On Finite Groups with the Cayley Isomorphism Property, II

For a positive integer m, a group G is said to have the m-DCIproperty if, for any Cayley digraphs Cay(G, S) and Cay(G, T) of G of valency m (that is, |S|=|T|=m), Cay(G, S)≅Cay(G, T) if and only if Sσ=T for some σ∈Aut(G). This paper is one of a series of papers towards characterizing finite groups with the m-DCI property. It is shown that, for infinitely many values of m, there exist Frobenius groups with the m-DCI property but not with the k-DCI property for any k<m. Further, it is conjectured that for relative small values of m, these groups and an explicit list of groups given by C. H. Li, C. E. Praeger, and M. Y. Xu (1998, J. Combin. Theory Ser. B73, 164–183) contain all finite groups with the m-DCI property. This conjecture is verified for the case m⩽4.