ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.07
https://doi.org/10.26493/2590-9770.1284.3ad
(Also available at http://adam-journal.eu)
Reflexible complete regular dessins and
antibalanced skew morphisms of cyclic groups
Kan Hu∗
Department of Mathematics, Zhejiang Ocean University,
Zhoushan, Zhejiang 316022, P.R. China
Young Soo Kwon†
Department of Mathematics, Yeungnam University,
Kyeongsan, 712-749, Republic of Korea
Received 27 December 2018, accepted 19 September 2019, published online 3 August 2020
Abstract
A skew morphism of a finite group A is a bijection ϕ on A fixing the identity element
of A and for which there exists an integer-valued function π on A such that ϕ(ab) =
ϕ(a)ϕπ(a)(b), for all a, b ∈ A. In addition, if ϕ−1(a) = ϕ(a−1)−1, for all a ∈ A,
then ϕ is called antibalanced. In this paper we develop a general theory of antibalanced
skew morphisms and establish a one-to-one correspondence between reciprocal pairs of
antibalanced skew morphisms of the cyclic additive groups and isomorphism classes of
reflexible regular dessins with complete bipartite underlying graphs. As an application,
reflexible complete regular dessins are classified.
Keywords: Graph embedding, antibalanced skew morphism, reciprocal pair.
Math. Subj. Class. (2020): 20B25, 05C10, 14H57
1 Introduction
A mapM is a 2-cell embedding i : Γ ↪→ S of a connected graph Γ, possibly with loops
or multiple edges, into a closed surface S such that each component of S \ i(Γ) is home-
omorphic to an open disc. A map is orientable if its supporting surface S is orientable,
∗Author was supported by Natural Science Foundation of Zhejiang Province (LY16A010010, LQ17A010003)
and National Natural Science Foundation of China (11801507).
†Author is supported by the Basic Science Research Program through the National Research Foundation of
Korea (NRF) funded by the Ministry of Education (2018R1D1A1B05048450).
E-mail addresses: hukan@zjou.edu.cn (Kan Hu), ysookwon@ynu.ac.kr (Young Soo Kwon)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 3 (2020) #P1.07
otherwise it is non-orientable. Throughout the paper, maps considered are all orientable. A
map with a 2-coloured bipartite underlying graph is called a dessin. An automorphism of
a dessin D is a permutation of the edges of the underlying bipartite graph which preserves
the graph structure and vertex colouring, and extends to an orientation-preserving self-
homeomorphism of the supporting surface. It is well known that the automorphism group
of a dessin acts semi-regularly on its edges. In the case where this action is transitive, and
hence regular, the dessin is called regular as well.
A regular dessin is reflexible if it is isomorphic to its mirror image, otherwise it is called
chiral. Moreover, a regular dessin is symmetric if it has an external symmetry transposing
the vertex colors. Thus, a symmetric regular dessin may be viewed as a regular map, that
is, a map whose orientation-preserving automorphism group acts transitively on the arcs.
A regular dessin is complete if its underlying graph is the complete bipartite graph
Km,n. Due to its important connection to generalized Fermat curves, the classification
problem of complete regular dessins has attracted much attention. A full classification of
the symmetric complete regular dessins was obtained in a series of papers [8, 9, 17, 18,
19, 22]. For the general case, complete bipartite graphs which underly a unique regular
dessin were determined by Fan and Li [10], and complete regular dessins of odd prime
power order have been recently classified by Hu, Nedela and Wang [13]. These results
were proved by group-theoretic methods through a translation of complete regular dessins
to exact bicyclic groups with two distinguished generators.
Recently, Feng et al discovered an alternative approach to this problem by establishing
a surprising correspondence between complete regular dessins and reciprocal pairs of skew
morphisms of cyclic groups [12]. A skew morphism of a finite group A is a bijection ϕ on
A fixing the identity element of A and for which there exists an integer function π on A
such that ϕ(ab) = ϕ(a)ϕπ(a)(b), for all a, b ∈ A. Suppose that ϕ and ϕ̃ are a pair of skew
morphisms of the cyclic additive groups Zn and Zm, and π and π̃ are associated power
functions, respectively. The skew morphism pair (ϕ, ϕ̃) is called reciprocal if they satisfy
the following conditions:
(a) the order of ϕ divides m and the order of ϕ̃ divides n,
(b) π(x) ≡ ϕ̃x(1) (mod |ϕ|) and π̃(y) ≡ ϕy(1) (mod |ϕ̃|) for all x ∈ Zn and y ∈ Zm.
In [12, Theorem 5] the authors proved that the isomorphism classes of complete regular
dessins with underlying graphs Km,n are in one-to-one correspondence with the reciprocal
pairs of skew morphisms of the cyclic groups Zn and Zm.
The aim of this paper is to classify the reflexible complete regular dessins. Employ-
ing methods used in [12] we are led naturally to introduce a new concept of antibalanced
skew morphism. More precisely, a skew morphism of a finite group A is antibalanced
if ϕ−1(a) = ϕ(a−1)−1, for all a ∈ A. We note that there is a big difference between
antibalanced skew morphisms and skew morphisms arising from antibanlanced regular
Cayley maps, because the latter have a generating orbit which is closed under taking in-
verses [4, 16], while the former do not have such a restriction.
In Section 3 we develop a general theory of antibalanced skew morphisms, and present
a classification and enumeration of antibalanced skew morphisms of cyclic groups, extend-
ing the results obtained by Conder, Jajcay and Tucker [4, Theorem 7.1]. In Section 4, we
establish a one-to-one correspondence between reflexible complete regular dessins and re-
ciprocal pairs of antibalanced skew morphisms of cyclic groups. In Section 5 all reciprocal
pairs of antibalanced skew morphisms of cyclic groups are completely determined.
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 3
2 Preliminaries
The theory of skew morphisms has been developed and expanded by various authors. In
this section we summarize and prove some preliminary results for future reference.
Let ϕ be a skew morphism of a finite group A, and let π be a power function of ϕ. In
general, the function π is not uniquely determined by ϕ. However, if ϕ has order k, then π
may be regarded as a function π : A → Zk, which is unique. In this case we will refer to
π as the power function of ϕ. A subgroup N of A is ϕ-invariant if ϕ(N) = N , in which
case the restriction of ϕ to N is a skew morphism of N . Moreover, it is well known [16]
that Kerϕ and Fixϕ defined by
Kerϕ = {a ∈ A | π(a) = 1} and Fixϕ = {a ∈ A | ϕ(a) = a}
are subgroups of A, and in particular, Fixϕ is ϕ-invariant. Note that, for any two elements
a, b ∈ A, π(a) = π(b) if and only if ab−1 ∈ Kerϕ, so the power function π of ϕ takes
exactly |A : Kerϕ| distinct values in Zk. The index |A : Kerϕ| will be called the skew
type of ϕ. It follows that a skew morphism of A is an automorphism if and only if it has
skew type 1. A skew morphism which is not an automorphism will be called a proper skew
morphism.
Moreover, define
Coreϕ =
k⋂
i=1
ϕi(Kerϕ).
Then Coreϕ is a ϕ-invariant normal subgroup of A, and it it is the largest ϕ-invariant
subgroup of A contained in Kerϕ. In particular, if A is abelian, then Coreϕ = Kerϕ [4,
Lemma 5.1].
The following properties of skew morphisms are fundamental.
Proposition 2.1 ([16]). Let ϕ be a skew morphism of a finite group A, let π be the power
function of ϕ, and let k be the order of ϕ. Then the following hold:
(a) for any positive integer ` and for any a, b ∈ A, ϕ`(ab) = ϕ`(a)ϕσ(a,`)(b), where
σ(a, `) =
∑̀
i=1
π(ϕi−1(a));
(b) for all a, b ∈ A, π(ab) ≡
π(a)∑
i=1
π(ϕi−1(b)) (mod k).
Proposition 2.2 ([1]). Let ϕ be a skew morphism of a finite group A, let π be the power
function of ϕ, and let k be the order of ϕ. Then µ = ϕ` is a skew morphism of A if and
only if the congruence `x ≡ σ(a, `) (mod k) is soluble for every a ∈ A, in which case
πµ(a) is the solution
Proposition 2.3 ([15]). If ϕ is a skew morphism of a finite group A, then O−1a = Oa−1 for
any a ∈ A, where Oa and Oa−1 denote the orbits of ϕ containing a and a−1, respectively.
Proposition 2.4 ([1, 25]). Let ϕ be a skew morphism of a finite group A, and let π be the
power function of ϕ. Then for any automorphism θ of A, µ = θ−1ϕθ is a skew morphism
of A with power function πµ = πθ.
4 Art Discrete Appl. Math. 3 (2020) #P1.07
Proposition 2.5 ([26, 25]). Let ϕ be a skew morphism of a finite group A, and let π be
the power function of ϕ. If A = 〈a1, a2, . . . , ar〉, then |ϕ| = lcm(|Oa1 |, |Oa2 |, . . . , |Oar |).
Moreover, the skew morphism ϕ and its power function π are completely determined by the
action of ϕ and the values of π on the generating orbits Oa1 , Oa2 , . . . , Oar , respectively.
Proposition 2.6 ([26]). Let ϕ be a skew morphism of a finite group A, and let π be the
power function of ϕ. IfN is a ϕ-invariant normal subgroup ofA, then ϕ̄ defined by ϕ̄(x̄) =
ϕ(x) is a skew morphism of Ā := A/N and the power function π̄ of ϕ̄ is determined by
π̄(x̄) ≡ π(x) (mod |ϕ̄|).
Since Coreϕ is a ϕ-invariant normal subgroup of A, by Proposition 2.6, ϕ induces a
skew morphism ϕ̄ of Ā = A/Coreϕ. It is shown in [25] that the ϕ̄-invariant subgroup
Fix ϕ̄ of Ā lifts to a ϕ-invariant subgroup Smoothϕ of A, namely,
Smoothϕ = {a ∈ A | ϕ̄(ā) = ā}.
In particular, if Smoothϕ = A, then the skew morphism ϕ is called a smooth skew mor-
phism.
Proposition 2.7 ([25]). Let ϕ be a skew morphism of a finite group A, and let π be the
power function of ϕ. Then ϕ is smooth if and only if π(ϕ(a)) = π(a) for all a ∈ A.
The most important properties of smooth skew morphisms are summarized as follows.
Proposition 2.8 ([25]). Let ϕ be a smooth skew morphism of A, |ϕ| = k, and let π be the
power function of ϕ. Then the following hold:
(a) ϕ(Kerϕ) = Kerϕ;
(b) π : A → Zk is a group homomorphism of A into the multiplicative group Z∗k with
Kerπ = Kerϕ;
(c) for any ϕ-invariant normal subgroupN ofA, the induced skew morphism ϕ̄ onA/N
is also smooth, and in particular, if N = Kerϕ then ϕ̄ is the identity permutation;
(d) for any positive integer `, µ = ϕ` is a smooth skew morphism of A;
(e) for any automorphism θ of A, µ = θ−1ϕθ is a smooth skew morphism of A.
Lemma 2.9. Let ϕ be a skew morphism of a finite group A. Then ϕ is smooth if and only
if there exists a ϕ-invariant normal subgroup N of A contained in Kerϕ such that the
induced skew morphism ϕ̄ of Ā = A/N is the identity permutation.
Proof. If ϕ is smooth, then by Proposition 2.8(a), ϕ is kernel-preserving, and so Kerϕ =
Coreϕ. Take N = Kerϕ, then by Proposition 2.8(c) the induced skew morphism ϕ̄ of
A/N is the identity permutation.
Conversely, suppose that there exists a ϕ-invariant normal subgroup N of A contained
in Kerϕ such that the induced skew morphism ϕ̄ of Ā = A/N is the identity permutation.
