Regular Cayley Maps Research Articles

Regular Cayley maps of elementary abelian p-groups: Classification and enumeration

Recently, regular Cayley maps of cyclic groups and dihedral groups have been classified in [7] and [20], respectively. A nature question is to classify regular Cayley maps of elementary abelian p-groups Zpn. In this paper, a complete classification of regular Cayley maps of Zpn is given and moreover, the number of these maps and their genera are enumerated.

Skew product groups for monolithic groups

Skew morphisms, which generalise automorphisms for groups, provide a fundamental tool for the study of regular Cayley maps and, more generally, for finite groups with a complementary factorisation $G=BY$, where $Y$ is cyclic and core-free in $G$. In this paper, we classify all examples in which $B$ is monolithic (meaning that it has a unique minimal normal subgroup, and that subgroup is not abelian) and core-free in $G$. As a consequence, we obtain a classification of all proper skew morphisms of finite non-abelian simple groups.

Skew-morphisms of nonabelian characteristically simple groups

A skew-morphism of a finite group G is a permutation σ on G fixing the identity element such that the product of 〈σ〉 with the left regular representation of G forms a permutation group on G. This permutation group is called the skew-product group of σ. The skew-morphism was introduced as an algebraic tool to investigate regular Cayley maps. In this paper, we characterize skew-products of skew-morphisms of finite nonabelian characteristically simple groups (see Theorem 1.2) and the corresponding regular Cayley maps (see Theorem 1.6).

Automorphism groups of Cayley graphs generated by block transpositions and regular Cayley maps

This paper deals with the Cayley graph Cay(Symn,Tn), where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that Aut(Cay(Symn,Tn)) is the product of the left translation group and a dihedral group Dn+1 of order 2(n+1). The proof uses several properties of the subgraph Γ of Cay(Symn,Tn) induced by the set Tn. In particular, Γ is a 2(n−2)-regular graph whose automorphism group is Dn+1,Γ has as many as n+1 maximal cliques of size 2, and its subgraph Γ(V) whose vertices are those in these cliques is a 3-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique cyclic subgroup of Dn+1 of order n+1 with regular Cayley maps on Symn is also discussed. It is shown that the product of the left translation group and the latter group can be obtained as the automorphism group of a non-t-balanced regular Cayley map on Symn.

On the skew-morphisms of dihedral groups

Abstract A skew-morphism φ of a finite group G is a permutation on G such that φ ⁢ ( 1 ) = 1 ${\varphi(1)\hskip-0.853583pt=\hskip-0.853583pt1}$ and φ ⁢ ( g ⁢ h ) = φ ⁢ ( g ) ⁢ φ π ⁢ ( g ) ⁢ ( h ) ${\varphi(gh)=\varphi(g)\varphi^{\pi(g)}(h)}$ for all g , h ∈ G ${g,h\in G}$ , where π is a function from G to 𝐙 | φ | ${\mathbf{Z}_{|\varphi|}}$ , called the power function of φ. Furthermore, we say φ is of skew-type k, provided π takes on exactly k values in 𝐙 | φ | ${\mathbf{Z}_{|\varphi|}}$ , and call the set Ker ⁡ φ = { g ∈ G ∣ π ⁢ ( g ) = 1 } ${\operatorname{Ker}\varphi=\{g\in G\mid\pi(g)=1\}}$ the kernel of φ, which is actually a subgroup of G (see [6]). Though skew-morphism is a pure group theoretical concept, it is closely related to regular Cayley maps. We have a long term goal to classify all regular Cayley maps for dihedral groups. In this paper, we mainly work on the skew-morphisms of dihedral groups. Let φ be a skew-morphism of the dihedral groups D 2 ⁢ n = 〈 a , b ∣ a n = b 2 = ( a ⁢ b ) 2 = 1 〉 ${D_{2n}=\langle a,b\mid a^{n}=b^{2}=(ab)^{2}=1\rangle}$ . We prove that the kernel of φ is contained in the cyclic subgroup 〈 a 〉 ${\langle a\rangle}$ if and only if φ is of skew-type 4 and preserves 〈 a 〉 ${\langle a\rangle}$ . In the case of φ preserving 〈 a 〉 ${\langle a\rangle}$ , we give some characterizations of φ, in particular, we prove that φ is of skew-type 1, 2 or 4. Working from these characterizations, an infinite family of skew-morphisms of D 2 ⁢ n ${D_{2n}}$ are constructed, all of which are of skew-type 4 and have kernel 〈 a 2 〉 ${\langle a^{2}\rangle}$ , which gives a positive answer to an open problem raised recently by M. Conder [1].

