Vesel https://orcid.org/0000-0003-3705-0071
282 Ars Math. Contemp. 21 (2021) #P2.07 / 271–282
References
[1] M. Aı̈der, S. Gravier and K. Meslem, Isometric embeddings of subdivided connected graphs
into hypercubes, Discrete Math. 309 (2009), 6402–6407, doi:10.1016/j.disc.2008.10.030.
[2] J. Azarija, S. Klavžar, Y. Rho and S. Sim, On domination-type invariants of Fibonacci cubes
and hypercubes, Ars Math. Contemp. 14 (2018), 387–395, doi:10.26493/1855-3974.1172.bae.
[3] P. Codara and O. M. D’Antona, Generalized Fibonacci and Lucas cubes arising from powers
of paths and cycles, Discrete Math. 339 (2016), 270–282, doi:10.1016/j.disc.2015.08.012.
[4] R. Hammack, W. Imrich and S. Klavžar, Handbook of product graphs, Discrete Mathematics
and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2nd edition, 2011, with a
foreword by Peter Winkler.
[5] W.-J. Hsu, Fibonacci cubes-a new interconnection topology, IEEE Trans. Parallel Distr. Sys-
tems 4 (1993), 3–12, doi:10.1109/71.205649.
[6] W. J. Hsu and M. J. Chung, Generalized fibonacci cubes, in: 1993 International Conference on
Parallel Processing - ICPP’93, volume 1, 1993 pp. 299–302, doi:10.1109/icpp.1993.95.
[7] A. Ilić, S. Klavžar and Y. Rho, Generalized Fibonacci cubes, Discrete Math. 312 (2012), 2–11,
doi:10.1016/j.disc.2011.02.015.
[8] A. Ilić and M. Milošević, The parameters of Fibonacci and Lucas cubes, Ars Math. Contemp.
12 (2017), 25–29, doi:10.26493/1855-3974.915.f48.
[9] S. Klavžar and M. Mollard, Daisy cubes and distance cube polynomial, European J. Combin.
80 (2019), 214–223, doi:10.1016/j.ejc.2018.02.019.
[10] H. M. Mulder, The interval function of a graph, volume 132 of Mathematical Centre Tracts,
Mathematisch Centrum, Amsterdam, 1980.
[11] E. Saygı, On the domination number and the total domination number of Fibonacci cubes, Ars
Math. Contemp. 16 (2019), 245–255, doi:10.26493/1855-3974.1591.92e.
[12] A. Taranenko, Daisy cubes: a characterization and a generalization, European J. Combin. 85
(2020), 103058, 10, doi:10.1016/j.ejc.2019.103058.
[13] A. Vesel, Cube-complements of generalized Fibonacci cubes, Discrete Math. 342 (2019),
1139–1146, doi:10.1016/j.disc.2019.01.008.
[14] P. Žigert Pleteršek, Resonance graphs of kinky benzenoid systems are daisy cubes, MATCH
Commun. Math. Comput. Chem. 80 (2018), 207–214.
[15] J. Wei, Y. Yang and G. Wang, Circular embeddability of isometric words, Discrete Math. 343
(2020), 112024, 11, doi:10.1016/j.disc.2020.112024.
[16] P. M. Winkler, Isometric embedding in products of complete graphs, Discrete Appl. Math. 7
(1984), 221–225, doi:10.1016/0166-218x(84)90069-6.
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P2.08 / 283–299
https://doi.org/10.26493/1855-3974.2398.7d5
(Also available at http://amc-journal.eu)
The polynomial method for list-colouring
extendability of outerplanar graphs
Przemysław Gordinowicz * , Paweł Twardowski
Institute of Mathematics, Lodz University of Technology, Al. Politechniki 10, Łódź, Poland
Received 3 August 2020, accepted 28 June 2021, published online 25 October 2021
Abstract
We restate theorems of Hutchinson [4] on list-colouring extendability for outerplanar
graphs in terms of non-vanishing monomials in a graph polynomial, which yields an Alon-
Tarsi equivalent for her work. This allows to simplify her proofs as well as obtain more
general results.
Keywords: Outerplanar graph, list colouring, paintability, Alon-Tarsi number.
Math. Subj. Class. (2020): 05C10, 05C15, 05C31
1 Introduction
In his famous paper [8] Thomassen proved that every planar graph is 5-choosable. Actually,
to proceed with an inductive argument, he proved the following stronger result.
Theorem 1.1 ([8]). Let G be any plane near-triangulation (every face except the outer one
is a triangle) with outer cycle C. Let x, y be two consecutive vertices on C. Then G can
be coloured from any list of colours such that the length of lists assigned to x, y, any other
vertex on C and any inner vertex is 1, 2, 3, and 5, respectively.
In other words vertices x and y can be precoloured in different colours. Basically,
this theorem implies that any outerplanar graph is 3-choosable. Moreover, lists of any two
neighbouring vertices can have a deficiency. To formalise this fact we say that a triple
(G, x, y), where G is outerplanar graph, x, y ∈ V (G) are neighbouring vertices is (1, 2)-
extendable in the sense that G is colourable from any lists whose length is 1, 2 and 3 for
vertex x, y and any other vertex, respectively.
*Corresponding author.
E-mail addresses: pgordin@p.lodz.pl (Przemysław Gordinowicz), pawel.twardowski@dokt.p.lodz.pl
(Paweł Twardowski)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
284 Ars Math. Contemp. 21 (2021) #P2.08 / 283–299
Hutchinson [4] analysed extendability of outerplanar graphs, in the case when the se-
lected vertices are not adjacent, showing that for any two vertices x, y of outerplanar graph
G a triple (G, x, y) is (2, 2)-extendable. Of course, it is enough to prove this for outerplane
2-connected near-triangulation only, as each outerplane graph can be extended to such a
graph just by adding some edges. The main theorem was the following.
Theorem 1.2 ([4]). Let G be outerplane 2-connected near-triangulation and x, y ∈ V (G),
x ̸= y. Let C : V (G) → {1, 2, 3} be any proper 3-colouring of G. Then
(i) (G, x, y) is not (1, 1)-extendable;
(ii) (G, x, y) is (1, 2)-extendable if and only if C(x) ̸= C(y);
(iii) (G, x, y) is (2, 2)-extendable.
Indeed, it is enough to prove the above theorem for near-triangulations with exactly 2
vertices of degree 2 and to let x and y be these degree 2 vertices. Hutchinson called such
configurations fundamental subgraphs. Such a configuration can be obtained by succes-
sively shrinking the outerplane near-triangulation along some chord (inner edge) that sepa-
rates the component of the graph not containing vertices x and y (in case when xy ∈ E(G)
this reduces to an edge xy). The general result follows now by succesive colouring of
shrank parts using Theorem 1.1 — the chord is an outer edge of the shrank component
and its endpoins (already coloured) are these 2 precoloured vertices. The details are in [4].
Also in [4], Hutchinson provided further results about extendability of general outerplanar
graphs, for which the conditions are more relaxed than those of Theorem 1.2, allowing for
(1, 1)-extendability.
One important thing is that the proper 3-colouring C mentioned in the theorem above is
not in any way connected to possible list colouring of G, but is rather an inherent property
of the graph. This is due to the fact that every 2-connected outerplane near triangulation has
an unique (up to permutation) 3-colouring, i.e the vertices graph can be uniquely partitioned
into 3 groups so that in every proper 3-colouring of the graph the vertices in the same group
will always have the same colour (the groups in this partition are called colour classes, as
the partition defines an equivalence relation). The reason for this is that the graph consists
entirely of triangles, and every vertex of a given triangle needs to be of different colour.
The situation of particular importance is when two vertices are in the same colour class.
This can be forced in two ways. One, mentioned in [4], is the so called chain of diamonds,
where the diamond is understood as K4 minus an edge. It is obviously a 2-connected
outerplane near triangulation, and the two non-neighbouring vertices are always of the same
colour. Therefore is we link diamonds together glueing them by the vertices of degree 2,
each of the linking vertices will have the same colour. The second way is to attach a
diamond to diamond along the common edge (cf. [6]). Both of those ways can be seen on
Figure 1.
Recently, Zhu [10] strengthened the theorem of Thomassen in the language of graph
polynomials showing that Alon-Tarsi number of any planar graph G satisfies AT (G) ≤ 5.
His approach utilizes a certain polynomial arising directly from the structure of the graph.
This graph polynomial is defined as:
P (G) =
∏
uv∈E(G),u<v
(u− v),
P. Gordinowicz and P. Twardowski: The polynomial method for list-colouring extendability . . . 285
where the relation < fixes an arbitrary orientation of graph G. Here we understand u and
v both as the vertices of G and variables of P (G), depending on the context. Notice that
the orientation affects the sign of the polynomial only. Therefore individual monomials
and the powers of the variables in each monomials are orientation-invariant. We refer the
reader to [1, 2, 7] for the connection between list colourings and graph polynomials. The
approach of Zhu may be described in the following form, analogous to Theorem 1.1.
Theorem 1.3 ([10]). Let G be any plane near-triangulation, let e = xy be a boundary edge
of G. Denote other boundary vertices by v1, . . . , vk and inner vertices by u1, . . . , um.
Then the graph polynomial of G − e contains a non-vanishing monomial of the form
ηx0y0vα11 . . . v
αk
k u
β1
1 . . . u
βm
m with αi ≤ 2, βj ≤ 4 for i ≤ k, j ≤ m.
The main tool connecting graph polynomials with list colourings is Combinatorial Null-
stellensatz [1]. It implies that for every non-vanishing monomial of P (G), if we assign to
each vertex of G a list of length greater than the exponent of corresponding variable in that
monomial, then such list assignment admits a proper colouring.
We note that this approach can be continued, allowing one to obtain stronger equivalents
of already known results for list-colouring. Moreover, in [3] where it is proven that every
planar graph G contains a matching M such that AT (G−M) ≤ 4, one can find an example
that with this approach it is possible to get results that are not known (or hard to prove) for
ordinary list colouring.
In this paper we provide a graph polynomial analogue to the result of Hutchinson,
obtaining a characterisation of polynomial extendability for outerplanar graphs, which may
be presented in the form of the following theorem.
Theorem 1.4. Let G be any outerplanar graph with V (G) = {x, y, v1, . . . , vn}. Then
in P (G) there is a non-vanishing monomial of the form ηxβyγ
∏n
i=1 v
αi
i with αi ≤ 2,
β, γ ≤ 1 satisfying:
(i) β = γ = 1 when every proper 3-colouring C of G forces C(x) = C(y);
(ii) β + γ = 1 when every proper 3-colouring C of G forces C(x) ̸= C(y);
(iii) β = γ = 0 otherwise.
We note that our proofs are simpler than the ones of Hutchinson, which show the
strength of the graph polynomial method for graph colouring problems. All considered
graphs are simple, undirected, and finite. For background in graph theory see [9].
2 Outerplane near-triangulations
In this section we provide a graph polynomial analogue to Theorem 1.2. The main tool is
the following theorem.
Theorem 2.1. Let G be a triangle or any 2-connected, outerplane near-triangulation with
exactly two vertices of degree 2. Let z ∈ V (G) be any neighbour of a degree 2 vertex.
Denote V (G) = {x, y, z, v1, . . . , vn}, where deg(x) = deg(y) = 2, yz ∈ E(G), y, z ̸= x.
Then
P (G) = Q(G) + η1xv
2
1 . . . v
2
ny
0z2 + η2xv
2
1 . . . v
2
ny
1z1 + η3xv
2
1 . . . v
2
ny
2z0,
where {η1, η2, η3} = {−1, 0, 1}, while Q(G) is a sum of monomials of the form
ηxαxvα11 . . . v
αn
n y
αyzαz , η ̸= 0, with (αx, α1, . . . , αn) ̸= (1, 2, . . . , 2).
286 Ars Math. Contemp. 21 (2021) #P2.08 / 283–299
Figure 1: An example of a graph satisfying conditions of point ii) of Theorem 1.4. When
3-colouring the graph, vertices a and b need to be in different colours. Vertices x and c are
in the same colour class as a (an example of the chain of diamonds), while y and d are in
the same colour class as b (the diamonds are linked along an edge). Therefore x and y have
different colours in every proper 3-colouring of the graph. The black vertices are yet to be
coloured.
Proof. The proof is done by induction on n. For the base step (n = 0), let G be a triangle
on vertices {x, y, z}. It is easy to check, that:
P (G) = (x− y)(y − z)(x− z)
= x2y1z0 − x2y0z1 + x1y0z2 − x1y2z0 + x0y2z1 − x0y1z2
= Q(G) + x1y0z2 − x1y2z0,
hence we have η2 = 0 and {η1, η3} = {1,−1}, and with Q(G) having necessary form, G
is concordant with the assertion.
We now proceed with the induction. Let n ∈ N and suppose the theorem holds for
graphs on at most n+3 vertices. Let G′ be any 2-connected, outerplane near-triangulation
on n+ 4 vertices and x, y ∈ V (G′) be the only two vertices of degree 2. Notice that x and
y cannot be adjacent (their common neighbour would then be a cutvertex, thus violating
2-connectivity). Let z and vn+1 be the neighbours of y. There is deg(z), deg(vn+1) ≥ 3
and (because G′ is triangulated) zvn+1 ∈ E(G′). Now consider G = G′ − y. Note that
G remains 2-connected outerplane near-triangulation. As outerplanar graph should have
at least 2 vertices of degree at most 2, one of neighbours of y has now degree 2. Let us
name it ỹ, while the second one — z̃. Notice that due to triangularity and 2-connectiveness,
we have deg(z̃) > 2 (with an exception when G is a triangle), as ỹ and z̃ have a common
neighbour. Now, we may consider P (G) using the inductive assumption. There are three
possible cases:
1. η̃1 = 0. As η̃1 = 0 and {η̃2, η̃3} = {−1, 1}, we know that:
P (G) = Q(G) + η̃2xv
2
1 . . . v
2
nỹ
1z̃1 + η̃3xv
2
1 . . . v
2
nỹ
2z̃0
= Q(G) + η̃2xv
2
1 . . . v
2
nỹ
1z̃1 − η̃2xv21 . . . v2nỹ2z̃0
= Q(G) + η̃2xv
2
1 . . . v
2
n(ỹ
1z̃1 − ỹ2z̃0).