Then, for each a ∈ A, there is an element u ∈ N ≤ Kerϕ such that ϕ(a) = ua. Thus,
π(ϕ(a)) = π(a), and therefore ϕ is smooth by Proposition 2.7.
There is a fundamental correspondence between skew morphisms and groups with
cyclic complements.
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 5
Proposition 2.10 ([5]). If G = AC is a factorisation of a finite group G into a product
of a subgroup A and a cyclic subgroup C = 〈c〉 with A ∩ C = 1, then c induces a skew
morphism ϕ of A via the commuting rule ca = ϕ(a)cπ(a), for all a ∈ A; in particular
|ϕ| = |C : CG|, where CG = ∩g∈GCg .
Conversely, if ϕ is a skew morphism of a finite groupA, thenG = LA〈ϕ〉 is a transitive
permutation group on A with LA ∩ 〈ϕ〉 = 1 and 〈ϕ〉 core-free in G, where LA is the left
regular representation of A.
3 Antibalanced skew morphisms
In this section we develop a theory of antibalanced skew morphisms and classify all an-
tibalanced skew morphisms of cyclic groups.
A skew morphism ϕ of a finite group A will be called antibalanced if
ϕ−1(a) = ϕ(a−1)−1, for all a ∈ A.
Since 1 = ϕ(aa−1) = ϕ(a)ϕπ(a)(a−1), we have ϕ(a)−1 = ϕπ(a)(a−1). Thus, ϕ
is antibalanced if and only if ϕπ(a)(a−1) = ϕ−1(a−1), or equivalently, π(a) ≡ −1
(mod |Oa−1 |), for all a ∈ A. By Proposition 2.3, |Oa| = |Oa−1 |. It follows that ϕ
is antiblanced if and only if π(a) ≡ −1 (mod |Oa|) for all a ∈ A. Note that for any
a ∈ Kerϕ, |Oa| is 1 or 2.
Remark 3.1. It was proved in [16, Theorem 1] that a Cayley map CM(A,X, p) is regular
(on the arcs) if and only if there is a skew morphism ϕ of A such that the restriction of ϕ
to X is equal to p. Since X is a generating set of A and is closed under taking inverses, the
associated skew morphism ϕ has a generating orbit which is closed under taking inverses.
For brevity, such a skew morphism will be called a Cayley skew morphism.
Moreover, a regular Cayley map CM(A,X, p) was termed antibalanced if p−1(x) =
p(x−1)−1 for all x ∈ X [24]. It follows that a regular Cayley map is antibalanced if and
only if the associated Cayley skew morphism is antibalanced. However, neither generat-
ing orbit, nor inverse-closed orbit are assumed in the preceding definition of antibalanced
skew morphisms. Therefore, antibalanced skew morphisms may be regarded as a natural
generalization of the skew morphisms arising from antibalanced regular Cayley maps.
We give an example to clarify the concept.
Example 3.2. The cyclic group Z12 has exactly eight skew morphisms, four of which are
proper:
ϕ = (0)(2)(4)(6)(8)(10)(1, 3, 5, 7, 9, 11), πϕ = [1][1][1][1][1][1][5, 5, 5, 5, 5, 5];
ψ = (0)(2)(4)(6)(8)(10)(1, 11, 9, 7, 5, 3), πψ = [1][1][1][1][1][1][5, 5, 5, 5, 5, 5];
µ = (0)(2)(4)(6)(8)(10)(1, 5, 9)(3, 7, 11), πµ = [1][1][1][1][1][1][2, 2, 2][2, 2, 2];
γ = (0)(2)(4)(6)(8)(10)(1, 9, 5)(3, 11, 7), πγ = [1][1][1][1][1][1][2, 2, 2][2, 2, 2].
It is easily seen that all the above skew morphisms are antibalanced. Note that the first two
skew morphisms contain a generating orbit closed under taking inverses, but the last two
skew morphisms do not contain such an orbit. Therefore, ϕ and ψ are antibalanced Cayley
skew morphism, and µ and γ are antibalanced non-Cayley skew morphisms.
We summarise some properties of antibalanced skew morphisms as follows.
6 Art Discrete Appl. Math. 3 (2020) #P1.07
Lemma 3.3. Let ϕ be an antibalanced skew morphism of a finite group A, and let π be the
associated power function. Then the following hold:
(a) for any positive integer `, ϕ−`(a) = ϕ`(a−1)−1 for all a ∈ A;
(b) for any automorphism θ of A, the skew morphism µ = θ−1ϕθ is antibalanced;
(c) for any ϕ-invariant normal subgroup N of A, the induced skew morphism ϕ̄ of A/N
is antibalanced;
(d) for any c ∈ Kerϕ and a ∈ A, π(a) ≡ 1 (mod |Oc|).
Proof. (a) The case ` = 1 is the definition. Assume the result for `, i.e. ϕ−`(a) =
ϕ`(a−1)−1 for all a ∈ A. Then
ϕ−(`+1)(a) = ϕ−1(ϕ−`(a)) = ϕ−1(ϕ`(a−1)−1) = ϕ(ϕ`(a−1))−1 = ϕ`+1(a−1)−1,
and the result follows by induction.
(b) For any a ∈ A, we have
µ−1(a) = θ−1ϕ−1θ(a) = θ−1(ϕ(θ(a)−1))−1) =
(
θ−1(ϕ(θ(a−1))
)−1
= µ(a−1)−1,
so µ is antibalanced.
(c) Since ϕ−1(a) = ϕ(a−1)−1, we have ϕ̄−1(ā) = ϕ̄(ā−1)−1, and so ϕ̄ is antibal-
anced.
