Regular maps from Cayley graphs, part 1: Balanced Cayley maps

Škoviera, Martin; Širáň‬, Jozef

doi:10.1016/0012-365x(92)90296-r

Cited by 63 publications

(62 citation statements)

References 2 publications

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“…In Section 2, we place the problem in a purely algebraic setting and relate that setting to the viewpoint of skew morphisms and balanced Type I/Type II Cayley maps developed byŠiráň andŠkoviera [21]. We also show that many nonabelian groups, for example all finite nonabelian simple groups, have balanced regular Cayley maps, and that some abelian groups have only unbalanced regular Cayley maps.…”

Section: Introductionmentioning

confidence: 98%

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Regular Cayley maps for finite abelian groups

2006

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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.

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Section: Introductionmentioning

confidence: 98%

“…Following [21], we say that a Cayley map whose cyclic ordering of its generating set X has either of these forms is balanced , of Type I or Type II, respectively. In fact:…”

Section: Introductionmentioning

confidence: 99%

Regular Cayley maps for finite abelian groups

2006

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“…In Proposition 3.6, if the map M is represented as a Cayley map of the base group G, then it is balanced, and so a nice description of the embedding exists in terms of the base group G; see [34]. However, if M is represented as a Cayley map of the base group H, then the embedding does not admit a 'nice' description.…”

Section: Is a Cayley Graph Of A Group G Such Thatĝ ≤ X If And Only Ifmentioning

confidence: 99%

“…A map M is called a Cayley map of a group G if AutM contains the regular subgroupĜ. The study of orientable Cayley maps was initiated by Biggs [1] in 1972, and is a current active topic in topological and algebraic graph theory; see for example [3,8,33,34]. Here we apply Theorem 1.5 to give a characterisation of rotary Cayley maps of simple groups.…”

Section: Dual Cayley Graphs and Rotary Cayley Mapsmentioning

confidence: 99%

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Finite edge-transitive Cayley graphs and rotary Cayley maps

2006

Trans. Amer. Math. Soc.

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Abstract. 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 corefree), 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 nonCayley 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.

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Planar embeddings with infinite faces

Bonnington

Watkins

2003

Journal of Graph Theory

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We investigate vertex-transitive graphs that admit planar embeddings having infinite faces, i.e., faces whose boundary is a double ray. In the case of graphs with connectivity exactly 2, we present examples wherein no face is finite. In particular, the planar embeddings of the Cartesian product of the r-valent tree with K 2 are comprehensively studied and enumerated, as are the automorphisms of the resulting maps, and it is shown for r ¼ 3 that no vertex-transitive group of graph automorphisms is extendable to a group of homeomorphisms of the plane. We present all known families of infinite, locally finite, vertex-transitive graphs ------------------

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Regular maps from Cayley graphs, part 1: Balanced Cayley maps

Cited by 63 publications

References 2 publications

Regular Cayley maps for finite abelian groups

Regular Cayley maps for finite abelian groups

Finite edge-transitive Cayley graphs and rotary Cayley maps

Planar embeddings with infinite faces

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