On finite groups with the Cayley isomorphism property. II (Q1806211)

A group \(G\) has the \(m\)-DCI property if any graph theoretic isomorphism between two Cayley graphs for \(G\) with \(m\) generators arises from a conjugacy between the generating sets, see Part I by \textit{C. H. Li, C. E. Praeger}, and \textit{M. Y. Xu} [J. Graph Theory 27, No. 1, 21-31 (1998; Zbl 0889.05053)]. In the present paper the author shows that, for infinitely many values of \(m\), there are groups with the \(m\)-DCI property but lack the \(k\)-DCI property for all \(k < m\). A conjecture [cf. \textit{C. H. Li, C. E. Praeger}, and \textit{M. Y. Xu}, J. Comb. Theory, Ser. B 73, No. 2, 164-183 (1998; Zbl 0904.05067)] classifying the groups with the \(m\)-DCI property is verified for \(m=4\).