Regular Cayley maps on dihedral groups with the smallest kernel

Let $$\mathcal {M}={{\mathrm{CM}}}(D_n,X,p)$$M=CM(Dn,X,p) be a regular Cayley map on the dihedral group $$D_n$$Dn of order $$2n, n \ge 2,$$2n,nź2, and let $$\psi $$ź be the skew-morphism associated with $$\mathcal {M}$$M. In this paper it is shown that the kernel $${{\mathrm{Ker}}}\psi $$Kerź of the skew-morphism $$\psi $$ź is a dihedral subgroup of $$D_n$$Dn and if $$n \ne 3,$$nź3, then the kernel $${{\mathrm{Ker}}}\psi $$Kerź is of order at least 4. Moreover, all $$\mathcal {M}$$M are classified for which $${{\mathrm{Ker}}}\psi $$Kerź is of order 4. In particular, besides four sporadic maps on 4, 4, 8 and 12 vertices, respectively, two infinite families of non-t-balanced Cayley maps on $$D_n$$Dn are obtained.

Cyclic complements and skew morphisms of groups

A skew morphism of a group is a generalisation of an automorphism, arising in the context of regular Cayley maps or of groups expressible as a product AB of subgroups A and B with B cyclic and A∩B={1}. A skew morphism of a group A is a bijection φ:A→A fixing the identity element of A and having the property that φ(xy)=φ(x)φπ(x)(y) for all x,y∈A, where π(x) depends only on x. The kernel of φ is the subgroup of all x∈A for which π(x)=1.In this paper, we present a number of previously unknown properties of skew morphisms, one being that if A is any finite group, then the order of every skew morphism of A is less than |A|, and another being that the kernel of every skew morphism of a non-trivial finite group is non-trivial. We also prove a number of theorems about skew morphisms of finite abelian groups, some of which simplify or extend recent theorems of Kovács and Nedela [13]. For example, we determine all skew morphisms of the finite abelian groups whose order is prime, or the square of a prime, or the product of two distinct primes. In addition, we completely determine the finite abelian groups for which every skew morphism is an automorphism.

A classification of regular Cayley maps with trivial Cayley-core for dihedral groups

Let M=CM(G,X,p) be a regular Cayley map for the finite group G, and let Aut+(M) be the orientation-preserving automorphism group of M. Then G can be regarded as a subgroup of Aut+(M) in the sense that G acts on itself by left multiplication. The core of G in Aut+(M) is called the Cayley-core of M. In this paper, the regular Cayley maps with trivial Cayley-core for dihedral groups are classified. This work is a partial result for our long term goal to classify all regular Cayley maps for dihedral groups.

Regular Cayley maps of skew-type 3 for dihedral groups

Let M=CM(G,X,p) be a regular Cayley map, and φ be the corresponding skew-morphism of M. If the power function of φ takes on exactly i values in Z|X|, then M is called of skew-type i. In this paper, all regular Cayley maps of skew-type 3 for dihedral groups are determined up to isomorphism.

Regular Cayley maps of skew-type 3 for abelian groups

Let M=CM(G,X,p) be a regular Cayley map, and let φ be the corresponding skew-morphism. If the power function of φ assumes exactly i values in Z|X|, then M is called of skew-type i. Note that regular balanced Cayley maps are of skew-type 1, and regular t-balanced (including anti-balanced) Cayley maps are of skew-type 2. Those two classes of maps have been studied by many researchers. In this paper, some results on regular Cayley maps of skew-type 3 for abelian groups are given. As applications, many new regular Cayley maps for abelian groups are constructed.

Decomposition of skew-morphisms of cyclic groups

A skew-morphism of a group H is a permutation σ of its elements fixing the identity such that for every x , y ∈ H there exists an integer k such that σ ( xy )= σ ( x ) σ k ( y ). It follows that group automorphisms are particular skew-morphisms. Skew-morphisms appear naturally in investigations of maps on surfaces with high degree of symmetry, namely, they are closely related to regular Cayley maps and to regular embeddings of the complete bipartite graphs. The aim of this paper is to investigate skew-morphisms of cyclic groups in the context of the associated Schur rings. We prove the following decomposition theorem about skew-morphisms of cyclic groups Z n : if n = n 1 n 2 such that ( n 1 , n 2 ) = 1, and ( n 1 , φ ( n 2 )) = ( φ ( n 1 ), n 2 ) = 1 ( φ denotes Euler's function) then all skew-morphisms σ of Z n are obtained as σ = σ 1 × σ 2 , where σ i are skew-morphisms of Z n_i , i = 1, 2. As a consequence we obtain the following result: All skew-morphisms of Z n are automorphisms of Z n if and only if n = 4 or ( n , φ ( n )) = 1.