P. Gordinowicz and P. Twardowski: The polynomial method for list-colouring extendability . . . 287
Now, P (G′) = P (G)(ỹ − y)(z̃− y) = P (G)(ỹz̃− ỹy − z̃y + y2), thus:
P (G′) = (Q(G) + η̃2xv
2
1 . . . v
2
n(ỹ
1z̃1 − ỹ2z̃0))(ỹz̃− ỹy − z̃y + y2)
= Q(G)(ỹz̃− ỹy − z̃y + y2) + η̃2xv21 . . . v2n(ỹ2z̃2y0 − ỹ2z̃1y1−
− ỹ1z̃2y1 + ỹ1z̃1y2 − ỹ3z̃1 + ỹ3y1 + ỹ2z̃1y1 − ỹ2z̃0y2)
= Q′(G′) + η̃2xv
2
1 . . . v
2
n(ỹ
2z̃2y0 − ỹ1z̃2y1 − ỹ2z̃0y2)
Now either z = ỹ and vn+1 = z̃, respectively, or the inverse may occur. In the first case,
we have:
P (G′) = Q′(G′) + η̃2xv
2
1 . . . v
2
n(v
2
n+1y
0z2 − v2n+1y1z1 − v0n+1y2z2),
thus {η1, η2} = {−1, 1} and η3 = 0, with the last monomial going into Q′(G′). With
analogous calculations, in the second case we have {η1, η3} = {−1, 1} and η2 = 0. As
Q′(G′) obviously contains only monomials of the form ηxαxvα11 . . . v
αn+1
n+1 y
αyzαz , η ̸=
0, (αx, α1, . . . , αn+1) ̸= (1, 2, . . . , 2), it can assume the role of Q(G), and the case is
finished.
2. η̃2 = 0. As η̃2 = 0 and {η̃1, η̃3} = {−1, 1}, we know that:
P (G) = Q(G) + η̃1xv
2
1 . . . v
2
nỹ
0z̃2 + η̃3xv
2
1 . . . v
2
nỹ
2z̃0
= Q(G) + η̃1xv
2
1 . . . v
2
nỹ
0z̃2 − η̃1xv21 . . . v2nỹ2z̃0
= Q(G) + η̃1xv
2
1 . . . v
2
n(ỹ
0z̃2 − ỹ2z̃0).
And then:
P (G′) = (Q(G) + η̃1xv
2
1 . . . v
2
n(ỹ
0z̃2 − ỹ2z̃0))(ỹz̃− ỹy − z̃y + y2)
= Q(G)(ỹz̃− ỹy − z̃y + y2) + η̃1xv21 . . . v2n(ỹ1z̃3y0 − ỹ1z̃2y1 − ỹ0z̃3y1+
+ ỹ0z̃2y2 − ỹ3z̃1y0 + ỹ3z̃0y1 + ỹ2z̃1y1 − ỹ2z̃0y2)
= Q′(G′) + η̃1xv
2
1 . . . v
2
n(ỹ
0z̃2y2 − ỹ1z̃2y1 − ỹ2z̃0y2 + ỹ2z̃1y1)
Continuing as in case 1, when z = ỹ and vn+1 = z̃, respectively, we have {η2, η3} =
{−1, 1} and η1 = 0. In the inverse case, when vn+1 = ỹ and z = z̃, there is {η2, η3} =
{1,−1} and η1 = 0. Q′(G′) can again assume the role of Q(G), and this case is also done.
3. η̃3 = 0. This case is handled analogously as η̃1 = 0, interchanging the roles of ỹ and z̃.
Here we have:
P (G) = Q(G) + η̃1xv
2
1 . . . v
2
nỹ
0z̃2 + η̃2xv
2
1 . . . v
2
nỹ
1z̃1
= Q(G) + η̃2xv
2
1 . . . v
2
nỹ
1z̃1 − η̃2xv21 . . . v2nỹ0z̃2
= Q(G) + η̃2xv
2
1 . . . v
2
n(ỹ
1z̃1 − ỹ0z̃2).
And then:
P (G′) = (Q(G) + η̃2xv
2
1 . . . v
2
n(ỹ
1z̃1 − ỹ0z̃2))(ỹz̃− ỹy − z̃y + y2)
= Q(G)(ỹz̃− ỹy − z̃y + y2) + η̃2xv21 . . . v2n(ỹ2z̃2y0 − ỹ2z̃1y1 − ỹ1z̃2y1+
+ ỹ1z̃1y2 − ỹ1z̃3 + ỹ1z̃2y1 + z̃3y1 − ỹ0z̃2y2)
= Q′(G′) + η̃2xv
2
1 . . . v
2
n(ỹ
2z̃2y0 − ỹ2z̃1y1 − ỹ0z̃2y2)
288 Ars Math. Contemp. 21 (2021) #P2.08 / 283–299
Finally, when z = ỹ and vn+1 = z̃, respectively, we have {η1, η3} = {−1, 1} and η2 = 0.
In the inverse case, when vn+1 = ỹ and z = z̃, there is {η1, η2} = {−1, 1} and η3 = 0.
Therefore, in each case we have the desired form of the polynomial, thus completing
the inductive argument.
Recall that by Combinatorial Nullstellensatz, (i, j)-extendability of (G, x, y) can be
expressed as the fact that there is a non-vanishing monomial in P (G) where exponents of
x and y are i−1 and j−1, respectively, and every other exponent is less than 3. We obtain
an analogue to Theorem 1.2 as the following
Corollary 2.2. Let G be any 2-connected, outerplane near-triangulation with V (G) =
{x, y, v1, . . . , vn}. Let C : V (G) → {1, 2, 3} be any proper 3-colouring of G. Then in the
graph polynomial P (G)
(i) there is no monomial of the form ηx0y0
∏n
i=1 v
αi
i with αi ≤ 2;
(ii) the monomial of the form ηx1y0
∏n
i=1 v
αi
i with αi ≤ 2 does not vanish if and only if
C(x) ̸= C(y);
(iii) there is non-vanishing monomial of the form ηxβyγ
∏n
i=1 v
αi
i with αi ≤ 2, β, γ ≤ 1.
Proof. For the first point, simply note that outerplane near-triangulation on n + 2 vertices
has 2n+ 1 edges, while the sum of the exponents of the given monomial is at most 2n.
For the second point and for the third one: when x and y are adjacent one may apply
Theorem 1.3 directly; otherwise, by the Hutchinson’s shrinking argument it is enough to
verify an existence of a suitable monomial for G having exactly 2 vertices of degree 2,
when x and y are these vertices.
Indeed, suppose otherwise and consider any chord (inner edge) ab of G that separates
the component H of the graph not containing vertices x and y. Such a chord exists, unless
x and y are the only degree 2 vertices of G. Let G1 = G[V (G) \ V (H)] and G2 =
G[V (H) ∪ {a, b}]. By Theorem 1.3 P (G2 − ab) contains non-vanishing monomial of
the form s2 = ηa0b0vα11 . . . v
αk
k with αi ≤ 2. Note, that common variables in P (G1)
and P (G2 − ab) are a and b only and that the sum of the exponents in any monomial in
P (G2 − ab) is fixed. Hence, any other monomial in P (G2 − ab) has different exponents
for some of v1, . . . vk. Therefore, as there is P (G) = P (G1)P (G2 − ab), G with x and
y satisfies the second (or the third one, respectively) point of the corollary if and only if
G1 with x and y does. Actually, the existence of desired monomials s in P (G) and s1 in
P (G1), respectively, is equivalent by identity s = s1s2.
Repeating the above argument until there is no separating chord one can shrink G to
the claimed form. By Theorem 2.1 this finishes the proof of the third point as then one
has either η1 ̸= 0 or η2 ̸= 0. For the second point it is enough to notice that under the
assumption of Theorem 2.1 there is η1 = 0 if and only if C(x) = C(y). Note that there is
also η3 = 0 if and only if C(z) = C(x) and then η2 = 0 if and only if x, y and z have 3
different colours. One may prove this fact by a simple analysis of the inductive step in the
proof of Theorem 2.1.
Indeed, in the base case (a triangle xyz) we have η2 = 0. Further, when G is extended
to G′ by a triangle ỹz̃y then
1. η̃1 = 0 (C(ỹ) = C(x)) forces η3 = 0 (when z = ỹ) or η2 = 0 (when z = z̃),
P. Gordinowicz and P. Twardowski: The polynomial method for list-colouring extendability . . . 289
2. η̃2 = 0 forces η1 = 0 (C(x) = C(y)),
3. η̃3 = 0 (C(z̃) = C(x)) forces η3 = 0 (when z = z̃) or η2 = 0 (when z = ỹ).
3 Poly-extendability of general outerplanar graphs
The results of the previous section can be of course applied to any outerplanar graph, not
necessarily triangulated. This, however, leads to loss of information, as usually there is
more than one way to triangulate the graph, and different triangulations may lead to dif-
ferent types of extendability. Moreover, in the case of non-triangulated graphs, as well
as those that are not 2-connected, the counting argument behind point (i) of Corollary 2.2
does not work any more. Hence, it is possible for a general outerplanar graph to be (1, 1)-
extendable. At first, a formal definition of fundamental subgraphs is provided, followed by
three instrumental lemmas.
Definition 3.1. Let G be a 2-connected outerplane graph, x, y ∈ V (G) and let T (G) be
the weak dual of G. The fundamental x − y subgraph of G is the subgraph of G induced
by the vertices belonging to faces that have vertices representing them in T (G) lying on
the shortest path between vertices representing faces on which x and y lie. If xy ∈ E(G),
then the fundamental subgraph reduces to an edge xy.
Here, the assumption that the graph is outerplane is needed, as the construction of
weak dual requires a particular embedding to be chosen. Notice however that in case of 2-
connected outerplanar graphs there is, up to isomorphism, just one outerplane embedding,
hence every 2-connected outerplanar graphs has essentially a single weak dual. Therefore
in the rest of the paper we will assume the graphs to be outerplanar, as the choice of an
embedding is irrelevant for our purpose.
Definition 3.2. Let G be a connected outerplanar graph with cutvertices, and let BC(G)
be the block-cutvertex graph of G. Let x, y ∈ V (G) be vertices lying in two different
blocks of G. The fundamental x − y subgraph of G consists of all blocks that have ver-
tices representing them in BC(G) lying on the shortest path between vertices representing
blocks containing x and y, and each of those blocks is restricted to the fundamental a − b
subgraph, where a, b ∈ V (G) are the two cutvertices belonging to the given block and to
the shortest path between blocks containing x and y in BC(G).
Definition 3.3. An outerplanar graph G with x, y ∈ V (G) is xy-fundamental if its fun-
damental x − y subgraph is equal to G. An outerplanar graph G is fundamental if it is
xy-fundamental for some x, y ∈ V (G).
Lemma 3.4. Let G be a 2-connected xy-fundamental near-triangulation, such that C(x) =
C(y), where C : V (G) → {1, 2, 3} is any proper 3-colouring of G. Let v0 be the vertex
of G that has degree 2 in G − y, and v1, . . . , vn be the remaining vertices. Then in P (G)
there is a non-vanishing monomial of the form ηx0y2v10v
2
1 . . . v
2
n, with η ∈ {−1, 1}.
Proof. As C(x) = C(y), then C(x) ̸= C(v0). Hence by the second case of Corollary 2.2,
there is a non-vanishing monomial ηx0v10v
2
1 . . . v
2
n, with η ∈ {−1, 1} in P (G−y). Adding
y back, thus multiplying P (G − y) by (y − v0)(y − vn) = y2 − yv0 − yvn + v0vn, we
get the monomial specified in the statement, and as it is the only way to obtain it, it is
non-vanishing.
290 Ars Math. Contemp. 21 (2021) #P2.08 / 283–299
Figure 2: Top: a connected, outerplanar graph G; Bottom: a fundamental x − y subgraph
of G.
Lemma 3.5. Let G,G′ be any two graphs, such that V (G) = {x, v1, . . . , vn}, V (G′) =
{x′, u1, . . . , um}. Let G′′ be the graph obtained from G and G′ by identifying x with x′,
thus creating vertex x′′, and carrying neighbouring relations from G,G′. Suppose there
are non-vanishing monomials ηxαΠvαii and η
′x′βΠu
βj
j in P (G) and P (G
′) respectively.
Then in P (G′′) there is a non-vanishing monomial A(G′′) = ηη′x′′α+βΠvαii Πu
βj
j .
Proof. As both η and η′ are non-zero, then the only way A(G′′) would vanish is that there
were a monomial A′(G′′) = νν′x′′α
′+β′Πvαii Πu
βj
j , where νν
′ = −ηη′ and α′ + β′ =
α + β. But then in P (G) and P (G′) there would have to be respective non-vanishing
monomials νxα
′
Πvαii and ν
′x′β
′
Πu
βj
j , and as the sum of exponents in every monomial in
a polynomial of given graph is fixed, we have that α = α′ and β = β′, a contradiction.
