Isomorphisms of Finite Cayley Digraphs of Bounded Valency, II

For a finite group G and a subset S of G which does not contain the identity of G, denote by Cay(G, S) the Cayley digraph of G with respect to S. An automorphism σ of the group G induces a graph isomorphism from Cay(G, S) to Cay(G, Sσ). In this paper, we investigate groups G and Cayley digraphs Cay(G, S) of G for which the following condition holds: for any T⊂G, Cay(G, S)≅Cay(G, T) if and only if Sσ=T for some σ∈Aut(G). For a positive integer m, a group G is called an m-DCI-group if the condition holds for all Cayley digraphs of valency at most m; while G is called a connected m-DCI-group if it holds for all connected digraphs of valency at most m. This paper contributes towards a complete classification of finite m-DCI-groups for m⩾2. It was previously proved by C. H. Li et al. (1998, J. Combin. Theory Ser. B74, 164–183) that finite m-DCI-groups for m⩾2 belong to an explicitly determined list DCI(m) of groups. However, it is still an open problem to determine which members of DCI(m) are really m-DCI-groups. We reduce this problem to the problem of determining whether all subgroups of groups in DCI(m) are connected m-DCI-groups. Then we give a complete classification of finite 2-DCI-groups.