(d) For any c ∈ Kerϕ and any a ∈ A, we have
ϕ(c)a−1 = ϕ(c)[ϕ−1(ϕ(a))]−1 = ϕ(c)ϕ(ϕ(a)−1) = ϕ(cϕ(a)−1)
= ϕ−1(ϕ(a)c−1)−1 =
(
ϕ−1(ϕ(a))ϕ−πϕ
−1(ϕ(a))(c−1)
)−1
= (aϕ−π(a)(c−1))−1 = (aϕπ(a)(c)−1)−1 = ϕπ(a)(c)a−1,
so ϕπ(a)(c) = ϕ(c), and hence π(a) ≡ 1 (mod |Oc|).
Lemma 3.4. Let ϕ be an automorphism of a finite group A. Then ϕ is antibalanced if and
only if ϕ2 = 1, that is, ϕ is an involutory automorphism.
Proof. By definition, ϕ is antibalanced if and only if for all a ∈ A, ϕ−1(a) = ϕ(a−1)−1.
Since ϕ is an automorphism, ϕ(a−1)−1 = ϕ(a), and hence ϕ is antibalanced if and only if
for all a ∈ A, ϕ−1(a) = ϕ(a), that is, ϕ2(a) = a.
Corollary 3.5. Every antibalanced automorphism of the cyclic additive group Zn is of the
form ϕ(x) = sx, x ∈ Zn, where s2 ≡ 1 (mod n).
Let ϕ be a skew morphism of a finite groupA. Suppose that ϕ has an orbitX generating
A. The words of even length in the generators from X form a subgroup of A, which will
be called the even word subgroup of A with respect to X and denoted by A+X . Note that
the index of A+X in A is 1 or 2.
The following results generalize the properties of antibalanced Cayley skew morphisms
(or more precisely, antibalanced regular Cayley maps) obtained in [4]
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 7
Lemma 3.6. Let ϕ be a skew morphism of a finite group A containing an orbit X which
generates A, and let π be the associated power function. Then ϕ is antibalanced if and
only if π(x) ≡ −1 (mod |X|) for all x ∈ X and ϕ restricted to A+X is an involutory
automorphism. Furthermore, if ϕ is antibalanced, then ϕ is a smooth skew morphism of
skew type 1 or 2.
Proof. Since X is a generating orbit of ϕ, we have |ϕ| = |X| by Proposition 2.5, and the
value of π onA is completely determined by the value of π onX . Suppose that π(x) ≡ −1
(mod |X|) for all x ∈ X . Then by Proposition 2.1, for any x, y ∈ X ,
π(xy) =
π(x)∑
i=1
π(ϕi−1(y)) ≡ π(x)π(y) ≡ 1 (mod |X|).
SinceA = 〈X〉, every element ofA is expressible as a word of finite length in the elements
ofX . By induction, π(a) ≡ 1 (mod |X|) if a is an even word, and π(a) ≡ −1 (mod |X|)
if a is an odd word. Note that if a is an even word (resp. an odd word), then both ϕ(a) and
a−1 are also even words (resp. odd words). Thus, ϕ is a smooth skew morphism of skew
type 1 or 2 and ϕ restricted to A+X is an automorphism of A
+
X .
If ϕ is antibalanced, then it is evident that π(x) ≡ −1 (mod |X|) for all x ∈ X .
This implies that ϕ restricted to A+X is an automorphism of A
+
X , and hence, for any a ∈
A+X , ϕ(a
−1) = ϕ(a)−1. Since ϕ is antibalanced, we have ϕ(a−1)−1 = ϕ−1(a), and
hence, for any a ∈ A+X , ϕ−1(a) = ϕ(a). Therefore, ϕ restricted to A
+
X is an involutory
automorphism.
Conversely, assume that π(x) ≡ −1 (mod |X|) for all x ∈ X and ϕ restricted to A+X
is an involutory automorphism. For any even word a ∈ A+X , ϕ−1(a) = ϕ(a) = ϕ(a−1)−1.
For any odd word b,
1 = ϕ(b−1b) = ϕ(b−1)ϕπ(b
−1)(b) = ϕ(b−1)ϕ−1(b),
and so ϕ−1(b) = ϕ(b−1)−1. Therefore, ϕ is antibalanced.
Remark 3.7. Let ϕ be a skew morphism of a finite group A containing an orbit X which
generates A, and let π be the associated power function. Lemma 3.6 implies that
(a) if |A : A+X | = 1, then the skew morphism ϕ is antibalanced if and only if ϕ is an
involutory automorphism of A;
(b) if |A : A+X | = 2, then ϕ is antibalanced if and only if π(a) ≡ 1 (mod |ϕ|) for all
a ∈ A+X , π(a) ≡ −1 (mod |ϕ|) for all a ∈ A \ A
+
X and ϕ restricted to A
+
X is an
involutory automorphism.
The following lemma deals with antibalanced skew morphisms of abelian groups.
Lemma 3.8. Let ϕ be a skew morphism of a finite abelian group A containing an orbit X
which generates A, and let π be the associated power function. Then ϕ is antibalanced if
and only if π(x) ≡ −1 (mod |X|) for all x ∈ X .
Proof. By Lemma 3.6, it suffices to prove the sufficient part. Assume that π(x) ≡ −1
(mod |X|) for all x ∈ X . Then, π(a) ≡ 1 (mod |X|) if a is an even word, and π(a) ≡ −1
8 Art Discrete Appl. Math. 3 (2020) #P1.07
(mod |X|) if a is an odd word. Furthermore ϕ restricted toA+X is an automorphism ofA
+
X .
For any a ∈ A+X and for any odd word b,
ϕ(b)ϕ−1(a) = ϕ(ba) = ϕ(ab) = ϕ(a)ϕ(b) = ϕ(b)ϕ(a),
and hence ϕ2(a) = a. Thus, by Lemma 3.6, ϕ is antibalanced.
Now we are ready to determine antibalanced skew morphisms of cyclic groups.
Theorem 3.9. Let ϕ be an antibalanced skew morphism of the cyclic additive group Zn.
(a) If n is odd, then ϕ is an involutory automorphism of the form ϕ(x) = sx, x ∈ Zn,
where s2 ≡ 1 (mod n).
