The Cayley isomorphism property for Cayley maps

Muzychuk, M. and Somlai, Gábor (2018) The Cayley isomorphism property for Cayley maps. ELECTRONIC JOURNAL OF COMBINATORICS, 25 (1). ISSN 1097-1440

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Abstract

The Cayley Isomorphism property for combinatorial objects was introduced by L. Babai in 1977. Since then it has been intensively studied for binary relational structures: graphs, digraphs, colored graphs etc. In this paper we study this prop- erty for oriented Cayley maps. A Cayley map is a Cayley graph provided by a cyclic rotation of its connection set. If the underlying graph is connected, then the map is an embedding of a Cayley graph into an oriented surface with the same cyclic rotation around every vertex. Two Cayley maps are called Cayley isomorphic if there exists a map isomorphism between them which is a group isomorphism too. We say that a finite group H is a CIM-group1 if any two Cayley maps over H are isomorphic if and only if they are Cayley isomorphic. The paper contains two main results regarding CIM-groups. The first one provides necessary conditons for being a CIM-group. It shows that a CIM-group should be one of the following ℤm × ℤr 2, ℤm × ℤ4, ℤm × ℤ8, ℤm × Q8, ℤm ⋊ ℤ2e, e = 1, 2, 3, where m is an odd square-free number and r a non-negative integer2. Our second main result shows that the groups ℤm × ℤr 2, ℤm × ℤ4, ℤm × Q8 contained in the above list are indeed CIM-groups. © 2018, Australian National University. All rights reserved.

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