The existence of unexpected automorphisms in direct product graphs

A pair of graphs $(\Gamma,\Sigma)$ is called unstable if their direct product $\Gamma\times\Sigma$ admits automorphisms not from $\mathrm{Aut}(\Gamma)\times\mathrm{Aut}(\Sigma)$, and such automorphisms are said to be unexpected. The stability of a graph $\Gamma$ refers to that of $(\Gamma,K_2)$. While the stability of individual graphs has been relatively well studied, much less is known for graph pairs. In this paper, we propose a conjecture that provides the best possible reduction of the stability of a graph pair to the stability of a single graph. We prove one direction of this conjecture and establish partial results for the converse. This enables the determination of the stability of a broad class of graph pairs, with complete results when one factor is a cycle.