Stability of graph pairs involving cycles

A graph pair $(\Gamma, \Sigma)$ is called stable if $\aut(\Gamma)\times\aut(\Sigma)$ is isomorphic to $\aut(\Gamma\times\Sigma)$ and unstable otherwise, where $\Gamma\times\Sigma$ is the direct product of $\Gamma$ and $\Sigma$. A graph is called $R$-thin if distinct vertices have different neighbourhoods. $\Gamma$ and $\Sigma$ are said to be coprime if there is no nontrivial graph $\Delta$ such that $\Gamma \cong \Gamma_1 \times \Delta$ and $\Sigma \cong \Sigma_1 \times \Delta$ for some graphs $\Gamma_1$ and $\Sigma_1$. An unstable graph pair $(\Gamma, \Sigma)$ is called nontrivially unstable if $\Gamma$ and $\Sigma$ are $R$-thin connected coprime graphs and at least one of them is non-bipartite. This paper contributes to the study of the stability of graph pairs with a focus on the case when $\Sigma = C_n$ is a cycle. We give two sufficient conditions for $(\Gamma, C_n)$ to be nontrivially unstable, where $n \ne 4$ and $\Gamma$ is an $R$-thin connected graph. In the case when $\Gamma$ is an $R$-thin connected non-bipartite graph, we obtain the following results: (i) if $(\Gamma, K_2)$ is unstable, then $(\Gamma, C_{n})$ is unstable for every even integer $n \geq 4$; (ii) if an even integer $n \ge 6$ is compatible with $\Gamma$ in some sense, then $(\Gamma, C_{n})$ is nontrivially unstable if and only if $(\Gamma, K_2)$ is unstable; (iii) if there is an even integer $n \ge 6$ compatible with $\Gamma$ such that $(\Gamma, C_{n})$ is nontrivially unstable, then $(\Gamma, C_{m})$ is unstable for all even integers $m \ge 6$. We also prove that if $\Gamma$ is an $R$-thin connected graph and $n \ge 3$ is an odd integer compatible with $\Gamma$, then $(\Gamma, C_{n})$ is stable.