(b) If n is even, then ϕ and the associated power function π are of the form
ϕ(x) =
{
xs, x is even,
(x− 1)s+ 2r + 1, x is odd,
and π(x) =
{
1, x is even,
−1, x is odd,
(3.1)
where r, s are integers in Zn/2 such that
s2 ≡ 1 (mod n/2) and (r + 1)(s− 1) ≡ 0 (mod n/2). (3.2)
In this case, the order of ϕ is equal to n/ gcd(n, r(s+ 1)), and in particular ϕ is an
automorphism if and only if s ≡ 2r + 1 (mod n/2) and (2r + 1)2 ≡ 1 (mod n).
Proof. First suppose that ϕ is an antibalanced skew morphism of Zn with the associated
power function π. Note that the orbit X of ϕ containing 1 generates Zn. Let Z+n be the
even word subgroup of Zn with respect to X . Then |Zn : Z+n | = 1 or 2. By Lemma 3.6, ϕ
restricted to Z+n is an involutory automorphism.
If n is odd, then Z+n = Zn, so ϕ is an involutory automorphism of Zn, and the result
follows from Corollary 3.5.
Now assume that n is an even number. By Lemma 3.6, ϕ is a smooth skew morphism
of skew type 1 or 2, so 〈2〉 is a ϕ-invariant normal subgroup of Zn contained in Kerϕ,
and the induced skew morphism ϕ̄ of Zn/〈2〉 is the identity permutation. Thus, there are
integers r, s ∈ Zn/2 such that
ϕ(1) ≡ 2r + 1 (mod n) and ϕ(2) ≡ 2s (mod n),
where gcd(s, n/2) = 1. By Lemma 3.6,
π(x) = 1 if x is even, π(x) = −1 if x is odd.
It follows that
ϕ(x) =
{
xs, x is even,
ϕ(x− 1) + ϕ(1) = (x− 1)s+ 2r + 1, x is odd.
(3.3)
Since ϕ restricted to Z+n is an involutory automorphism, we have s2 ≡ 1 (mod n/2).
Furthermore, we have
2s = ϕ(2) = ϕ(1) + ϕ−1(1) = 2r + 1− 2rs+ 1 (mod n),
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 9
and hence (r + 1)(s− 1) ≡ 0 (mod n/2). Moreover, by induction we have
ϕj(1) ≡ 1 + 2r
j∑
i=1
si−1 (mod n).
Let k be the smallest positive integer such that ϕk(1) = 1. By Proposition 2.5, k = |ϕ|.
If k is odd, then s ≡ 1 (mod n/2), since the length of the orbit of ϕ containing 2 di-
vides k. Upon substitution we get k = n/ gcd(n, 2r). If k is even, then the congruence
2r
∑k
i=1 s
i−1 ≡ 0 (mod n) reduces to rk(s+ 1) ≡ 0 (mod n), so k = n/ gcd(n, r(s+
1)). Note that in either cases k = n/ gcd(n, r(s + 1)). In particular, if ϕ is an automor-
phism, then for any x ∈ Zn, ϕ(x) = xϕ(1) = x(2r + 1) and (2r + 1)2 ≡ 1 (mod n).
Conversely, we need to verify that ϕ given by (3.1) is an antibalanced skew morphism
of Zn, provided that the stated numerical conditions in (3.2) are fulfilled. It is easily seen
that ϕ(0) = 0 and ϕ is a bijection on Zn. Now for any x ∈ Zn and for any y ∈ Zn, if x is
even, then one can easily show that
ϕ(x) + ϕ(y) = ϕ(x+ y).
If x is odd and y is even, then
ϕ(x) + ϕ−1(y) = (x− 1)s+ 2r + 1 + ys = (x+ y − 1)s+ 2r + 1 = ϕ(x+ y).
Finally, if both x and y are odd, then ϕ(−2rs + (y − 1)s + 1) = y, and so ϕ−1(y) =
−2rs+ (y − 1)s+ 1. From the condition (r + 1)(s− 1) ≡ 0 (mod n/2) we deduce that
−2rs ≡ 2s− 2r − 2 (mod n) and hence ϕ−1(y) = (y + 1)s− 2r − 1. Consequently,
ϕ(x) + ϕ−1(y) = (x− 1)s+ 2r + 1 + (y + 1)s− 2r − 1 = (x+ y)s = ϕ(x+ y).
Therefore, ϕ is a skew morphism of Zn. By Lemma 3.8, it is antibalanced.
From the proof of Theorem 3.9 we obtain the following corollary.
Corollary 3.10. Let ϕ be an antibalanced skew morphism of Zn. If ϕ is of odd order, then
the restriction of ϕ to Kerϕ is the identity automorphism of Kerϕ.
Theorem 3.11. Let n = 2αpα11 p
α2
2 · · · p
α`
` be the prime power factorization of a positive
integer n. Then the number ν(n) of antibalanced skew morphisms of the cyclic additive
group Zn is determined by the following formula:
ν(n) =

2`, α = 0,∏̀
i=1
(pαii + 1), α = 1,
2
∏̀
i=1
(pαii + 1), α = 2,
6
∏̀
i=1
(pαii + 1), α = 3,
(4 + 2α−2 + 2α−1)
∏̀
i=1
(pαii + 1), α ≥ 4.
10 Art Discrete Appl. Math. 3 (2020) #P1.07
Proof. If α = 0, then n is odd. By Theorem 3.9(a), every antibalanced skew morphism of
Zn is an automorphism of the form ϕ(x) = xs, x ∈ Zn, where s2 ≡ 1 (mod n). It follows
that the number ν(n) is equal to the number of solutions of the quadratic congruence s2 ≡ 1
(mod n), which is equal to 2`.
Now assume α ≥ 1, so n is an even number. By Theorem 3.9(b), the number of
antibalanced skew morphisms of Zn is equal to the number of integer solutions (r, s) in
Zn/2 of the system {
s2 ≡ 1 (mod n/2),
(r + 1)(s− 1) ≡ 0 (mod n/2).
By the Chinese Remainder Theorem, (r, s) is a solution of the system if and only if it is a
solution of each of the following `+ 1 systems{
s2 ≡ 1 (mod 2α−1),
(r + 1)(s− 1) ≡ 0 (mod 2α−1)
(3.4)
and {
s2 ≡ 1 (mod pαii ),
(r + 1)(s− 1) ≡ 0 (mod pαii ),
i = 1, 2, . . . , `. (3.5)
We first determine the solutions of (3.5). By assumption, for each i, i = 1, 2, . . . , `,
pi is an odd prime. It follows from the congruence s2 ≡ 1 (mod pαii ) that either s ≡
1 (mod pαii ) or s ≡ −1 (mod p
αi
i ). If s ≡ 1 (mod p
αi
i ), then upon substitution the
congruence (r+ 1)(s− 1) ≡ 0 (mod pαii ) holds for every r ∈ Zpαii . On the other hand, if
s ≡ −1 (mod pαii ), then upon substitution the congruence (r+ 1)(s−1) ≡ 0 (mod p
αi
i )
reduces to r ≡ −1 (mod pαii ). Therefore, for each odd prime pi, the system (3.5) has
precisely (pαii + 1) solutions in Zpαii .
Now we turn to solutions of (3.4). If α = 1, then it only has the trivial solution
(r, s) = (1, 1). If α = 2, then (r, s) = (0, 1), (1, 1) in Z2. If α = 3, then (r, s) =
(0, 1), (1, 1), (2, 1), (3, 1), (1, 3), (3, 3) in Z4. If α ≥ 4, then by the congruence s2 ≡ 1
(mod 2α−1) we have s ≡ ±1, 2α−2±1 (mod 2α−1). Combining this with the congruence
(r+ 1)(s− 1) ≡ 0 (mod 2α−1) we obtain the following solutions (r, s) in Z2α−1 : (a) r ∈
Z2α−1 and s = 1; (b) r = 2α−1 − 1, 2α−2 − 1 and s = −1; (c) r = 2α−1 − 1, 2α−2 − 1
and s = 2α−2 − 1; (d) r ≡ 1 (mod 2) and s = 2α−2 + 1.
Finally, multiplying the numbers of solutions for the prime power cases we obtain the
number ν(n), as required.
4 Correspondence
A correspondence between complete regular dessins and pairs of certain skew morphisms
of cyclic groups has been established in [12, Theorem 5]. In this section we extend the
correspondence to reflexible complete regular dessins.
Definition 4.1. Let ϕ : Zn → Zn and ϕ̃ : Zm → Zm be a pair of skew morphisms of the
cyclic additive groups Zn and Zm, associated with power functions π : Zn → Z|ϕ| and
π̃ : Zm → Z|ϕ̃|, respectively. The pair (ϕ, ϕ̃) will be called reciprocal if they satisfy the
following conditions:
(a) |ϕ| divides m and |ϕ̃| divides n,
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 11
(b) π(x) ≡ ϕ̃x(1) (mod |ϕ|) and π̃(y) ≡ ϕy(1) (mod |ϕ̃|) for all x ∈ Zn and y ∈ Zm.
Suppose that D is a complete regular dessin with underling graph Km,n. Take an
arbitrary pair of vertices u and v of valency m and n, respectively. Then the stabilizers Gu
and Gv of G = Aut(D) are cyclic of orders m and n, respectively. Assume Gu = 〈a〉 and
Gv = 〈b〉. Then by the regularity we have G = 〈a, b〉 and |G| = mn. Since the underlying
graph Km,n is simple, 〈a〉 ∩ 〈b〉 = 1. Consequently, from the product formula we deduce
that G = 〈a〉〈b〉. Thus each complete regular dessin determines a triple (G, a, b) such that
G = 〈a〉〈b〉 and 〈a〉 ∩ 〈b〉 = 1.
Now each of the cyclic factors 〈a〉 and 〈b〉 of G can be taken as the complement of the
other, so in the spirit of Proposition 2.10, there are a pair of skew morphisms ϕ and ϕ̃ of
the cyclic additive groups Zn and Zm such that
a−1bx = bϕ(x)a−π(x) and b−1ay = aϕ̃(y)b−π̃(y) (4.1)
where x ∈ Zn and y ∈ Zm. By induction we deduce that
a−kbx = bϕ
k(x)a−σ(x,k) and b−lay = aϕ̃
l(y)b−σ̃(y,l), (4.2)
where
σ(x, k) =
k∑
i=1
π(ϕi−1(x)) and σ̃(y, l) =
l∑
i=1
π̃(ϕ̃i−1(y)).
Inverting the above identities yields
b−xak = aσ(x,k)b−ϕ
k(x) and a−ybl = bσ̃(y,l)a−ϕ̃
l(y). (4.3)
Substituting for x = 1 and k = y we obtain b−1ay = aσ(1,y)b−ϕ
y(1). By comparing this
with the second identity in (4.1) we obtain
π̃(y) ≡ ϕy(1) (mod n) and ϕ̃(y) ≡ σ(1, y) (mod m).
Similarly, inserting y = 1 and l = x into the second identity in (4.3) we have a−1bx =
bσ̃(1,x)a−ϕ̃
x(1). A similar comparison with the first identity in (4.1) yields
π(x) ≡ ϕ̃x(1) (mod m) and ϕ(x) ≡ σ̃(1, x) (mod n).
