The Cayley isomorphism property for Cayley maps

Mikhail Muzychuk, Gábor Somlai

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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.

Original languageEnglish
Article number#P1.42
JournalElectronic Journal of Combinatorics
Issue number1
StatePublished - 2 Mar 2018


  • CI property
  • Cayley maps

ASJC Scopus subject areas

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


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