This is the fourth one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups of minus type.
We start up the study of the stability of general graph pairs. This notion is a generalization of the concept of the stability of graphs. We say that a pair of graphs $(Gamma,Sigma)$ is stable if $Aut(GammatimesSigma) cong Aut(Gamma)times Aut(Sigma)$
and unstable otherwise, where $GammatimesSigma$ is the direct product of $Gamma$ and $Sigma$. An unstable graph pair $(Gamma,Sigma)$ is said to be a nontrivially unstable graph pair if $Gamma$ and $Sigma$ are connected coprime graphs, at least one of them is non-bipartite, and each of them has the property that different vertices have distinct neighbourhoods. We obtain necessary conditions for a pair of graphs to be stable. We also give a characterization of a pair of graphs $(Gamma, Sigma)$ to be nontrivially unstable in the case when both graphs are connected and regular with coprime valencies and $Sigma$ is vertex-transitive. This characterization is given in terms of the $Sigma$-automorphisms of $Gamma$, which are a new concept introduced in this paper as a generalization of both automorphisms and two-fold automorphisms of a graph.
