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 $m$CI- and $m$PCI-digraphs, and correspondingly, $m$DCI- and $m$PDCI-groups. Analogues to Babai's criterion for CI-digraphs are given for $m$CI- and $m$PCI-digraphs, respectively. With these we then classify finite $m$DCI-groups for each $m\geq 2$, and finite $m$PDCI-groups for each $m\geq 4$. 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.