Skew-morphisms of regular Cayley maps (original) (raw)
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Discrete Mathematics, 2007
The class of t-balanced Cayley maps [J. Martino, M. Schultz, Symmetrical Cayley maps with solvable automorphism groups, abstract in SIGMAC '98, Flagstaff, AR, 1998] is a natural generalisation of balanced and antibalanced Cayley maps introduced and studied by Širáň and Škoviera [Regular maps from Cayley graphs II: antibalanced Cayley maps, Discrete Math. 124 (1994) 179-191; Groups with sign structure and their antiautomorphisms, Discrete Math. 108 (1992) 189-202]. The present paper continues this study by investigating the distribution of inverses, automorphism groups, and exponents of t-balanced Cayley maps. The methods are based on the use of t-automorphisms of groups with sign structure which extend the notion of an antiautomorphism crucial for antibalanced Cayley maps. As an application, a new series of nonstandard regular embeddings of complete bipartite graphs K n,n is constructed for each n divisible by 8.
Journal of Combinatorial Theory, 2005
We present a theory of Cayley maps, i.e., embeddings of Cayley graphs into oriented surfaces having the same cyclic rotation of generators around each vertex. These maps have often been used to encode symmetric embeddings of graphs. We also present an algebraic theory of Cayley maps and we apply the theory to determine exactly which regular or edge-transitive tilings of the sphere or plane are Cayley maps or Cayley graphs. Our main goal, however, is to provide the general theory so as to make it easier for others to study Cayley maps.
Automorphisms and Enumeration of Maps of Cayley Graphs of a Finite Group
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A map is a connected topological graph Γ cellularly embedded in a surface. In this paper, applying Tutte's algebraic representation of map, new ideas for enumerating non-equivalent orientable or non-orientable maps of graph are presented. By determining automorphisms of maps of Cayley graph Γ = Cay(G : S) with AutΓ ∼ = G × H on locally, orientable and non-orientable surfaces, formulae for the number of non-equivalent maps of Γ on surfaces (orientable, non-orientable or locally orientable) are obtained . Meanwhile, using reseults on GRR graph for finite groups, we enumerate the non-equivalent maps of GRR graph of symmetric groups, groups generated by 3 involutions and abelian groups on orientable or non-orientable surfaces.
Automorphisms and Enumeration of Maps of Cayley Graph of a Finite Group
2006
A map is a connected topological graph Gamma\GammaGamma cellularly embedded in a surface. In this paper, applying Tutte's algebraic representation of map, new ideas for enumerating non-equivalent orientable or non-orientable maps of graph are presented. By determining automorphisms of maps of Cayley graph Gamma=rmCay(G:S)\Gamma={\rm Cay}(G:S)Gamma=rmCay(G:S) with rmAutGammacongGtimesH{\rm Aut} \Gamma\cong G\times HrmAutGammacongGtimesH on locally, orientable and non-orientable surfaces, formulae for the number of non-equivalent maps of Gamma\GammaGamma on surfaces (orientable, non-orientable or locally orientable) are obtained . Meanwhile, using reseults on GRR graph for finite groups, we enumerate the non-equivalent maps of GRR graph of symmetric groups, groups generated by 3 involutions and abelian groups on orientable or non-orientable surfaces.
On automorphism groups of Cayley graphs
Periodica Mathematica Hungarica, 1976
It is well-knot~n tha~ its automorphism group A(X o H) must contain the regular subgroup L G corresponding to the set of left multiplication~ by elements of G. This paper is concerned with minimizing the index [A(Xo, t/):L a] for given G, in particular when this index is always greater than 1. If G is a.beli~n but not one of seven exceptional groups, then a Cayley graph of G exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.
Decomposition of skew-morphisms of cyclic groups
Ars Mathematica Contemporanea, 2011
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 ni , 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.
Endomorphisms of Cayley digraphs of rectangular groups
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Let \(\mathrm{Cay}(S,A)\) denote the Cayley digraph of the semigroup \(S\) with respect to the set \(A\), where \(A\) is any subset of \(S\). The function \(f : \mathrm{Cay}(S,A) \to \mathrm{Cay}(S,A)\) is called an endomorphism of \(\mathrm{Cay}(S,A)\) if for each \((x,y) \in E(\mathrm{Cay}(S,A))\) implies \((f(x),f(y)) \in E(\mathrm{Cay}(S,A))\) as well, where \(E(\mathrm{Cay}(S,A))\) is an arc set of \(\mathrm{Cay}(S,A)\). We characterize the endomorphisms of Cayley digraphs of rectangular groups \(G\times L\times R\), where the connection sets are in the form of \(A=K\times P\times T\).
Skew endomorphisms on some n-ary groups
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We characterize n-ary groups dened on a cyclic group and describe a group of their automorphisms induced by the skew operation. Finally, we consider splitting automorphisms.