Thus A(G′′) is non-vanishing.
Lemma 3.6. Let G be a path of length n, n ≥ 2, where x, y are the endpoints and
v1, . . . , vn−1 are the internal vertices of G. Then in P (G) there is a non-vanishing mono-
mial of the form ηx0y0v21v
1
2 . . . v
1
n−1, where η ∈ {−1, 1}.
Proof. Suppose at first that n = 2. Then P (G) = (x−v1)(y−v1) = xy−xv1−yv1+v21 ,
and the last monomial is the one fulfilling the assertion. Now suppose that the lemma
holds for n = k − 1. Then in P (G), where G is a path xv1 . . . vk−1, there is a monomial
ηx0v21v
1
2 . . . v
1
k−2v
0
k−1. Now adjoining vk to vk−1, thus multiplying P (G) by (vk−1 − vk)
we obtain a monomial ηx0v21v
1
2 . . . v
1
k−2v
1
k−1v
0
k for path of length k, hence completing the
induction.
3.1 Near-triangulations with cutvertices
The following theorem is a polynomial analogue of [4, Theorem 5.3] that characterizes
extendability of outerplanar near-triangulations with cutvertices.
P. Gordinowicz and P. Twardowski: The polynomial method for list-colouring extendability . . . 291
Figure 3: Illustration for Lemma 3.5. Top: graphs G (left) and G′ (right); Bottom:
graph G′′.
Theorem 3.7. Let G be a fundamental x − y subgraph with cutvertices {v1, . . . , vj−1},
CV (G) = (x, v1, . . . , vj−1, y) be the sequence consisting of x, y and the cutvertices of
G in order that they occur on any of the paths from x to y, and ui,k being the remaining
vertices in the i-th block. Then in P (G):
(i) there is a non-vanishing monomial of the form η1x1y1Πvαmm Πu
βi,k
i,k , αm, βi,k ≤ 2 if
every vertex from CV (G) is in the same colour class;
(ii) there is a non-vanishing monomial of the form η2x0y1Πvαmm Πu
βi,k
i,k , αm, βi,k ≤ 2
if there is a single pair of successive vertices in CV (G) that are in different colour
classes;
(iii) there is a non-vanishing monomial of the form η3x0y0Πvαmm Πu
βi,k
i,k , αm, βi,k ≤ 2
if there are at least two pairs of successive vertices in CV (G) that are in different
colour classes;
Figure 4: An example of labelling described in Theorem 3.7.
292 Ars Math. Contemp. 21 (2021) #P2.08 / 283–299
Proof. Start with partitioning G by its cutvertices into separate, 2-connected, vi−1vi-funda-
mental outerplanar near-triangulations B1, . . . , Bj . To each of these graphs, Theorem 2.1
applies, and P (G) = P (B1) . . . P (Bj). If in each of those blocks the colour class of
degree 2 vertices is the same, then in each of their polynomials there is a non-vanishing
monomial such that exponents of degree 2 vertices are equal to 1, with other exponents no
larger than 2. Thus case 1 is just a repeated use of Lemma 3.5.
In the second case, let Bi be the block with degree 2 vertices in different colour classes.
If i = 1, then in P (B1) there is a non-vanishing monomial of the form η0x0v11Πu
2
1,k.
Hence again by Lemma 3.5 we get the desired monomial. If i > 1, then we apply
Lemma 3.4 to each block B1 to Bi−1, thus by Lemma 3.5 obtaining monomial with x0
and v2i−1. As vi−1 and vi are in different colour classes, P (Bi) contains a non-vanishing
monomial ηiv0i−1v
1
iΠu
2
i,k, hence through Lemma 3.5 we finish the case.
The last case is starts analogously to the second one, with Bi, Bl, i < l being two blocks
with endpoints in different colour classes. Let G′ be the vi−1vl-fundamental subgraph of
G. By Theorem 2.1 there is a non-vanishing monomial in P (Bi) with v0i−1 and v
1
i and a
monomial in P (Bl) with v1l−1 and v
0
l . As every block between Bi and Bl has a monomial
with endpoints in power 1, by Lemmas 3.4 and 3.5 there is a monomial in P (G′) with both
vi−1 and vl in power 0. Again by Lemmas 3.4 and 3.5 we can now adjoin remaining parts
of G to G′, with their suitable monomials creating a desired monomial in P (G).
3.2 2-connected outerplanar graphs with non-triangular faces
The following three theorems are jointly analogous to [4, Theorem 4.3].
Theorem 3.8. Let G be a 2-connected xy-fundamental graph with exactly one non-tri-
angular interior face, and that face contains x and does not contain y. Let V (G) =
{x, y, a, b, v1, . . . , vn}, where a, b are the two vertices of non-triangular face belonging to
the neighbouring interior face. Let C(v) be the colour class of vertex v in the 3-colouring
of the graph induced by all of the triangular faces. Then in P (G):
(i) there is a non-vanishing monomial of the form η1x0y1aαabαbΠvαii , αk ≤ 2 if
d(x, a) = 1 and C(a) = C(y) OR d(x, b) = 1 and C(b) = C(y);
(ii) there is a non-vanishing monomial of the form η2x0y0aαabαbΠvαii , αk ≤ 2 other-
wise.
Proof. Suppose that d(x, a) = 1 and C(a) = C(y). Let G′ be the subgraph of G created
by deleting all the vertices on the non-triangular face except for a and b. As G′ is an
outerplanar near-triangulation Theorem 2.1 applies, and as C(a) = C(y), then in P (G′)
there is a non-vanishing monomial with a1 and y1. If we now adjoin vertex x to a, creating
graph G′′, then it P (G′′) there is a non-vanishing monomial with x0, a2 and y1. Now
adding a path between x and b, thus reconstructing G (notice that the length of this path is
at least 2, as the face is not a triangle), by Lemma 3.6 we obtain a desired monomial. The
case when d(x, b) and C(b) = C(y) is handled analogously.
If this is not the case, then either d(x, a) > 1 and d(x, b) > 1, or d(x, a) = 1 and
C(a) ̸= C(y) (or analogously d(x, b) = 1 and C(b) ̸= C(y)). In the first case, then
by Theorem 2.1 and Lemma 3.4 in P (G′) (with G′ defined as previously) there is a non-
vanishing monomial with y0 and all other powers less than 3. Now as we join x with a and
b with previously deleted paths, Lemma 3.6 gives us a monomial with x0, y0 and all other
P. Gordinowicz and P. Twardowski: The polynomial method for list-colouring extendability . . . 293
Figure 5: Examples of labelling as in Theorem 3.8. Left: example to point (i); Right:
example to point (ii).
powers less than 3. In the second case, as C(a) ̸= C(y), by 2.1 there is a monomial in
P (G′) where y has power 0 and a has power 1. Adjoining x to a, we obtain a monomial
with x0, y0 and a2, and as we join x with b by a path, Lemma 3.6 gives us a desired
monomial. Case when d(x, b) = 1 and C(b) ̸= C(y) is again analogous to the last one.
Theorem 3.9. Let G be a 2-connected xy-fundamental graph with exactly one non-trian-
gular interior face, and that face does not contain x nor y. Let V (G) = {x, y, a, b, c, v1,
. . . , vn}, where a, b and a, c are the two pairs of vertices of non-triangular face belonging
to the neighbouring interior faces, and let C(v) be the colour class of vertex v in the 3-
colouring of the subgraph of G created by deleting the path connecting b and c. Then in
P (G):
(i) there is a non-vanishing monomial of the form η1x1y1aαabαbcαcΠvαii , αk ≤ 2, if
C(x) = C(a) = C(y);
(ii) there is a non-vanishing monomial of the form η2x0y1aαabαbcαcΠvαii , αk ≤ 2, if
C(x) ̸= C(a) = C(y) or C(x) = C(a) ̸= C(y);
(iii) there is a non-vanishing monomial of the form η3x0y0aαabαbcαcΠvαii , αk ≤ 2, if
C(x) ̸= C(a) ̸= C(y);
Proof. Let G′ be the subgraph of G obtained by deleting path connecting b and c from G.
Obviously G′ is an outerplanar near-triangulation with a single cutvertex a, hence Theo-
rem 3.7 applies to it. Notice moreover, that the first case of the above theorem leads to the
first case of Theorem 3.7, and the second and third case also relate similarly. As Theo-
rem 3.7 gives us suitable monomials, when we add back the path we previously deleted, an
application of Lemma 3.6 finishes the proof.
Theorem 3.10. Let G be a 2-connected xy-fundamental graph with exactly one non-
triangular interior face, and that face does not contain x nor y. Let V (G) = {x, y, a, b, c, d,
v1, . . . , vn}, where a, b and c, d are the two pairs of vertices of the non-triangular face be-
longing to the neighbouring interior faces with ab ∈ E(G) and cd ∈ E(G), and let C(v)
be the colour class of vertex v in the 3-colouring of the subgraphs of G created by deleting
the paths connecting a with c and b with d. Then in P (G):
294 Ars Math. Contemp. 21 (2021) #P2.08 / 283–299
Figure 6: An example of labelling described in Theorem 3.9.
(i) there is a non-vanishing monomial of the form η1x0y1aαabαbcαcdαdΠvαii , αk ≤ 2,
if d(a, c) = 1, C(x) = C(a) and C(y) = C(c) OR d(b, d) = 1, C(x) = C(b) and
C(y) = C(d);
(ii) there is a non-vanishing monomial of the form η2x0y0aαabαbcαcdαdΠvαii , αk ≤ 2
otherwise;
Figure 7: An example of labelling described in Theorem 3.10.
Proof. Suppose at first that C(x) = C(a) and C(y) = C(c). We can connect vertex a
with d, and if d(b, d) > 1, also with every interior vertex on the path connecting b with
d, thus obtaining an xy-fundamental 2-connected near triangulation G′. If d(a, c) = 1,
then C(a) ̸= C(c), thus C(x) ̸= C(y), and by Corollary 2.2 P (G′) contains a non-
vanishing monomial with x0, y1 and every other exponent equals 2. As neither x nor y
were affected by addition of edges to G, P (G) contains a non-vanishing monomial of the
form η1x0y1aαabαbcαcdαdΠvαii , αk ≤ 2. If d(a, c) > 1, then G′ fulfils the conditions
of Theorem 3.9, with d serving as vertex a in the statement of that theorem. Moreover, as
C(x) = C(a) and C(y) = C(c), and d neighbours both a and c in G′, then in colouring
of G′ C(x) ̸= C(d) and C(y) ̸= C(d). Hence by Theorem 3.9 P (G′) contains a non-
vanishing monomial with x0, y0 and every other exponent no larger than 2, and this again
P. Gordinowicz and P. Twardowski: The polynomial method for list-colouring extendability . . . 295
implies that there is a non-vanishing monomial of the form η2x0y0aαabαbcαcdαdΠvαii ,
αk ≤ 2 in P (G). The case when C(x) = C(b) and C(y) = C(d) is analogous.
Suppose now that C(x) ̸= C(a) and C(y) = C(c). Start by removing the paths from
a to c and b to d from G. This leaves us with two separate, 2-connected near triangu-
lations G′ and G′′ with {x, a, b} ∈ V (G′) and {y, c, d} ∈ V (G′′). As C(y) = C(c),
then C(y) ̸= C(d), and by Corollary 2.2 in P (G′′) there is a non-vanishing monomial
of the form η1y0d1c2Πv2i . Now as C(x) ̸= C(a), there exists a non-vanishing monomial
η1x
0a1b2Πu2i in P (G
′), as the polynomial of xa-fundamental subgraph of G′ contains a
non-vanishing monomial with x0 and a1, and as G′ is a 2-connected near triangulation,
every other exponent must be equal to 2. Now add back the previously removed paths.
Each of them contains in its graph polynomial a non-vanishing monomial with every ex-
ponent equal to 1, except for one of its endpoints, which has power 0. We will call that
monomial oriented towards the endpoint with non-zero exponent. Add paths connecting a
with c and b with d to G′ and G′′, and by multiplication of the monomials described above
we obtain a monomial of the form η2x0y0aαabαbcαcdαdΠvαii , αk ≤ 2 in P (G), where
exponent of each of the vertices a, b, c, d is equal to 2. This monomial does not vanish,
as the only other way to get this monomial would require us to orient both of the paths
in the opposite direction, but this would imply that there were a non-vanishing monomial
η1y
0d2c1Πv2i in P (G
′′), which is not the case as C(y) = C(c). Cases where C(x) = C(a)
and C(y) ̸= C(c), C(x) ̸= C(b) and C(y) = C(d) or C(x) = C(b) and C(y) ̸= C(d) are
sorted out in the same manner.
The last case is when C(x) ̸= C(a) and C(y) ̸= C(c). Observe at first, that we can
also assume that C(x) ̸= C(b) and C(y) ̸= C(d), as all the other cases were already
solved in previous arguments due to symmetries. Let G′ and G′′ be as in previous case. As
C(b) ̸= C(x) ̸= C(a), then in P (G′) there are non-vanishing monomials η1x0a1b2Πv2i
and −η1x0a2b1Πv2i . Similarly, there are non-vanishing monomials η2y0c1d2Πu2i and
−η2y0c2d1Πu2i in P (G′′). Now reconstruct G as previously, orienting path connecting
a and c towards a and path connecting b and d towards d. To comply with requirements
of the assertion, we have to use the first and fourth monomial from those specified above,
thus in P (G) we have a monomial −η1η2x0y0a2b2c2d2Πv2i . The only other way to reach
this set of exponents is to use the second and third monomial, and orient paths in opposite
directions, but as a simultaneous switch of orientations preserves sign, we again obtain
−η1η2x0y0a2b2c2d2Πv2i , so those monomials do not annihilate each other, but rather dou-
ble the coefficient. As all cases are now addressed, the proof is complete.
