The cyclic groups with the \(m\)-DCI property (Q1367594)

Let \(G\) be a finite group and \(S\) a subset of \(G^\#=G-\{1\}\). The Cayley graph \(\text{Cay}(G,S)\) of \(G\) with respect to \(S\) has as vertices the elements of \(G\) and as edges the pairs \((g,sg)\) where \(g\in G\) and \(s\in S\). If \(\sigma\in\text{Aut}(G)\), then \(\sigma\) induces an isomorphism from \(\text{Cay}(G,S)\) to \(\text{Cay}(G,S^\sigma)\). A Cayley graph \(\text{Cay}(G,S)\) is called a CI-graph if for any subset \(T\) of \(G^\#\), \(\text{Cay}(G,S)\cong\text{Cay}(G,T)\) implies \(S^\sigma=T\) for some \(\sigma\in\text{Aut}(G)\). If all Cayley graphs of \(G\) of valence \(m\) are CI-graphs then \(G\) is said to have the \(m\)-DCI property. The aim of this paper is to give a reasonably complete classification of cyclic groups with the \(m\)-DCI property. The main theorems proved are the following: Theorem 1: Let \(G\) be a cyclic group of order \(p^2\), where \(p\) is an odd prime, and let \(m\) be a positive integer with \(1\leq m\leq (p^2-1)/2\). Then \(G\) has the \(m\)-DCI property iff either \(m<p\), or \(m\equiv 0\) or 1 (mod \(p\)). Theorem 2: Let \(G\) be a cyclic group, and let \(p\) be a prime divisor of \(|G|\) and \(G_p\), the Sylow \(p\)-subgroup of \(G\). Suppose that \(G\) has the \(m\)-DCI property, where \(p+1\leq m\leq(|G|-1)/2\). Then one of the following holds: (i) \(G=Z_{p^2}\) and \(m\equiv 0\) or 1 (mod p), (ii) \(p\) is odd and \(G_p=Z_p\), (iii) \(p=2\) and \(G_2=Z_2\) or \(Z_4\). The author conjectures that the converse of theorem 2 is also true. If the conjecture is true, then these theorems together with other known results would provide a complete classification of cyclic groups with the \(m\)-DCI property.