$Z_{3}$-Connectivity of Wreath Product of Graphs

Abstract

Let $G$ and $G'$ be two simple connected nontrivial graphs. Denote by $G\Box G^{'}$ and $G\rho G^{'}$ the Cartesian product and the Wreath product of $G$ and $G^{'}$, respectively. In this paper, we will show two results: (1) if $G^{'}$ is a triangularly connected simple graph with $|V(G^{'})|\ge 3$ which is neither $K_{2,t}^{+}$ nor one specific graph, then $G\Box G^{'}$ is $Z_{3}$-connected; (2) if $G^{'}$ is a vertex-transitive graph, then $G\rho G^{'}$ is $Z_{3}$-connected.