Regular [formula omitted]-balanced Cayley maps for abelian groups

The inherent high symmetry of Cayley maps makes them an excellent source of orientably regular maps, and the regularity of a Cayley map has been shown to be equivalent to the existence of a skew-morphism of its underlying group that has a generating orbit closed under inverses. We set to investigate the properties of the so-called t-balanced skew-morphisms of abelian groups with the aim of providing the basis for a complete classification of t-balanced regular Cayley maps of abelian groups. In the case of cyclic groups, we show that the only t-balanced regular Cayley maps for the groups Z2r, Z2pr and Z4pr, p an odd prime, r≥1, are the well understood balanced and antibalanced Cayley maps.

A CLASSIFICATION OF PRIME-VALENT REGULAR CAYLEY MAPS ON ABELIAN, DIHEDRAL AND DICYCLIC GROUPS

A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface with the same local orientation induced by a cyclic permutation of generators at each vertex. In this paper, we provide classifications of prime-valent regular Cayley maps on abelian groups, dihedral groups and dicyclic groups. Consequently, we show that all prime-valent regular Cayley maps on dihedral groups are balanced and all prime-valent regular Cayley maps on abelian groups are either balanced or anti-balanced. Furthermore, we prove that there is no prime-valent regular Cayley map on any dicyclic group.

Reflexibility of regular Cayley maps for abelian groups

In this paper, properties of reflexible Cayley maps for abelian groups are investigated, and as a result, it is shown that a regular Cayley map of valency greater than 2 for a cyclic group is reflexible if and only if it is anti-balanced.

Regular Cayley maps for finite abelian groups

A regular Cayley map for a finite group A is an orientable map whose orientation-preserving automorphism group G acts regularly on the directed edge set and has a subgroup isomorphic to A that acts regularly on the vertex set. This paper considers the problem of determining which abelian groups have regular Cayley maps. The analysis is purely algebraic, involving the structure of the canonical form for A. The case when A is normal in G involves the relationship between the rank of A and the exponent of the automorphism group of A, and the general case uses Ito's theorem to analyze the factorization G = AY, where Y is the (cyclic) stabilizer of a vertex.

Regular t-balanced Cayley maps

The concept of a t-balanced Cayley map is a natural generalization of the previously studied notions of balanced and anti-balanced Cayley maps (the terms coined by [J. Širáň, M. Škoviera, Groups with sign structure and their antiautomorphisms, Discrete Math. 108 (1992) 189–202. [12]]). We develop a general theory of t-balanced Cayley maps based on the use of skew-morphisms of groups [R. Jajcay, J. Širáň, Skew-morphisms of regular Cayley maps, Discrete Math. 244 (1–3) (2002) 167–179], and apply our results to the specific case of regular Cayley maps of abelian groups.

Finite edge-transitive Cayley graphs and rotary Cayley maps

This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and core-free), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various combinatorial objects, such as half-transitive graphs, (orientable and non-orientable) regular Cayley maps, vertex-transitive non-Cayley graphs, and permutation groups containing certain regular subgroups. In particular, a characterisation is given of locally primitive holomorph Cayley graphs, and a classification is given of rotary Cayley maps of simple groups. Also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup.

Skew-morphisms of regular Cayley maps

A Cayley map M is a 2-cell embedding of a Cayley graph in an orientable surface with the same orientation (the induced permutation of generators) at each vertex. The concept of a skew-morphism generalizes several concepts previously studied with respect to regular Cayley maps, and allows for a unified theory of regular Cayley maps and their automorphism groups. Using algebraic properties of skew-morphisms of groups we reprove or extend some previously known results and obtain several new ones.

Characterization and construction of Cayley graphs admitting regular Cayley maps

The paper characterizes Cayley graphs that underlie Cayley maps of the highest possible level of symmetry, the regular Cayley maps, in terms of their automorphism groups. Using the introduced characterization, we obtain some useful necessary conditions, as well as a constructive way of finding presentations of all Cayley graphs admitting regular Cayley maps.