3.3 General outerplanar graphs
The three theorems above can be combined with Theorem 3.7 to obtain a general character-
isation of (i, j)-extendability of outerplanar graphs. We will start with some technicalities.
Definition 3.11. Let G be an outerplanar graph. A non-triangular inner face of G will be
called type 0 if it is as defined in Theorem 3.8 (with possibly y belonging to that face instead
of x), type 1 if it is as defined in Theorem 3.9 and type 2 if it is as defined in Theorem 3.10.
In case of type 1 faces, the vertex belonging to the two neighbouring faces will be called
an apex of that face.
Lemma 3.12. Let G be a connected outerplanar graph with V (G) = {x, y, v1, . . . , vi}
and let G′ be a supergraph of G obtained by adding a path of the length 2 to G in a way
that preserves outerplanarity. Then the monomial xαxyαyΠvαii does not vanish in P (G) if
296 Ars Math. Contemp. 21 (2021) #P2.08 / 283–299
and only if the monomial xαxyαyΠvαii z
2 does not vanish in P (G′), where z is the middle
vertex of the added path.
Proof. The implication from P (G) to P (G′) is obvious and was shown to be true and uti-
lized multiple times in this paper. Suppose there is a non-vanishing monomial xαxyαyΠvαii z
2
in P (G′). As P (G′) = P (G)(ab−az−bz+z2), where a, b are the endpoints of the added
path, and none of the monomials from P (G) contains z due to the fact that z /∈ V (G), then
the only way to obtain the monomial above is by multiplying xαxyαyΠvαii by z
2, thus the
former must occur in P (G).
Definition 3.13. Let G be a 1-connected fundamental outerplanar graph. For every cutver-
tex of G that is not an endpoint of any bridge add a path of length 2, connecting the pair of
some neighbours of that cutvertex without disrupting outerplanarity, thus creating a non-
triangular face of type 0. Then for every bridge or chain of bridges of G add a path of the
length 2 connected to the pair of the neighbours of the endpoints of that bridge or chain
of bridges (or to the neighbour and the endpoint if it has degree 1) in a way that preserves
outerplanarity, creating a face of type 2 (or type 0). Finally, if G is a path, connect end-
points of that path with a path of length 2. The resulting supergraph of G will be called a
2-connection of G. The 2-connection of A 2-connected graph would be the graph itself.
Notice, that the 2-connection of a 1-connected graph is not unique — for example, the
graph on Figure 8 has 8 different 2-connections. However, each of the 2-connections has
the same relevant properties — namely the color classes of the cutvertices and types of the
newly created non-triangular faces.
Figure 8: Top: a connected, outerplanar graph G; Bottom: a possible 2-connection of G.
The following remark is a direct consequence of Lemma 3.12.
Remark 3.14. Let G be a connected xy-fundamental outerplanar graph, V (G) = {x, y,
v1, . . . , vm} and let G′ be its 2-connection, V (G′) = {x, y, v1, . . . , vm, u1, . . . , un}. There
is a non-vanishing monomial xαxyαyΠvαii in P (G) if and only if there is a non-vanishing
monomial xαxyαyΠvαii Πu
2
j in P (G
′).
P. Gordinowicz and P. Twardowski: The polynomial method for list-colouring extendability . . . 297
The following theorem presents a full characterisation of the polynomial extendability
of connected fundamental outerplanar graphs.
Theorem 3.15. Let G be a connected xy-fundamental outerplanar graph, V (G) = {x, y,
v1, . . . , vi}, and let G′ be a 2-connection of G. Then in P (G):
(i) there is a non-vanishing monomial of the form x1y1Πvαii , αk ≤ 2 if G is a 2-
connected near-triangulation with C(x) = C(y) OR G is as in point 1 of Theo-
rem 3.7 OR every non-triangular face of G′ is of type 1 and every apex, x and y have
the same colour in every 3-colouring of G.
(ii) there is a non-vanishing monomial of the form x0y1Πvαii , αk ≤ 2 if G is a 2-
connected near-triangulation with C(x) ̸= C(y) OR G is as in point 2 of Theo-
rem 3.7 OR G′ is as in point 1 of Theorem 3.8 OR G′ is as in point 1 of Theorem 3.10
OR every non-triangular face of G′ is of type 1 and in every 3-colouring of G′ there
is exactly one pair of consecutive apexes (or either x or y with the closest apex) with
different colours OR only one of the non-triangular faces of G′ is not of the type 1
and conditions of point 1 of Theorem 3.10 are fulfilled on that face.
(iii) there is a non-vanishing monomial of the form x0y0Πvαii , αk ≤ 2 otherwise.
Proof. We will omit every case that is covered already by previous theorems, leaving us
only with the cases when there are multiple non-triangular faces. Suppose all of those are
of type 1. It is easy to see (with some help of Lemma 3.6) that for every such face removal
of all vertices belonging only to this (and outer) face produces a cutvertex, simultaneously
changing nothing in terms of extendability-relevant monomials. Hence apply Theorem 3.7,
with each apex acting as a cutvertex.
Suppose now there is a face of type either 0 or 2 in G′. Theorems 3.8 and 3.10 show
that the only cases where there is no monomial in P (G′) (and thus in P (G)) with both x
and y in power 0 is when 3-colouring G′ we cannot avoid a situation described in point 1
of either of these theorems on any of such faces, and in those cases there is a non-vanishing
monomial with x0 and y1. Observe that this is not the case when there are at least two
faces of type 0 or 2, as we can avoid this situation by either permuting the colours, or by
changing them on vertices of degree 2 (as in case of type 0 faces at least one such vertex
other than x and y definitely exists). So there are only two cases when we cannot avoid
that. The first is when in G′ there is only one face of type 2, no faces of type 0, there is
a pair of neighbouring vertices belonging to this face such that the only other face of G′
they belong to simultaneously is the outer face, and in any 3-colouring of G (and thus also
G′) each of those vertices has the same colour as x or y, depending on which of those
vertices lies on the same ”side” of that face. Label the vertex from this pair lying closer to
x as vx, and the one being closer to y as vy . The case of C(x) = C(vx) can occur either
when on one side there are only triangular faces between x and vx, with the structure of
that triangulation forcing the same colour of those vertices, or when for every type 1 face
between those vertices, the triangular structure between neighbouring faces or between x
(or vx) and the nearest such face forces the same colour on each of those vertices. The same
is true for y and vy , with the restriction that the former situation cannot occur for both of
those pairs. The second case is when there is exactly one face of type 0 in G′ (without loss
of generality we can assume that x lies on that face), no faces of type 2, x has a neighbour
(v0) that lies also on adjacent inner face, and the colour of that vertex is the same as colour
298 Ars Math. Contemp. 21 (2021) #P2.08 / 283–299
of y in every 3-colouring of G′. This can be only caused by the fact that the apex of every
type 1 face is forced to have the same colour as the others, as well as y and v0.
Finally, we prove that Theorem 3.15 can be restated as Theorem 1.4.
Proof of Theorem 1.4. Neither the graph polynomial nor the colouring depends on a partic-
ular graph embedding. Therefore, let G be any outerplanar graph with V (G) = {x, y, v1,
. . . , vn}. At first notice, that if G is not connected and x and y are in different connected
components, one may use Theorem 1.3 directly to obtain a monomial with β = γ = 0, so
then obviously the third case occurs.
For x and y in one component observe that by the Hutchinson’s shrinking argument it is
enough to prove theorem for G being xy-fundamental graph. See the proof of Corollary 2.2
for details. Now consider consequences of each of the situations described in the statement
of Theorem 3.15 in terms of 3-colourings. In every case of point (i) we obviously have that
C(x) = C(y). Moving to the second point, the first condition again directly states that
C(x) ̸= C(y). If G is as in point 2 of Theorem 3.7 or every non-triangular face of G′ is
of type 1 and in every 3-colouring of G′ there is exactly one pair of consecutive apexes (or
either x or y with the closest apex) with different colours, as the colour class changes only
once on the cutvertices/apexes, then obviously classes of terminal vertices x and y have to
be different. If G′ is as in point 1 of Theorem 3.8, then it is directly stated that the colour
of one of terminal vertices is the same as the colour of one of the neighbours of the other
terminal vertex, thus the colours of terminal vertices have to be different. Finally, if G′ is
as in point 1 of Theorem 3.10 or only one of the non-triangular faces of G′ is not of the
type 1 and the conditions of point 1 of Theorem 3.10 are fulfilled on that face, the vertices
x and y are in the same colour class as vertices a and c (or b and d), respectively, and those
vertices are adjacent, hence their colours cannot possibly be the same.
Finally, observe that in any other case the colour classes of x and y are independent
— the structure of the graph permits the colours to be rearranged in some parts without
changing the colours in the other parts, therefore the graph can be properly 3-coloured with
both C(x) = C(y) and C(x) ̸= C(y). As an example consider point (ii) of Theorem 3.8,
other cases are analogous. Starting with the triangulated part of the graph (i.e. the graph
minus internal vertices of the path between a and b containing x) already coloured, analyse
possible proper 3-colourings of the path from a to b. If min(d(x, a), d(x, b)) > 1, then
we can colour x with any of the 3 colours. Otherwise, suppose without loss of generality
that d(x, a) = 1 and hence d(x, b) > 1. Then x can be coloured with any colour except
C(a), but there is C(a) ̸= C(y). Therefore, again x can get colour of y or some different
one.
4 Further work
Extendability is naturally transformed into plane graphs by allowing interior vertices to
have a list of colours of length 5. In [5] and [6] Postle and Thomas provided results that
may be summarized in the following theorem.
Theorem 4.1. Let G = (V,E) be any plane graph, let C ⊆ V be the set of vertices on the
outer face, x, y ∈ C, x ̸= y. Then
(i) (G, x, y) is (1, 2)-extendable if and only if there exists a proper colouring c : C →
{1, 2, 3} such that c(x) ̸= c(y);
P. Gordinowicz and P. Twardowski: The polynomial method for list-colouring extendability . . . 299
(ii) (G, x, y) is (2, 2)-extendable.
One may ask, whether is it possible to restate the above theorem in the terms of a graph
polynomial, i.e. to extend, at least partially Theorem 1.4 to planar graphs. Our partial
results suggest that it is possible.
ORCID iDs
Przemysław Gordinowicz https://orcid.org/0000-0001-9843-3928
Paweł Twardowski https://orcid.org/0000-0003-1259-4830
References
[1] N. Alon, Combinatorial Nullstellensatz, Combin. Probab. Comput. 8 (1999), 7–29, doi:10.
1017/s0963548398003411, recent trends in combinatorics (Mátraháza, 1995).
[2] N. Alon and M. Tarsi, Colorings and orientations of graphs, Combinatorica 12 (1992), 125–
134, doi:10.1007/bf01204715.
[3] J. Grytczuk and X. Zhu, The Alon-Tarsi number of a planar graph minus a matching, J. Combin.
Theory Ser. B 145 (2020), 511–520, doi:10.1016/j.jctb.2020.02.005.
[4] J. P. Hutchinson, On list-coloring extendable outerplanar graphs, Ars Math. Contemp. 5 (2012),
171–184, doi:10.26493/1855-3974.179.189.
[5] L. Postle and R. Thomas, Five-list-coloring graphs on surfaces I. Two lists of size two in planar
graphs, J. Combin. Theory Ser. B 111 (2015), 234–241, doi:10.1016/j.jctb.2014.08.003.
[6] L. Postle and R. Thomas, Five-list-coloring graphs on surfaces III. one list of size one and one
list of size two, J. Combin. Theory Ser. B 128 (2018), 1–16, doi:10.1016/j.jctb.2017.06.004.
[7] U. Schauz, Mr. Paint and Mrs. Correct, Electron. J. Combin. 16 (2009), Research Paper 77, 18,
doi:10.37236/166.
[8] C. Thomassen, Every planar graph is 5-choosable, J. Combin. Theory Ser. B 62 (1994), 180–
181, doi:10.1006/jctb.1994.1062.
[9] D. B. West, Introduction to Graph Theory, 2nd edition, Prentice Hall, Inc., Upper Saddle River,
NJ, 2001, https://faculty.math.illinois.edu/˜west/igt/.
[10] X. Zhu, The Alon-Tarsi number of planar graphs, J. Combin. Theory Ser. B 134 (2019), 354–
358, doi:10.1016/j.jctb.2018.06.004.

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P2.09 / 301–307
https://doi.org/10.26493/1855-3974.2388.928
(Also available at http://amc-journal.eu)
On generalized strong complete mappings and
mutually orthogonal Latin squares*
Amela Muratović-Ribić †
University of Sarajevo, Faculty of Science,
Zmaja od Bosne 33-35, Sarajevo, Bosnia and Herzegovina
Received 20 July 2020, accepted 1 April 2021, published online 2 November 2021
Abstract
We present an application of generalized strong complete mappings to construction of
a family of mutually orthogonal Latin squares. We also determine a cycle structure of
such mapping which form a complete family of MOLS. Many constructions of generalized
strong complete mappings over an extension of finite field are provided.
Keywords: Strong complete mapping, group, finite field, mutually orthogonal Latin squares (MOLS).