By Proposition 2.10, |ϕ| = |〈a〉 : 〈a〉G| and |ϕ̃| = |〈b〉 : 〈b〉G|. Thus |ϕ| divides m and
|ϕ̃| divides n. In particular, the above four congruences are reduced to
π(x) ≡ ϕ̃x(1) (mod |ϕ|), π̃(y) ≡ ϕy(1) (mod |ϕ̃|)
and
ϕ(x) ≡ σ̃(1, x) (mod |ϕ̃|), ϕ̃(y) ≡ σ(1, y) (mod |ϕ|). (4.4)
It follows that every complete regular dessin with underlying graphKm,n determines a pair
of reciprocal skew morphisms (ϕ, ϕ̃) of the cyclic additive groups Zn and Zm. Conversely,
it is shown in [12, Proposition 4] that given a pair of reciprocal skew morphisms of the
cyclic groups Zn and Zm, a complete regular dessin with underlying graph Km,n may be
reconstructed in a canonical way. Therefore, we obtain a correspondence between complete
regular dessins and pairs of reciprocal skew morphisms of cyclic groups. See [12, Theorem
5] for details.
The following theorem extends this correspondence to reflexible complete regular dessins.
12 Art Discrete Appl. Math. 3 (2020) #P1.07
Theorem 4.2. The isomorphism classes of reflexible regular dessins with complete bipar-
tite underlying graphs Km,n are in one-to-one correspondence with pairs of reciprocal
antibalanced skew morphisms (ϕ, ϕ̃) of the cyclic groups Zn and Zm.
Proof. It is proved in [12, Theorem 5] that the isomorphism classes of regular dessinsD =
(G, a, b) with complete bipartite underlying graphsKm,n are in one-to-one correspondence
with the pairs (ϕ, ϕ̃) of reciprocal skew morphisms of the cyclic groups Zn and Zm. It
remains to show thatD is reflexible if and only if the corresponding pair of skew morphisms
(ϕ, ϕ̃) are both antibalanced.
First suppose that D = (G, a, b) is reflexible, then the identities in (4.1) determine a
pair of reciprocal skew morphisms (ϕ, ϕ̃) of the cyclic groups Zn and Zm. Since D is
reflexible, the assignment θ : a 7→ a−1, b 7→ b−1 extends to an automorphism of G. By the
identities in (4.2) derived from (4.1) we have
ab−x = bϕ
−1(−x)a−σ(−x,−1) and ba−y = aϕ̃
−1(−y)b−σ̃(−y,−1).
Applying the automorphism θ of G to the above identities we obtain
a−1bx = θ(ab−x) = θ(bϕ
−1(−x)a−σ(−x,−1)) = b−ϕ
−1(−x)aσ(−x,−1)
and
b−1ay = θ(ba−y) = θ(aϕ̃
−1(−y)b−σ̃(−y,−1)) = a−ϕ̃
−1(−y)bσ̃(−y,−1).
By comparing these with the identities in (4.1) we get ϕ(x) = −ϕ−1(−x) and ϕ̃(y) =
−ϕ̃−1(−y). Thus both ϕ and ϕ̃ are antibalanced.
Conversely, suppose that ϕ : Zn → Zn and ϕ̃ : Zm → Zm form a pair of antibalanced
reciprocal skew morphisms. Denote
Zn = {0, 1, . . . , (n− 1)} and Zm = {0′, 1′, . . . , (m− 1)′},
so that Zn and Zm are disjoint sets. Define two cyclic permutations ρ and ρ̃ on the sets Zn
and Zm by setting
ρ = (0, 1, . . . , (n− 1)) and ρ̃ = (0′, 1′, . . . , (m− 1)′).
We extend the permutations ϕ, ϕ̃, ρ and ρ̃ to permutations on Ω = Zn ∪ Zm in a natural
way, still denoted by ϕ, ϕ̃, ρ and ρ̃. Set a = ϕρ̃, b = ϕ̃ρ and G = 〈a, b〉. It is proved in [12,
Proposition 4] that |a| = m, |b| = n, 〈a〉 ∩ 〈b〉 = 1 and G = 〈a〉〈b〉, so D = (G, a, b) is a
complete regular dessin with underlying graph Km,n. Now define a bijection γ : Ω → Ω
on Ω to be γ(x) = −x and γ(y′) = −y′ for all x ∈ Zn and y′ ∈ Zm. Since both ϕ and ϕ̃
are antibalanced, we have
γa(x) = γϕρ̃(x) = γϕ(x) = −ϕ(x) = ϕ−1(−x)
= ϕ−1γ(x) = ϕ−1ρ̃−1(γ(x)) = a−1γ(x)
and
γa(y′) = γϕρ̃(y′) = γϕ((y + 1)′) = γ((y + 1)′) = −(y + 1)′
= (−y − 1)′ = ϕ−1ρ̃−1(−y′) = a−1γ(y′).
Thus γa = a−1γ. Similarly, γb = b−1γ. Hence, (G, a, b) ∼= (G, a−1, b−1), where
(G, a−1, b−1) denotes the mirror image ofD. Therefore, (G, a, b) is reflexible, as required.
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 13
We summarize two properties of reciprocal skew morphisms.
Lemma 4.3 ([12, 14]). Let ϕ : Zn → Zn and ϕ̃ : Zm → Zm be a pair of reciprocal skew
morphisms of the cyclic additive groups Zn and Zm. Then
(a) ϕ(x) ≡
∑x
i=1 π̃(ϕ̃
i−1(1)) (mod |ϕ̃|) and ϕ̃(y) ≡
∑y
i=1 π(ϕ
i−1(1)) (mod |ϕ|),
(b) |Zm : Ker ϕ̃| divides |ϕ| and |Zn : Kerϕ| divides |ϕ̃|.
Lemma 4.4 ([14]). Let ϕ : Zn → Zn and ϕ̃ : Zm → Zm be a pair of reciprocal skew
morphisms of the cyclic additive groups Zn and Zm. If one of the skew morphisms is an
automorphism, then the other is smooth. In particular, if one of the skew morphism is the
identity automorphism, then the other is an automorphism.
5 Classification
By Theorem 4.2, the classification of reflexible complete regular dessins is reduced to the
classification of reciprocal pairs of antibalanced skew morphisms of cyclic groups. The
aim of this section is to present a classification of the latter.
