A graph pair \((\Gamma , \Sigma )\) is called stable if \(\textrm{Aut}\,(\Gamma )\times \textrm{Aut}\,(\Sigma )\) is isomorphic to \(\textrm{Aut}\,(\Gamma \times \Sigma )\) 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 introduce two key concepts in our study, namely the compatibility of n with \(\Gamma \) and an auxiliary graph \(\Gamma ^*\) on the same vertex set as \(\Gamma \) . We prove that for an R-thin connected graph \(\Gamma \) and an integer \(n \ge 3\) with \(n \ne 4\) such that at least one of \(\Gamma \) and \(C_n\) is non-bipartite, if n is compatible with \(\Gamma \) , or \(n \ge 5\) is odd and for every edge \(\{u, v\}\) of \(\Gamma ^*\) the set of common neighbours of u and v in \(\Gamma \) is not an independent set of \(\Gamma \) , then \((\Gamma , C_n)\) is nontrivially unstable if and only if at least one \(C_n\) -automorphism of \(\Gamma \) is nondiagonal. In the case when \(\Gamma \) is an R-thin connected non-bipartite graph, we obtain the following results: (i) \((\Gamma , K_2)\) is unstable if and only if \((\Gamma , C_{n})\) is unstable for every even integer \(n \ge 4\) ; (ii) if an even integer \(n \ge 6\) is compatible with \(\Gamma \) , 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 for an R-thin connected graph \(\Gamma \) and an odd integer \(n \ge 3\) , if n is compatible with \(\Gamma \) , or \(n \ge 5\) and for every edge \(\{u, v\}\) of \(\Gamma ^*\) the set of common neighbours of u and v in \(\Gamma \) is not an independent set of \(\Gamma \) , then \((\Gamma , C_{n})\) is stable. Three conjectures arisen from our study are proposed in this paper.