Math. Subj. Class. (2020): 11T06, 12Y05
1 Introduction
Let G be an additive group. A mapping θ : G → G is called a complete mapping if both
θ(x) and θ(x)+x are 1-to-1 and onto. If both θ(x) and θ(x)−x are 1-to-1 and onto, θ(x) is
called an orthomorphism. A strong complete mapping is a complete mapping which is also
an orthomorphism. These mappings are used for a construction of Knut Vic designs and
they exist only for the groups of order n where gcd(n, 6) = 1. An Abelian group admits
strong complete mappings if and only if its Sylow 2-subgroup is trivial or noncyclic, and
also, its Sylow 3-group is trivial or noncyclic (see [2]).
Let p be a prime, m be a positive integer and q = pm. Let Fq be a finite field of or-
der q. We consider complete and strong complete mappings (and orthomorphisms) over
(Fq(x),+). Polynomials induced by these mappings are called complete and strong com-
plete polynomials, respectively. In [1], strong complete mappings over finite fields are
called very complete mappings. Many results have been established on this topic. In the
*Results in this article were partially presented on the Carleton Finite Fields Workshop, May 21 – 24, 2019.
†The author gratefully thanks the anonymous referee for the constructive comments and recommendations
which definitely helped to improve the quality of the paper.
E-mail address: amela@pmf.unsa.ba (Amela Muratović-Ribić)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
302 Ars Math. Contemp. 21 (2021) #P2.09 / 301–307
sequel, f0(x) = x, f2(x) = f ◦ f(x), fk(x) = f ◦ fk−1(x) for k > 0. Generalized
complete polynomials were introduced in [6] with applications to the check digit systems.
There were considered polynomials over finite fields with a property that f(x), f(x) ± x
and f2(x) ± x are all permutation polynomials. Note that there exist monomials of the
form xℓ
q−1
m where m | q − 1 with this property (see [5]).
We turn our attention to mappings θ(x) such that θk(x) are strong complete mappings
for all k = 1, 2, ..., t. Here, t is a positive integer. Our point of interest is an application
of these mappings to construction of mutually orthogonal Latin squares (MOLS). Many
constructions of such mappings over finite fields will be presented.
2 Construction of MOLS
Theorem 2.1. LetG be an additive finite Abelian group of order n, where n is odd. Assume
that θ : G→ G is such that θk(x) are strong complete mappings for k = 1, 2, . . . , t where
t is a positive integer. For 1 ≤ k ≤ t and i, j ∈ G define
aki,j = i+ θ
k(j)
a−ki,j = i− θ
k(j)
a0
+
i,j = i+ j; a
0−
i,j = i− j.
A family of Latin squares Lk = (aki,j) where k = −t, . . . ,−1, 0−, 0+, 1 . . . t is a family
of pairwise mutually orthogonal Latin squares. Therefore, a family of 2(t + 1) MOLS is
obtained.
Proof. We use the following convention θ0
±
(x) = x. Assume (aki,j , a
s
i,j) = (a
k
u,v, a
s
u,v)
for k ̸= s and consider the following cases:
• If (0 < s < k) or (s = 0+ and 0 < k) we have that
i+ θk(j) = u+ θk(v) (2.1)
and
i+ θs(j) = u+ θs(v).
Subtracting these equalities we obtain
θk(j)− θs(j) = θk(v)− θs(v).
Thus
θk−s(θs(j))− θs(j) = θk−s(θs(v))− θs(v).
By assumption, θk−s(y) − y is a permutation. Hence, θs(j) = θs(v) and j = v.
Inserting this in (2.1) we obtain i = u.
• If (k < s < 0) or (k < 0 and s = 0−) then we have
i− θ|k|(j) = u− θ|k|(v) (2.2)
and
i− θ|s|(j) = u− θ|s|(v).
A. Muratović-Ribić: On generalized strong complete mappings and mutually orthogonal . . . 303
Subtracting these equalities, we obtain
θ|k|(j)− θ|s|(j) = θ|k|(v)− θ|s|(v).
Thus
θ|k|−|s|(θ|s|(j))− θ|s|(j) = θ|k|−|s|(θ|s|(v))− θ|s|(v).
Reasoning as above, we get j = v and i = u.
• If (−s < 0 < k), (s = 0− and k > 0) or (s < 0 and k = 0+) then we have that
i+ θk(j) = u+ θk(v) (2.3)
and
i− θ|s|(j) = u− θ|s|(v)
which implies
θk(j) + θ|s|(j) = θk(v) + θ|s|(v).
Assume first k > |s|. Then θk−|s|(θ|s|(j)) + θ|s|(j) = θk−|s|(θ|s|(v)) + θ|s|(v).
As θk−|s|(y) + y is a permutation, it follows that θ|s|(j) = θ|s|(v). Thus j = v.
Using this in (2.3), we obtain i = u. If k ≤ |s| then θ|s|−k(θk(j)) + θk(j) =
θ|s|−k(θk(v)) + θk(v) similarly implies j = v and i = u.
• If k = 0+ and s = 0− then i+ j = u+ v and i− j = u− v which implies 2i = 2u.
Then 2ki = 2ku for all integers k. By assumption, the order of the group G is an
odd integer. Then n + 1 is even and thus (n + 1)i = (n + 1)u. However, ni = nu
by Lagrange’s theorem. Hence, i = u and further j = v.
Lemma 2.2. Let G be a group of order n. Assume that θ : G → G is such that all θk(x)
are strong complete mappings for k = 1, 2, . . . , t. Then the permutation θ has exactly one
fixed element and lengths of all other cycles are greater than t.
Proof. Assume that ℓ is the length of a cycle (a1, a2, . . . , aℓ) of the permutation θ, where
1 < ℓ ≤ t. Then θℓ(a1) = a1 and θℓ(a2) = a2. Therefore θℓ(a1)−a1 = θℓ(a2)−a2 = 0.
It follows that θℓ(x) − x is not a permutation which is a contradiction. Therefore, there is
no cycle of the length 1 < ℓ ≤ t. Since θ(x) − x is a permutation, there is exactly one
solution of the equation θ(x)− x = 0 and thus exactly one fixed element of θ.
Theorem 2.3. If θ generates a complete set of MOLS over a finite Abelian group of order
n as in the Theorem 2.1, then θ has either one fixed element and one cycle of the length
n− 1 or one fixed element and two cycles of the length n−12 .
Proof. In this case all θk(x) are strong complete mappings for k = 1, 2, . . . , n−12 − 1. By
the Lemma 2.2, there is one fixed element in the permutation θ and the lengths of nontrivial
cycles are greater than n−12 − 1. It follows that there can either one such cycle with the
length n− 1 or two cycles of the length n−12 .
Remark 2.4. Let Zp be a field of order p, where p > 2 is a prime. Let d be a generator of
Z∗p. Then θk(s) = dks is a strong complete mapping for k = 1, 2, . . . ,
p−3
2 . The mapping
θ(s) has a fixed element s = 0 and one full cycle (1, d, d2, ..., dp−2) of the length p−1. On
the other hand, θ2(s) = d2s has a property that θ2k(s) = d2ks is also a strong complete
mapping for all k = 1, 2, . . . , p−32 since
p−1
2 is odd. This mapping has a fixed element
s = 0 and two cycles of the length p−12 .
304 Ars Math. Contemp. 21 (2021) #P2.09 / 301–307
Proposition 2.5. Assume that Ψ : G → G, is a permutation such that Ψ(x ± y) =
Ψ(x)±Ψ(y) for all x, y ∈ G. If θ(x) generates a complete set of MOLS as in Theorem 2.1,
then η(x) = Ψ ◦ θ ◦Ψ−1(x) also generates a complete set of MOLS.
Note: An example of the mapping is Ψ(x) = kx where k is an integer, which prove its
existence.
Proof. Since ηk(x) = Ψ ◦ θk ◦ Ψ−1(x) is a permutation we need to show that ηk(x) + x
and ηk(x)− x are permutations for all k = 1, 2, . . . |G|−12 . Using substitution y = ψ
−1(x)
we get
ηk(x)±x = Ψ[θk(Ψ−1(x))]±Ψ(Ψ−1(x)) = Ψ[θk(Ψ−1(x))±Ψ−1(x)] = Ψ(θk(y)±y).
This is a permutation since Ψ(x) and θk(x)±x are permutations. Therefore, η(x) generates
a complete set of MOLS.
Let Fq be a field with a prime subfield Zp. Linearized polynomials over Fq are of the
form L(x) =
∑m
k=0 akx
pk and these polynomials have property that L(ax) = aL(x) for
all a ∈ Zp and L(x + y) = L(x) + L(y) for all x, y ∈ Fq . Thus, if we consider Fq as a
vector space over Zp, then L(x) is a linear operator on Fq .
Corollary 2.6. Let Fq be a finite field of order q = pn where p is a prime. Let d be a
primitive element of Fq and L(x) be a linearized permutation polynomial of Fq . Then the
polynomial f(x) = L(dL−1(x)) generates a complete set of MOLS as in Theorem 2.1.
Proof. It is easy to see that sx is strong complete polynomial for s ∈ Fq \ {0,±1}. There-
fore, for g(x) = dx, gk(x) = dkx are strong complete mappings for all k ̸= q−12 , q − 1.
Since, L(x ± y) = L(x) ± L(y) we have that f(x) = L ◦ g ◦ L−1(x) = L(dL−1(x))
generates a complete set of MOLS as in Theorem 2.1.
Remark 2.7. Consider a family of strong complete polynomials over finite field Fq which
generate a complete set of MOLS as in Theorem 2.1 and which have one fixed element
and one cycle of the length q − 1. Let d be a generator of F∗q . Then f(x) = dx is in this
family and considering the cycle structure, all other polynomials are conjugate with f(x).
Therefore, all polynomials in this family are of the form Ψ(dΨ−1(x)) for some permutation
polynomial Ψ(x) over Fq .
If q−12 is odd, then g(x) = d
2x is a strong polynomial which generate a complete
family of MOLS as in Theorem 2.1 and which have one fixed element and two cycles of
the length q−12 . Similarly, all other strong complete mappings with a same cycle structure
induce a polynomial of the form Ψ(d2Ψ−1(x)) for some permutation polynomial Ψ(x)
over Fq .
3 Construction of the strong complete mappings over extension fields
Let n be a positive integer and Fqn be an extension field of Fq . Let {α1, α2, . . . , αn} be
a basis of the vector space Fqn over Fq . We shall use similar technique as in the proof of
Theorem 2.1 in [3] to obtain the following recursive constructions of many strong complete
polynomials over the extension field.
A. Muratović-Ribić: On generalized strong complete mappings and mutually orthogonal . . . 305
Theorem 3.1. Let fi(x) be strong complete polynomials over Fq for i = 1, 2, . . . , n. Let
ψi : Fiq → Fq be arbitrary functions for i = 1, 2, . . . , n− 1. Denote X = x1α1 + x2α2 +
· · ·+ xnαn. Then the function
F (X) = f1(x1)α1 + [f2(x2) + ψ1(x1)]α2 + · · ·+ [fn(x) + ψn−1(x1, x2, . . . , xn−1)]αn
is a strong complete polynomial over Fqn .
Proof. In the proof of Theorem 1 in [3], it was shown that F (X) is a complete polynomial.
To show that it is a strong complete polynomial, lets check that F (X) −X is 1 − to − 1.
Assume that F (X) − X = F (Y ) − Y for X = x1α1 + x2α2 + · · · + xnαn and Y =
y1α1 + y2α2 + · · ·+ ynαn. Then the coefficients with the basis elements on the two sides
of equation are identical.
Looking at the coefficient with α1 we see that f1(x1)− x1 = f1(y1)− y1. As f1(x) is
orthomorphism it follows that x1 = y1 .
Now, equating the coefficients with α2 we get f2(x2) + ψ1(x1) − x2 = f2(y2) +
ψ1(y1) − y2. Taking into account x1 = y1, this implies f2(x2) − x2 = f2(y2) −
y2. Hence, x2 = y2 since f2(x) is an orthomorphism. We proceed by induction. As-
sume that x1 = y1, x2 = y2, . . . , xi−1 = yi−1 which imply ψi−1(x1, x2, . . . , xi−1) =
ψi−1(y1, y2, . . . , yi−1). Comparing the coefficients with αi, we obtain
fi(xi) + ψi−1(x1, x2, . . . , xi−1)− xi = fi(yi) + ψi−1(y1, y2, . . . , yi−1)− yi.
Thus fi(xi)− xi = fi(yi)− yi. So, xi = yi since fi(x) is an orthomorphism. Therefore,
xi = yi for all i = 1, 2, . . . , n and X = Y . Now, F (X)−X being 1− to− 1 on the finite
set Fqn it is a bijection, i.e. a permutation.
In the case of linearized polynomials, we extend the same technique to the compositions
of mappings. The proofs of the next theorems are similar to the proof of the Theorem 3.1.
So, we may omit a number of details.
Theorem 3.2. Let fi(x), i = 1, 2, . . . , n, be linearized strong complete polynomials over
Fq such that fki (x) are also strong complete polynomials for k = 1, 2, ..., t. Let ψi : Fiq →
Fq be arbitrary functions for i = 1, 2, . . . , n−1. Denote X = x1α1+x2α2+ · · ·+xnαn.
Then function
F (X) = f1(x1)α1 + [f2(x2) + ψ1(x1)]α2 + · · ·+ [fn(x) + ψn−1(x1, x2, . . . , xn−1)]αn
is a strong complete polynomial over Fqn such that F (k)(X) are also strong complete
mappings for all k = 2, 3, ..., t.
Proof. By Theorem 3.1, F (X) is a strong complete polynomial. Since F (X) is permu-
tation, it follows that F (k)(X) are permutations for all k = 2, · · · , t. Assume now that
F (2)(X) +X = F (2)(Y ) + Y (or F (2)(X)−X = F (2)(Y )− Y ).
