Unexpected automorphisms in direct product graphs

A pair of graphs $(\Gamma ,\Sigma )$ is called unstable if their direct product $\Gamma \times \Sigma$ has automorphisms that do not come from $\text{Aut}\left(\Gamma \right)\times \text{Aut}\left(\Sigma \right)$, and such automorphisms are said to be unexpected. In the special case when $\Sigma =K_2$, the stability of $(\Gamma ,K_2)$ is well studied in the literature, where the so-called two-fold automorphisms of the graph Γ have played an important role. As a generalization of two-fold automorphisms, the concept of non-diagonal automorphisms was recently introduced to study the stability of general graph pairs. In this paper, we obtain, for a large family of graph pairs, a necessary and sufficient condition to be unstable in terms of the existence of non-diagonal automorphisms. As a byproduct, we determine the stability of graph pairs involving complete graphs or odd cycles, respectively. The former result in fact solves a problem proposed by Dobson, Miklavič and Šparl for undirected graphs, as well as confirms a recent conjecture of Qin, Xia and Zhou.