## 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 ℤ_{m} × ℤ^{r} _{2}, ℤ_{m} × ℤ_{4}, ℤ_{m} × ℤ_{8}, ℤ_{m} × Q_{8}, ℤ_{m} ⋊ ℤ_{2e}, e = 1, 2, 3, where m is an odd square-free number and r a non-negative integer^{2}. Our second main result shows that the groups ℤ_{m} × ℤ^{r} _{2}, ℤ_{m} × ℤ_{4}, ℤ_{m} × Q_{8} contained in the above list are indeed CIM-groups.

Original language | English |
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Article number | #P1.42 |

Journal | Electronic Journal of Combinatorics |

Volume | 25 |

Issue number | 1 |

DOIs | |

State | Published - 2 Mar 2018 |

## Keywords

- CI property
- Cayley maps

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics