On Isomorphisms of Connected Cayley Graphs, II

A Cayley graph or digraph Cay(G, S) of a finite groupGis called aCI-graphofGif, for anyT⊂G, Cay(G, S)≅Cay(G, T) if and only ifSσ=Tfor someσ∈Aut(G). We study the problem of determining which Cayley graphs and digraphs for a given group are CI-graphs. A finite groupGis called aconnected m-DCI-group(orconnected m-CI-group) if all connected Cayley digraphs (or connected Cayley graphs, respectively) ofGof (out)-valency at most m are CI-graphs. For a groupG, letp(G) be the smallest prime divisor of |G|. It was previously shown that all finite groupsGare connected (p(G)−1)-DCI-groups and all finite groupsGare connected 2(p(G)−1)-CI-groups. In this paper, for every primep, we construct infinitely many finite groupsGsuch thatp(G)=pandGis neither a connectedp-DCI-group nor a connected 2p-CI-group, which provides solutions for several open problems in this area.