2023/03/15 by Joy Morris, Morris, Joy
Engineering · Mathematics · #05C25 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2303.08742
openalex publication_date 2023/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Cayley graph Cay(G;S) has the CI (Cayley Isomorphism) property if for every isomorphic graph Cay(G;T), there is a group automorphism α of G such that Sα=T. The DCI (Directed Cayley Isomorphism) property is defined analogously on digraphs. A group G is a CI-group if every Cayley graph on G has the CI property, and is a DCI-group if every Cayley digraph on G has the DCI property. Since a graph is a special type of digraph, this means that every DCI-group is a CI-group, and if a group is not a CI-group then it is not a DCI-group, but there are well-known examples of groups that are CI-groups but not DCI-groups. In 2009, Spiga showed that \mathbb Z38 is not a DCI-group, by producing a digraph that does not have the DCI property. He also showed that \mathbb Z35 is a DCI-group (and therefore also a CI-group). Until recently the question of whether there are elementary abelian 3-groups that are not CI-groups remained open. In a recent preprint with Dave Witte Morris, we showed that \mathbb Z310 is not a CI-group. In this paper we show that with slight modifications, the underlying undirected graph of order 38 described by Spiga is does not have the CI property, so \mathbb Z38 is not a CI-group.