Equating the coefficients with α1 on the both sides, we get f
(2)
1 (x1)+x1 = f
(2)
2 (y1)+
y1 (or f
(2)
1 (x1) − x1 = f
(2)
2 (y1) − y1). This implies x1 = y1 because f
(2)
1 (x) is a strong
complete polynomial. With α2 we have
f2[f2(x2) + ψ1(x1)] + ψ1(f1(x1))± x2 = f2[f2(y2) + ψ1(y1)] + ψ1(f1(y1))± y2.
306 Ars Math. Contemp. 21 (2021) #P2.09 / 301–307
Since f2 is linearized, we obtain
f2(f2(x2))+f2(ψ1(x1))+ψ1(f1(x1))±x2 = f2(f2(y2))+f2(ψ1(y1))+ψ1(f1(y1))±y2.
Taking into account that x1 = y1, we get f2(f2(x2)) ± x2 = f2(f2(y2)) ± y2.This yields
x2 = y2 since f
(2)
2 (x2) is a strong complete polynomial. Proceeding by induction, we can
prove that x3 = y3, ..., xn = yn and thus X = Y . Therefore, F (2)(X) is strong complete.
We can also prove by induction that F (k)(X) are strong complete for all k = 2, 3, ..., t.
Proposition 3.3. Assume that f(x) is a permutation and that f(dx)+ f(x), f(dx)− f(x)
are also permutations where d ∈ Fq , d ̸= 0, d ̸= ±1. Then the polynomial gd(x) =
f(df−1(x)) is strong complete.
Proof. Assume that f(x), f(dx) + f(x) and f(dx) − f(x) are permutations. Let x =
f−1(y). Then f(df−1(y)), f(df−1(y)) + y and f(df−1(y))− y are permutations. There-
fore, gd(x) = f(df−1(x)) is a strong complete polynomial.
Note that g(2)d (x) = gd(f(df
−1(x)) = f(df−1(fdf−1(x))) = f(d2f−1(x)) = gd2(x)
and, by induction g(k)d (x) = gdk(x).
A permutation polynomial f(x) such that f(dx) − f(x) is also a permutation for all
d ∈ Fq , d ̸= 1, is called a Costas polynomial. The only Costas polynomial over a field of
the prime order p is xs where gcd(s, p− 1) = 1. The only known Costas polynomial over
Fq is L(xs) where gcd(s, q − 1) = 1 and L is a linearized permutation polynomial (see
[4]). The polynomial L(xs) satisfies the conditions of Proposition 2.5. Indeed, L(dxs) ±
L(xs) = L((d ± 1)xs) is permutation polynomial whenever d ± 1 ̸= 0 and d ̸= 0. Thus,
gd(x) = L(d
sL−1(x)) is strong complete polynomial for all ds ̸∈ {0, 1,−1}. Then,
g
(k)
d (x) = gdk(x) is the strong complete polynomial whenever d
ks ̸∈ {0, 1,−1}. If dsk1 +
dsk2 + · · ·+ dskt ̸∈ {0, 1,−1} for a set of positive integers K = {k1, k2, ..., kt} then
t∑
i=1
gkid (x)± x =
t∑
i=1
L(dskiL−1(x))± x = L((
t∑
i=1
dski)L−1(x))± x
is also a permutation. It follows that gd(x) is the K-strong complete mapping (see [6] ).
This class of K-strong complete polynomials is linearized. Now, we will present one more
construction of the nonlinearized generalized strong complete polynomials over extension
fields.
Theorem 3.4. Let fi(x) be permutation polynomials over Fq such that fi(dkx) ± fi(x)
are permutation polynomials for d ∈ F∗q , k = 1, 2, ..., t < q − 1 and i = 1, 2, ..., n. Let
ψi : Fiq → Fq be arbitrary functions for i = 1, 2, . . . , n− 1. Denote X = x1α1 + x2α2 +
· · ·+ xnαn. Then the mapping
F (X) = f1(x1)α1 + [f2(x2) + ψ1(x1)]α2 + · · ·+ [fn(x) + ψn−1(x1, x2, . . . , xn−1)]αn
is a permutation polynomial such that F (dkX) ± F (X) are permutation polynomials for
all k = 1, 2, ..., t.
Note: For functions fi(x) we can take L(xs) as discussed above.
A. Muratović-Ribić: On generalized strong complete mappings and mutually orthogonal . . . 307
Proof. As dk ∈ F∗q we have that dkX = dkx1α1 + dkx2α2 + ... + dkxnαn. Assume
F (dkX) ± F (X) = F (dkY ) ± F (Y ). Then, equating the coefficients with the basis
elements, we get f1(dkx1) ± f1(x1) = f1(dky1) ± f1(y1). Thus x1 = y1. Further,
f2(d
kx2)+ψ1(d
kx1)± (f2(x2)+ψ1(x1)) = f2(dky2)+ψ1(dky1)± (f2(y2)+ψ1(y1)).
Since x1 = y1, we have f2(dkx2)± f2(x2) = f2(dky2)± f2(y2). It follows that x2 = y2.
By induction, x3 = y3, ..., xn = yn. Hence, X = Y . Therefore, F (dkX) ± F (X) are
permutations for all k = 1, 2, ..., t.
Corollary 3.5. For a function F (X) defined in Theorem 3.4, the function Gd(X) =
F (dF−1(X)) is strong complete mapping with a property that G(k)d (X) are strong com-
plete mappings for all d = 1, 2, ..., t.
Proof. The result follows from Proposition 3.3 and G(k)d (X) = Gdk(X).
Note: If we put x1 = x2 = . . . = xn−1 = 0 and xn = 1, then in all constructions
presented in Section 3 we will form a cycle whose elements are of the form (0, 0, . . . , 0, s).
The length of this cycle is less or equals to q. Using Lemma 2.2, we obtain t < q. There-
fore, by means of Theorem 2.1 we can not obtain more than 2q of MOLS over Fqn using
constructions in the Section 3.
ORCID iDs
Amela Muratović-Ribić https://orcid.org/0000-0001-7903-4884
References
[1] W.-S. Chou, Permutation polynomials on finite fields and their combinatorial applications,
Ph.D. thesis, Penn State Universit, 1990, https://www.proquest.com/openview/
24d2ca00f8f48ffe88a97a7c431a2096/1?pq-origsite=gscholar&cbl=
18750&diss=y.
[2] A. B. Evans, The existence of strong complete mappings of finite groups: a survey, Discrete
Math. 313 (2013), 1191–1196, doi:10.1016/j.disc.2011.11.040.
[3] A. Muratović-Ribić and E. Pasalic, A note on complete polynomials over finite fields and their
applications in cryptography, Finite Fields Appl. 25 (2014), 306–315, doi:10.1016/j.ffa.2013.10.
008.
[4] A. Muratović-Ribić, A. Pott, D. Thomson and Q. Wang, On the characterization of a semi-
multiplicative analogue of planar functions over finite fields, in: Topics in finite fields, Amer.
Math. Soc., Providence, RI, volume 632 of Contemp. Math., pp. 317–325, 2015, doi:10.1090/
conm/632/12635.
[5] A. Muratović-Ribić and S. Surdulli, On strong complete monomials and its multiplicative ana-
logue over finite fields, Sarajevo J. Math. 16(29) (2020), 137–144, doi:10.5644/sjm.
[6] A. Winterhof, Generalizations of complete mappings of finite fields and some applications, J.
Symbolic Comput. 64 (2014), 42–52, doi:10.1016/j.jsc.2013.12.006.

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P2.10 / 309–317
https://doi.org/10.26493/1855-3974.2049.3db
(Also available at http://amc-journal.eu)
Point-primitive generalised hexagons and
octagons and projective linear groups*
Stephen P. Glasby † , Emilio Pierro ‡ , Cheryl E. Praeger
Centre for the Mathematics of Symmetry and Computation, Department of Mathematics
and Statistics, The University of Western Australia
Received 16 July 2019, accepted 8 April 2021, published online 11 November 2021
Abstract
We discuss recent progress on the problem of classifying point-primitive generalised
polygons. In the case of generalised hexagons and generalised octagons, this has reduced
the problem to primitive actions of almost simple groups of Lie type. To illustrate how
the natural geometry of these groups may be used in this study, we show that if S is a
finite thick generalised hexagon or octagon with G ⩽ Aut(S) acting point-primitively and
the socle of G isomorphic to PSLn(q) where n ⩾ 2, then the stabiliser of a point acts
irreducibly on the natural module. We describe a strategy to prove that such a generalised
hexagon or octagon S does not exist.
Keywords: Generalised hexagon, generalised octagon, generalised polygon, primitive permutation
group.
Math. Subj. Class. (2020): 51E12, 20B15, 05B25
1 Introduction
We show in this paper that the Aschbacher–Dynkin [2] classification of maximal subgroups
of classical groups is a potentially useful tool to investigate whether or not a finite thick
*This work was supported by the Australian Research Council. Discovery Project Grants DP140100416
and DP190100450. The research reported in the paper forms part of the Australian Research Council (ARC)
Discovery Project Grant DP140100416 of the first and third authors. The authors gratefully acknowledge sup-
port from the Australian Research Council Discovery Project Grants DP140100416 and DP190100450; and also
advice from an anonymous referee on exposition. This research was conducted at the University of Western Aus-
tralia whose campus we acknowledge is located on the traditional lands of the Whadjuk people of the Noongar
Nation. We pay our respects to Noongar Elders past, present and emerging.
†Corresponding author.
‡Author gratefully acknowledges the support of the ARC Discovery Project Grant DP140100416.
E-mail addresses: stephen.glasby@uwa.edu.au (Stephen P. Glasby), e.pierro@mail.bbk.ac.uk (Emilio
Pierro), cheryl.praeger@uwa.edu.au (Cheryl E. Praeger)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
310 Ars Math. Contemp. 21 (2021) #P2.10 / 309–317
generalised hexagon or octagon admits a large rank classical group as an automorphism
group with a point-primitive action.
The notion of a generalised polygon arose from the investigations of Tits [12] and is
connected with the groups of Lie type having twisted Lie rank 2. They belong to a wider
class of geometric objects known as buildings, which were also introduced by Tits, whose
motivation was to find natural geometric objects on which the finite groups of Lie type
act, in order to work towards a proof of the classification of finite simple groups. Indeed,
all families of simple groups of Lie type having twisted Lie rank 2 arise as automorphism
groups of generalised polygons.
An incidence geometry S = (P,L, I) of rank 2 consists of a point set P , a line set L
and an incidence relation I ⊆ P×L such that P and L are disjoint non-empty sets. We say
that S is finite if |P∪L| is finite. The dual of S is SD = (L,P, ID), where (p, ℓ) ∈ I if and
only if (ℓ, p) ∈ ID. We say that S is thick if each point is incident with at least three lines
and each line is incident with at least three points. A flag of S is a set {p, ℓ} with p ∈ P ,
ℓ ∈ L and (p, ℓ) ∈ I. The incidence graph of S is the bipartite graph whose vertices are
P ∪ L and whose edges are the flags of S. A generalised n-gon is, then, a thick incidence
geometry of rank 2 whose incidence graph is connected of diameter n and girth 2n such that
each vertex lies on at least three edges [13, Lemma 1.3.6]. It is not immediate, but if S is a
thick generalised n-gon, then there exist constants s, t ⩾ 2 such that each point is incident
with t + 1 lines and each line is incident with s + 1 points [13, Corollary 1.5.3]. We then
say that the order of S is (s, t). A collineation of S is a pair (α, β) ∈ Sym(P)× Sym(L)
that preserves the subset I ⊆ P ×L. The subset of all collineations of Sym(P)× Sym(L)
is a subgroup denoted Aut(S). A celebrated result of Feit and Higman [8] states that if
S is a finite thick generalised n-gon, then n ∈ {2, 3, 4, 6, 8}. We refer the reader to Van
Maldeghem’s book [13] both for further details about classical generalised polygons, and
for a full introduction to the theory of generalised polygons.
In this paper we shall only be concerned in the case that S is a finite thick generalised
hexagon or octagon. At present the only known examples of these are the split Cayley
hexagon H(q), the twisted triality hexagon T (q, q3), the Ree–Tits octagon O(22m+1) and
their duals. These correspond to the groups G2(q), 3D4(q) and 2F4(22m+1) and complete
descriptions of these can be found in [13].
The point graph of a generalised polygon S is the graph with points as vertices and
with two points adjacent if they are collinear. The classification of (not necessarily thick)
generalised polygons admitting an automorphism group which acts distance-transitively
on the point graph of S is due to Buekenhout and Van Maldeghem [6]. In addition, they
show that distance-transitivity implies that G acts primitively on P . If S is also thick, then
Buekenhout and Van Maldeghem show that the socle of G is a finite simple group of Lie
type having twisted Lie rank 2. The assumption of distance-transitivity for this graph is
strong, and in recent years there has been work by a number of authors to show that the
assumption of distance-transitivity can be relaxed.
Schneider and Van Maldeghem [11, Theorem 2.1] showed that if G ⩽ Aut(S) acts
flag-transitively, point-primitively and line-primitively, then G is an almost simple group
of Lie type. The following theorem, which significantly strengthened this result, provided
motivation for the present paper.
Theorem 1.1 ([3, Theorem 1.2]). Let S be a finite thick generalised hexagon or octagon.
If a subgroup G of Aut(S) acts point-primitively, then G is an almost simple group of Lie
type.
S. P. Glasby et al.: Point-primitive generalised hexagons and octagons and projective . . . 311
The proof of Theorem 1.1 relies on the classification of finite simple groups. In order to
rule out certain possibilities for soc(G), it is sufficient to consider the primitive actions of
the almost simple groups of Lie type, or equivalently, their maximal subgroups. For an ex-
ceptional Lie type group that has a faithful projective representation in defining characteris-
tic of degree at most 12, a complete classification of its maximal subgroups is summarised
in [4, Chapter 7]. Using this classification it was proved by Morgan and Popiel in [9] that
under the hypothesis of the above theorem, if in addition it is assumed that the socle of G is
isomorphic to one of the Suzuki–Ree groups, 2B2(22m+1)′, 2G2(32m+1)′ or 2F4(22m+1)′,
where m ⩾ 0, then up to point-line duality, S is the Ree–Tits octagon O(22m+1). For a
general classical group G, however, we appeal to the Aschbacher–Dynkin classification [2]
of its maximal subgroups. The maximal subgroups of G fall into eight families of “geo-
metric” subgroups, those which preserve a natural geometric structure, and a ninth class of
exceptions. These classes are denoted Ci for 1 ⩽ i ⩽ 9, and some authors denote C9 as
S . The class C1 consists of stabilisers of subspaces and includes the maximal parabolic
subgroups of G. Our main result is as follows.
Theorem 1.2. Let S be a finite thick generalised hexagon or octagon. If G ⩽ Aut(S) acts
point-primitively on S and the socle of G is isomorphic to PSLn(q) where n ⩾ 2, then
the stabiliser of a point of S is not the stabiliser in G of a subspace of the natural module
V = (Fq)n.
The subspace stabilisers considered in Theorem 1.2 are all maximal parabolic sub-
groups. Given this result, and in the light of the result of Morgan and Popiel [9] mentioned
above, it would in the first instance be good to handle all primitive coset actions of Lie type
groups on maximal parabolic subgroups.
Problem 1.3. Extend Theorem 1.2 to show that, if S is a finite thick generalised hexagon
or octagon and G ⩽ Aut(S) is an almost simple group of Lie type such that the stabiliser
Gx of a point x is a maximal parabolic subgroup, then (S,G) is one of the known classical
examples.
Problem 1.3 has been solved for the Suzuki–Ree groups in [9], and it has also been
solved by Popiel and the second author [10] for the groups G2(q)′. It would be especially
interesting to have a solution to Problem 1.3 for the groups of (twisted or untwisted) Lie
rank 2, and in particular for the family 3D4(q)′ which is the only untreated case where
the groups are known to act on a generalised hexagon or octagon. Moreover, it would be
even more interesting to have a characterisation of all point-primitive actions of groups
with socle G2(q)′ or 3D4(q)′ on a thick generalised hexagon or octagon (not just the coset
actions on maximal parabolic subgroups). This, however, seems to be a substantially harder
problem.
Maximal parabolic subgroups mentioned in Problem 1.3 are examples of large sub-
groups, a notion introduced by Alavi and Burness in [1], namely a subgroup H of a finite
group G is large if |H|3 > |G|. In [1] all large subgroups of all finite simple groups are
determined. In our view the next level of attack on the general classification problem would
be to handle actions on cosets of large subgroups.
Problem 1.4. Extend Theorem 1.2 to show that, if S is a finite thick generalised hexagon
or octagon and G ⩽ Aut(S) is an almost simple group of Lie type such that the stabiliser Gx
of a point is a large maximal subgroup, then (S,G) is one of the known classical examples.
312 Ars Math. Contemp. 21 (2021) #P2.10 / 309–317
Popiel and the second author [10] have almost solved Problem 1.4 for groups G with
socle G2(q)′. The only unresolved point-primitive action is on a generalised hexagon with
stabiliser satisfying Gx ∩ soc(G) ∼= G2(q1/2).
The result of Alavi and Burness [1, Theorem 4] for groups G ∼= PSLn(q), taking into
account Theorem 1.2 for parabolic actions and using properties of the parameters of a gen-
eralised n-gon, shows that a solution to Problem 1.4 for these groups involves consideration
of just four kinds of point actions. We follow Alavi and Burness in using type to denote a
rough approximation of the structure of a subgroup.
Proposition 1.5. Let S be a finite thick generalised hexagon or octagon of order (s, t).
Suppose that G ⩽ Aut(S) with G ∼= PSLn(q), and G acts point-primitively on S such that
the stabiliser Gx of a point x is a large subgroup. Then one of the following holds:
(a) Gx is a C2-subgroup of type GLn/k(q) ≀ Sk, where k = 2 or k = 3;
(b) Gx is a C3-subgroup of type GLn/k(qk), where k = 2 or k = 3;
(c) Gx is a C5-subgroup of type GLn(q0) with q = qk0 , and either k = 2 or k = 3, or;
(d) Gx ∈ C8 of type Spn(q) (n even), SUn(q0) (q = q20), SOn(q) (nq odd), or SO
ϵ
n(q)
(n even, ϵ = ±).
Proof of Proposition 1.5. In addition to the classes asserted in the statement of the propo-
sition, Alavi and Burness show that either Gx ∈ C1, which is excluded by Theorem 1.2, or
Gx is one of finitely many cases belonging to classes C6 or C9 [1, Proposition 4.7 and The-
orem 4(ii)]. Of these, the cases where G is a group appearing in the Atlas [7] are excluded
by [5, Theorems 1.1 and 1.2]. The remaining possibilities for (G,Gx) are:
G PSL5(3) PSL4(5) PSL4(7) PSL2(q) q ∈ {41, 49, 59, 61, 71}
Gx M11 24. A6 PSU4(2) A5
The number |P| of points is the polynomial f(s, t) = (s + 1)(s2t2 + st + 1) if S is
a generalised hexagon, and f(s, t) = (s + 1)(s3t3 + s2t2 + st + 1) if S is a generalised
octagon. Running through the possibilities for |P| = |G : Gx| from the table above, we find
that there are no solutions to the equation |P| = f(s, t) with s, t ⩾ 2. This completes the
proof.
Extending Theorem 1.2 to include the large subgroups in class C2 has also proven to
be unexpectedly challenging to the authors.
2 The proof of Theorem 1.2
To prove Theorem 1.2, we assume for a contradiction that S is a thick generalised hexagon
or octagon and G ⩽ Aut(S), with soc(G) = PSLn(q), is such that a point stabiliser is
maximal in G and is the stabiliser of a k-subspace of the natural module V = (Fq)n, where
0 < k < n. Hence we may identify the point set P of S with the set of k-subspaces of
V , which we denote by
(
V
k
)
. If G contains a graph automorphism then k = n/2 and, for
its index 2 subgroup H = G ∩ PΓLn(q), the stabiliser HU is maximal in H. Thus we
may assume that PSLn(q) P G ⩽ PΓLn(q). It is convenient in the proofs to work with a
S. P. Glasby et al.: Point-primitive generalised hexagons and octagons and projective . . . 313
group G such that SLn(q) P G ⩽ ΓLn(q) acting linearly on V , with the scalar matrices
acting trivially on
(
V
k
)
, so G = G/Z where Z is the subgroup of scalars. Since a graph
automorphism of G maps
(
V
k
)
to
(
V
n−k
)
, and hence maps S to an isomorphic generalised
polygon with point set identified with
(
V
n−k
)
, we may assume further that 1 ⩽ k ⩽ n/2,
and so the following hypotheses hold.
Hypothesis 2.1. Let S = (P,L) be a finite thick generalised hexagon or octagon of order
(s, t), such that P is identified with the set
(
V
k
)
of k-subspaces of V = (Fq)n, where
1 ⩽ k ⩽ n/2. Suppose that SLn(q) P G ⩽ ΓLn(q) and that G induces a group of
automorphisms of S acting naturally on P , (so that a point stabiliser belongs to class C1).
Our proof of Theorem 1.2 uses the following three lemmas. The first is from [3].
Lemma 2.2 ([3, Lemma 2.1(iv)]). Let S be a finite thick generalised hexagon or octagon
of order (s, t), and let P denote the set of points of S. Let x, y1, y2 ∈ P such that x ∼ y1
and x ∼ y2, and let g ∈ Aut(S) such that xg ̸= x. If g fixes y1 and y2, then x, y1, y2, xg
all lie on a common line.
The second lemma is not difficult to prove, and its proof is left to the reader.
Lemma 2.3. Suppose SLn(q) P G ⩽ ΓLn(q), V = (Fq)n and k ⩽ n/2. Then, if
dim(V ) = n and k ⩽ n/2, then the orbits of G on
(
V
k
)
×
(
V
k
)
are
Γi =
{
(x, y) ∈
(
V
k
)
×
(
V
k
)
| dim(x ∩ y) = i
}
where 0 ⩽ i ⩽ k. (2.1)
Moreover, for x ∈
(
V
k
)
the orbits of Gx, are
Γi(x) =
{
y ∈
(
V
k
)
| dim(x ∩ y) = i
}
where 0 ⩽ i ⩽ k.
The third lemma allows us to characterise adjacency in S.
Lemma 2.4. Assume Hypothesis 2.1 and let x, y ∈ P . Then the following properties hold.
(F1) For every i ∈ {0, . . . , k}, if x, y are collinear and dim(x ∩ y) = i, then any x′, y′ ∈
P with dim(x′ ∩ y′) = i are also collinear.
(F2) For every i ∈ {0, . . . , k − 1}, if x, y are collinear and dim(x ∩ y) = i, then there
exists y′ ∈ P such that dim(x ∩ y′) = i and y′ ̸∼ y.
Proof. Property (F1) follows from Lemma 2.3. For (F2), suppose towards a contradiction
that every point y′ with dim(x ∩ y′) = i is collinear with y. By (F1), every such point y′
is also collinear with x, and hence lies on the line ℓ through x and y (because otherwise
{x, y, y′} would form a triangle and S contains no triangles). Let J = J(n, k)i denote
the generalised Johnson graph with vertex set V (J) =
(
V
k
)
and two vertices adjacent if
and only if they intersect in an i-subspace. Since G acts primitively on
(
V
k
)
, and since the
connected components are G-invariant, J is a connected graph. Note that Property (F1)
implies that adjacency in J implies collinearity, but the converse is not necessarily true. By
definition of J , y, y′ ∈ J1(x), the set of vertices adjacent to x in J . By the above argument,
{x} ∪ J1(x) is contained in the line ℓ. Since G acts transitively on J and since adjacency
314 Ars Math. Contemp. 21 (2021) #P2.10 / 309–317
is preserved by this action, it is true for all u ∈ P that {u} ∪ J1(u) is contained in a line of
S. Since S has more than one line, the diameter of J is at least 2.
We now prove by induction on the distance d, where 2 ⩽ d ⩽ diam(J), that, for any
vertices u, v of J , if the distance d = δ(u, v) and (u0, u1, . . . , ud) is a path of length d
in J from u = u0 to v = ud, then {u0, . . . , ud} is contained in the line ℓ containing
{u} ∪ J1(u). First we prove this for d = 2. Suppose that δ(u, v) = 2 and let (u,w, v)
be a path of length 2 in J from u to v. Note that w ∈ J1(u) ⊆ ℓ. Also u, v both lie in
{w} ∪ J1(w) which, as we have shown, is contained in some line ℓ′ of S. Then u,w are
contained in both ℓ and ℓ′, and since two points lie in at most one line of S it follows that
ℓ′ = ℓ, and so u,w, v all lie in ℓ and the inductive assertion is proved for d = 2. Now
suppose inductively that 3 ⩽ d ⩽ diam(J) and that the assertion is true for all integers
from 2 to d − 1. Suppose that δ(u, v) = d and that (u0, u1, . . . , ud) is a path in J from
u = u0 to v = ud. Then δ(u, ud−1) = d − 1, so by induction {u0, . . . , ud−1} ⊆ ℓ. Also
ud−2, v ∈ {ud−1} ∪ J(ud−1), which we have shown to be contained in some line ℓ′; since
ud−2, ud−1 are contained in both ℓ and ℓ′, it follows that ℓ′ = ℓ, and the inductive assertion
is proved for d. Hence by induction the assertion holds for all d ⩽ diam(J). However, this
is a contradiction because the points of S do not all lie on a single line.
We are now in a position to prove Theorem 1.2.
Proof of Theorem 1.2. As discussed at the beginning of this section we may assume that
Hypothesis 2.1 holds. Thus P =
(
V
k
)
and k ⩽ n/2.
CLAIM 1: k ⩾ 4. Consider the action of G on P × P . For each i with 0 ⩽ i ⩽
k − 1, G acts transitively on the set Γi defined in (2.1) by Lemma 2.3. It is a standard
result in the theory of permutation groups that the orbits of G on P × P are in one-to-one
correspondence with the orbits of Gx on P , and there must be at least one Gx-orbit for
each possible distance from x in the point graph of S. If k < 3, then the number of orbits
of Gx is less than four, so no point of P \{x} is at distance 3 from x in S , contradicting the
assumption that S is either a generalised hexagon or a generalised octagon. If k = 3, then
for the same reason S is not a generalised octagon, and so S is a generalised hexagon and
G acts distance transitively on the point graph. By the main result of Buekenhout and Van
Maldeghem in [6], S is a classical generalised hexagon and its distance transitive group has
socle G2(rf ) for some prime power rf , which is a contradiction. Hence k ⩾ 4 as claimed.
Now let {e1, . . . , en} be a basis of V and take x = ⟨e1, . . . , ek⟩. Let k1 < k be
maximal such that there exists a point y ∼ x with (x, y) ∈ Γk1 (as defined in (2.1)). Note
that, by Claim 1, n ⩾ 2k ⩾ 8.
