On isomorphisms of m-Cayley digraphs

The isomorphism problem for digraphs is a fundamental problem in graph theory. This problem for Cayley digraphs has been extensively investigated over the last half a century. In this paper, we consider this problem for m-Cayley digraphs which are generalization of Cayley digraphs. Let m be a positive integer. A digraph admitting a group G of automorphisms acting semiregularly on the vertices with exactly m orbits is called an m-Cayley digraph of G. In particular, 1-Cayley digraph is just the Cayley digraph. We first characterize the normalizer of G in the full automorphism group of an m-Cayley digraph of a finite group G. This generalizes a similar result for Cayley digraph achieved by Godsil in 1981. Then we use this to study the isomorphisms of m-Cayley digraphs. The CI-property of a Cayley digraph (CI stands for `Cayley isomorphism') and the DCI-groups (whose Cayley digraphs are all CI-digraphs) are two key topics in the study of isomorphisms of Cayley digraphs. We generalize these concepts into m-Cayley digraphs by defining mCI- and mPCI-digraphs, and correspondingly, mDCI- and mPDCI-groups. Analogues to Babai's criterion for CI-digraphs are given for mCI- and mPCI-digraphs, respectively. With these we then classify finite mDCI-groups for each m 2, and finite mPDCI-groups for each m 4. Similar results are also obtained for m-Cayley graphs. Note that 1DCI-groups are just DCI-groups, and the classification of finite DCI-groups is a long-standing open problem that has been worked on a lot.