On the automorphism group of direct product of digraphs

Abstract

Determining the conditions under which the direct product of graphs $G$ and $H$ satisfies $\mathrm{Aut}(G\times H)=\mathrm{Aut}(G)\times\mathrm{Aut}(H)$ has been a problem of considerable interest since Sabidussi's classic work in the 1950s. We call such a pair $(G,H)$ stable, and unstable otherwise. Although much progress has been made for graph pairs, the general digraph case has remained completely open. In this paper, we initiate the study of the stability of digraph pairs, and then focus on the stability of a single digraph $G$. This is defined as the stability of the pair $(G,K_2)$ and has been studied extensively when $G$ is undirected. We establish a necessary and sufficient condition for a connected digraph to be unstable, and use it to derive four sufficient conditions for circulant digraphs to be unstable. Moreover, we prove the nonexistence of nontrivially unstable finite arc-transitive circulant digraphs and nontrivially unstable Cayley digraphs of abelian groups of odd order.