CLAIM 2: k1 < k − 1. For a contradiction, assume that k1 = k − 1 and without loss
of generality that y = ⟨e1, . . . , ek−1, ek+1⟩. By (F2) there exists a point y′ ∈ P such that
(x, y′) ∈ Γk−1 and y ≁ y′ and by (F1) we have x ∼ y′ and so dim(x ∩ y′) = k − 1
and dim(y ∩ y′) ⩽ k − 2. Now dim(x ∩ y) = dim(x ∩ y′) = k − 1 implies that
dim(x ∩ y ∩ y′) ⩾ k − 2, and hence dim(y ∩ y′) = dim(x ∩ y ∩ y′) = k − 2. We may
assume without loss of generality that y′ = ⟨e2, . . . , ek, ek+2⟩. But now the permutation
matrix corresponding to (1, k+1)(k, k+2) leaves y and y′ fixed, but not x. By Lemma 2.2,
this implies that y ∼ y′, a contradiction. Hence k1 < k − 1, as required.
CLAIM 3: k1 = 0. Assume to the contrary that k1 > 0. Recall that k1 < k is maximal
such that there exists a point y ∼ x with y ∈ Γk1 . Thus we may assume that
y = ⟨e1, . . . , ek1 , ek+1, . . . , e2k−k1⟩.
S. P. Glasby et al.: Point-primitive generalised hexagons and octagons and projective . . . 315
If 2k − k1 + 1 ⩽ n, let
z = ⟨e1, . . . , ek1 , ek+2, . . . , e2k−k1+1⟩
so that dim(x∩z) = k1 and dim(y∩z) = k−1 > k1. It then follows from (F1) that x ∼ z
and from Claim 2 and the maximality of k1 that y ≁ z. Since 1 ⩽ k1 ⩽ k − 2, we have
k + 2 ⩽ 2k − k1 and hence the permutation matrix corresponding to (1, k + 2)(k, k − 1)
fixes y and z but not x. But once again Lemma 2.2 implies that y ∼ z, a contradiction.
Therefore k1 = 0 as claimed.
An immediate corollary of Claim 3 and (F1) is that G acts flag-transitively on S.
CLAIM 4: n = 2k or 2k + 1. For a contradiction, suppose that 2k + 1 < n and recall
k ⩽ n/2. Let y = ⟨ek+1, . . . , e2k⟩ and z = ⟨ek+2, . . . , e2k+1⟩. Observe that x ∼ y, x ∼ z
by (F1); furthermore dim(y ∩ z) = k − 2 > 0, so y ≁ z by the maximality of k1. Since
k ⩾ 4 by Claim 1, the permutation matrix corresponding to (1, 2k + 2)(2, 3) fixes y and z
but not x, contradicting Lemma 2.2.
CLAIM 5: n = 2k. Assume n = 2k + 1. Let y be as in Claim 4 and let z =
⟨ek+1, . . . , e2k−1, e1+ e2k+1⟩. Then x ∼ y and x ∼ z by Claim 3 and since dim(y∩ z) =
k − 1 > 0, we see that y ≁ z by Claim 3. Once again we apply Lemma 2.2 by noting that
since k ⩾ 4, the permutation matrix for (1, 2k + 1)(k + 1, k + 2) leaves y and z fixed but
not x, contradicting Lemma 2.2. Hence n = 2k and Claim 5 is true.
To complete the proof let
x = ⟨e1, . . . , ek⟩, y = ⟨ek+1, . . . , e2k⟩,
z = ⟨e1 + ek+1, . . . , ei + ei+k, . . . , ek + e2k⟩.
(2.2)
Then dim(x ∩ y) = dim(x ∩ z) = dim(y ∩ z) = 0, and so x, y and z are pairwise
collinear by Claim 3. Then, since S does not contain any triangles, x, y and z lie on a
line of S, say ℓ. Consider the stabiliser Gℓ. Note that ℓ is the unique line containing any
pair of the elements x, y or z and so in particular, Gℓ ⩾ ⟨Gxy, Gxz, Gyz⟩. Writing vectors
in V as n-dimensional row vectors over Fq relative to the basis e1, . . . , en, and writing
matrices relative to this basis, we see that x consists of all vectors of the form (X, 0),
where X, 0 denote k-dimensional row vectors, and the stabiliser Gx consists of all matrices
M =
(
A B
C D
)
∈ G for which (I | 0)M has the form (X | 0) where X ∈ GLk(Fq). Our aim
is to show that ⟨Gxy, Gxz, Gyz⟩ contains SLn(q). Let H = SLn(q) and let Hx = H ∩Gx
and define Hy , Hz , Hxy , Hxz , Hyz and Hℓ analogously. Let Mk(q) denote the ring of all
k × k matrices over Fq , and recall that k = n/2. Then
Hx = ⟨
(
A 0
C D
)
∈ SLn(q) | A,D ∈ GLk(q), C ∈ Mk(q)⟩.
Similarly,
Hy = ⟨
(
A B
0 D
)
∈ SLn(q) | A,D ∈ GLk(q), B ∈ Mk(q)⟩
and
Hz = ⟨
(
A B
C D
)
∈ SLn(q) | A+ C = B +D⟩.
From this we see that
Hxy = ⟨
(
A 0
0 D
)
∈ GLn(q) | A,D ∈ GLk(q),det(AD) = 1⟩.
Similarly,
Hxz = ⟨
(
A 0
D−A D
)
∈ GLn(q) | A,D ∈ GLk(q),det(AD) = 1⟩
316 Ars Math. Contemp. 21 (2021) #P2.10 / 309–317
and
Hyz = ⟨
(
A A−D
0 D
)
∈ GLn(q) | A,D ∈ GLk(q),det(AD) = 1⟩.
Our aim is now to show that Hℓ := ⟨Hxy, Hxz, Hyz⟩ is equal to H . We interrupt our proof
of Theorem 1.2 to prove this in the following lemma.
Lemma 2.5. The group generated by all matrices of the form
(
A 0
0 D
)
,
(
A A−D
0 D
)
and(
A 0
D−A D
)
where A,D ∈ GLk(q) and det(AD) = 1 equals H := SL2k(q).
Proof. Let L = ⟨
(
A 0
0 D
)
,
(
A A−D
0 D
)
,
(
A 0
D−A D
)
| A,D ∈ GLk(q),det(AD) = 1⟩ and
let x, y, z ∈
(
V
k
)
be as in (2.2). Then, L contains the matrix
(
A 0
D−A D
)(
A−1 0
0 D−1
)
=(
I 0
DA−1−I I
)
. In particular, choosing A = I and D = I+E1,2 we have DA−1 = I+E1,2
so that L contains the matrix M =
(
I 0
E1,2 I
)
. An element h = hA,D =
(
A 0
0 D
)
conjugates
M =
(
I 0
E1,2 I
)
to
( I 0
D−1E1,2A I
)
. Since L contains
(
A 0
0 A
)
for each permutation matrix
A, it follows that L contains
(
I 0
Ei,j I
)
for each i, j. Hence L contains Hx. Similarly, L
contains Hy . Since Hx is maximal in H and Hx ̸= Hy , we conclude that L = H .
Resuming our proof: Lemma 2.5 implies that Gℓ contains SLn(q) and since G is primi-
tive on points it follows that SLn(q) and hence also Gℓ, is transitive on points. This implies
that ℓ is incident with all points, which is a contradiction. This completes the proof of
Theorem 1.2.
ORCID iDs
Stephen P. Glasby https://orcid.org/0000-0002-0326-1455
Emilio Pierro https://orcid.org/0000-0003-1300-7984
Cheryl E. Praeger https://orcid.org/0000-0002-0881-7336
References
[1] S. H. Alavi and T. C. Burness, Large subgroups of simple groups, J. Algebra 421 (2015), 187–
233, doi:10.1016/j.jalgebra.2014.08.026.
[2] M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76
(1984), 469–514, doi:10.1007/bf01388470.
[3] J. Bamberg, S. P. Glasby, T. Popiel, C. E. Praeger and C. Schneider, Point-primitive generalised
hexagons and octagons, J. Comb. Theory Ser. A 147 (2017), 186–204, doi:10.1016/j.jcta.2016.
11.008.
[4] J. N. Bray, D. F. Holt and C. M. Roney-Dougal, The Maximal Subgroups of the Low-
Dimensional Finite Classical Groups, volume 407 of London Mathematical Society Lecture
Note Series, Cambridge University Press, Cambridge, 2013, doi:10.1017/cbo9781139192576.
[5] F. Buekenhout and H. Van Maldeghem, Remarks on finite generalized hexagons and octagons
with a point-transitive automorphism group, in: A. Beutelspacher, F. Buekenhout, J. Doyen,
F. De Clerck, J. A. Thas and J. W. P. Hirschfeld (eds.), Finite Geometry and Combinatorics,
Cambridge University Press, Cambridge, volume 191 of London Mathematical Society Lec-
ture Note Series, pp. 89–102, 1993, doi:10.1017/cbo9780511526336.011, Proceedings of the
Second International Conference held in Deinze, May 31 – June 6, 1992.
[6] F. Buekenhout and H. Van Maldeghem, Finite distance-transitive generalized polygons, Geom.
Dedicata 52 (1994), 41–51, doi:10.1007/bf01263523.
S. P. Glasby et al.: Point-primitive generalised hexagons and octagons and projective . . . 317
[7] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, An Atlas of Finite
Groups, Oxford University Press, Eynsham, 1985.
[8] W. Feit and G. Higman, The nonexistence of certain generalized polygons, J. Algebra 1 (1964),
114–131, doi:10.1016/0021-8693(64)90028-6.
[9] L. Morgan and T. Popiel, Generalised polygons admitting a point-primitive almost simple group
of Suzuki or Ree type, Electron. J. Combin. 23 (2016), #P1.34 (12 pages), doi:10.37236/5510.
[10] E. Pierro and T. Popiel, On the action of G2(q) on generalised hexagons and octagons, preprint.
[11] C. Schneider and H. Van Maldeghem, Primitive flag-transitive generalized hexagons and oc-
tagons, J. Comb. Theory Ser. A 115 (2008), 1436–1455, doi:10.1016/j.jcta.2008.02.004.
[12] J. Tits, Sur la trialité et certains groupes qui s’en déduisent, Inst. Hautes Études Sci. Publ. Math.
2 (1959), 13–60, http://www.numdam.org/item/PMIHES_1959__2__13_0/.
[13] H. Van Maldeghem, Generalized Polygons, volume 93 of Monographs in Mathematics,
Birkhäuser, 1998, doi:10.1007/978-3-0348-8827-1.

Author Guidelines
Before submission
Papers should be written in English, prepared in LATEX, and must be submitted as a PDF
file. The title page of the submissions must contain:
• Title. The title must be concise and informative.
• Author names and affiliations. For each author add his/her affiliation which should
include the full postal address and the country name. If avilable, specify the e-mail
address of each author. Clearly indicate who is the corresponding author of the paper.
• Abstract. A concise abstract is required. The abstract should state the problem stud-
ied and the principal results proven.
• Keywords. Please specify 2 to 6 keywords separated by commas.
• Mathematics Subject Classification. Include one or more Math. Subj. Class. (2020)
codes – see https://mathscinet.ams.org/mathscinet/msc/msc2020.html.
After acceptance
Articles which are accepted for publication must be prepared in LATEX using class file
amcjoucc.cls and the bst file amcjoucc.bst (if you use BibTEX). If you don’t use BibTEX,
please make sure that all your references are carefully formatted following the examples
provided in the sample file. All files can be found on-line at:
https://amc-journal.eu/index.php/amc/about/submissions/#authorGuidelines
Abstracts: Be concise. As much as possible, please use plain text in your abstract and
avoid complicated formulas. Do not include citations in your abstract. All abstracts will be
posted on the website in fairly basic HTML, and HTML can’t handle complicated formulas.
It can barely handle subscripts and greek letters.
Cross-referencing: All numbering of theorems, sections, figures etc. that are referenced
later in the paper should be generated using standard LATEX \label{. . . } and \ref{. . . }
commands. See the sample file for examples.
Theorems and proofs: The class file has pre-defined environments for theorem-like state-
ments; please use them rather than coding your own. Please use the standard \begin{proof}
. . . \end{proof} environment for your proofs.
Spacing and page formatting: Please do not modify the page formatting and do not use
\medbreak, \bigbreak, \pagebreak etc. commands to force spacing. In general, please
let LATEX do all of the space formatting via the class file. The layout editors will modify the
formatting and spacing as needed for publication.
Figures: Any illustrations included in the paper must be provided in PDF format, or
via LATEX packages which produce embedded graphics, such as TikZ, that compile with
PdfLATEX. (Note, however, that PSTricks is problematic.) Make sure that you use uniform
lettering and sizing of the text. If you use other methods to generate your graphics, please
provide .pdf versions of the images (or negotiate with the layout editor assigned to your
article).
xiii
Subscription
Yearly subscription: 150 EUR
Any author or editor that subscribes to the printed edition will receive a complimentary
copy of Ars Mathematica Contemporanea.
Subscription Order Form
Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E-mail: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Postal Address: . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I would like to subscribe to receive . . . . . . copies of each issue of
Ars Mathematica Contemporanea in the year 2021.
I want to renew the order for each subsequent year if not cancelled by e-mail:
□ Yes □ No
Signature: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Please send the order by mail, by fax or by e-mail.
By mail: Ars Mathematica Contemporanea
UP FAMNIT
Glagoljaška 8
SI-6000 Koper
Slovenia
By fax: +386 5 611 75 71
By e-mail: info@famnit.upr.si
xiv

Printed in Slovenia by IME TISKARNE