Proposition 5.1. Let ϕ : Zn → Zn and ϕ̃ : Zm → Zm be a reciprocal pair of antibalanced
skew morphisms of the cyclic additive groups Zn and Zm, respectively. If both n and m
are odd, then (ϕ, ϕ̃) = (idn, idm) where idk denotes the identity automorphism of Zk,
k = n,m.
Proof. By Theorem 3.9(a), both ϕ and ϕ̃ are involutory automorphisms. The divisibility
condition on reciprocality implies that both ϕ and ϕ̃ are the identity automorphisms.
Theorem 5.2. Let ϕ : Zn → Zn and ϕ̃ : Zm → Zm be a reciprocal pair of antibalanced
skew morphisms of the cyclic additive groups Zn and Zm, respectively. If n is odd and
m is even, then ϕ is an automorphism of the form ϕ(x) ≡ sx (mod n), and ϕ̃ is a skew
morphism of the form
ϕ̃(y) =
{
y, y is even,
y + 2u, y is odd
and π̃(y) =
{
1, y is even,
−1, y is odd
where s ∈ Zn and u ∈ Zm/2 are integers such that
gcd(n, s+ 1) gcd(m/2, u) ≡ 0 (mod m/2) and s2 ≡ 1 (mod n). (5.1)
Proof. By assumption, both ϕ and ϕ̃ are antibalanced. Since n is odd and m is even, by
Theorem 3.9, ϕ is an automorphism of the form ϕ(x) = sx, where s2 ≡ 1 (mod n) and
ϕ̃ is a skew morphism of the form
ϕ̃(y) =
{
ty, y is even,
t(y − 1) + 2u+ 1, y is odd,
for some t, u ∈ Zm/2 satisfying the following conditions:
t2 ≡ 1 (mod m/2) and (u+ 1)(t− 1) ≡ 0 (mod m/2).
14 Art Discrete Appl. Math. 3 (2020) #P1.07
Note that the order of ϕ is equal to the multiplicative order of s in Zn, which is a divisor of
2, and the order of ϕ̃ is the smallest positive integer ` such that
2u
∑̀
i=1
ti−1 ≡ 0 (mod m).
Now we employ the reciprocality to simplify these numerical conditions. By Defini-
tion 4.1(a), |ϕ̃| divides n. Since n is odd, |ϕ̃| is also odd, so by Corollary 3.10, t = 1, and
consequently, ϕ̃ reduces to the stated form and |ϕ̃| = m/ gcd(m, 2u). By Definition 4.1(b),
−1 ≡ π̃(1) ≡ ϕ(1) = s (mod m
gcd(m, 2u)
).
Thus, |ϕ̃| = m/ gcd(m, 2u) is a common divisor of (s + 1) and n. Since m is even,
gcd(m, 2u) = 2 gcd(m/2, u), and we obtain the first condition in (5.1), as required.
By exchanging the roles of ϕ and ϕ̃, and the associated parameters, we obtain all recip-
rocal pairs of antibalanced skew morphisms of Zn and Zm where n is even and m is odd.
The details are left to the reader.
Theorem 5.3. Let ϕ : Zn → Zn and ϕ̃ : Zm → Zm be a reciprocal pair of antibalanced
skew morphisms of the cyclic additive groups Zn and Zm, respectively. If both n and m
are even, then
ϕ(x) =
{
sx, x is even,
s(x− 1) + 2r + 1, x is odd,
π(x) =
{
1, x is even,
−1, x is odd
and
ϕ̃(y) =
{
ty, y is even,
t(y − 1) + 2u+ 1, y is odd,
π̃(y) =
{
1, y is even,
−1, y is odd
where r, s ∈ Zn/2 and u, t ∈ Zm/2 are integers such that
s2 ≡ 1 (mod n/2),
(r + 1)(s− 1) ≡ 0 (mod n/2),
gcd(m/2, u+ 1) gcd(n, r(s+ 1)) ≡ 0 (mod n/2)
(5.2)
and 
t2 ≡ 1 (mod m/2),
(u+ 1)(t− 1) ≡ 0 (mod m/2),
gcd(n/2, r + 1) gcd(m,u(t+ 1)) ≡ 0 (mod m/2).
(5.3)
Proof. By Theorem 3.9(b), the skew morphisms ϕ and ϕ̃ may be represented by the stated
form, where the parameters r, s ∈ Zn/2 and u, t ∈ Zm/2 are integers such that
s2 ≡ 1 (mod n/2), (r + 1)(s− 1) ≡ 0 (mod n/2)
and
t2 ≡ 1 (mod m/2), (u+ 1)(t− 1) ≡ 0 (mod m/2).
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 15
In particular,
|ϕ| = n/ gcd(n, r(s+ 1)) and |ϕ̃| = m/ gcd(m,u(t+ 1)).
We now employ the reciprocality to simplify the numerical conditions. By Defini-
tion 4.1, we have |ϕ| = n/ gcd(n, r(s+ 1)) divides m, |ϕ̃| = m/ gcd(m,u(t+ 1)) divides
n,
−1 ≡ π(1) ≡ ϕ̃(1) ≡ 2u+ 1 (mod n/ gcd(n, r(s+ 1)))
and
−1 ≡ π̃(1) ≡ ϕ(1) ≡ 2r + 1 (mod m/ gcd(m,u(t+ 1))).
Thus, n/ gcd(n, r(s + 1)) divides gcd(m, 2(u + 1)) and m/ gcd(m,u(t + 1)) divides
gcd(n, 2(r + 1)), or equivalently,
gcd(n, r(s+ 1)) gcd(m/2, u+ 1) ≡ 0 (mod n/2)
and
gcd(m,u(t+ 1)) gcd(n/2, r + 1) ≡ 0 (mod m/2),
as required.
ORCID iDs
Kan Hu https://orcid.org/0000-0003-4775-7273
Young Soo Kwon https://orcid.org/0000-0002-1765-0806
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