Stability of nontrivial graph pairs

Abstract

A graph pair $(Γ, Σ)$ is called stable if every automorphism of the direct product $Γ\timesΣ$ is induced componentwise by automorphisms of $Γ$ and $Σ$. A graph is twin-free if no two distinct vertices share the same neighbourhood in the graph. Two graphs $Γ$ and $Σ$ are coprime with respect to the direct product if there is no graph $Δ$ of order greater than $1$ such that $Γ\congΓ'\timesΔ$ and $Σ\congΣ'\timesΔ$ for some graphs $Γ'$ and $Σ'$. A graph pair $(Γ,Σ)$ is nontrivial if $Γ$ and $Σ$ are coprime connected twin-free graphs and exactly one of them is bipartite. In this paper, we prove that if $Γ$ is non-bipartite, stable, and factor-loopless, then each nontrivial graph pair $(Γ,Σ)$ is stable. This gives a partial answer to [Question~19, Qin, Xia and Zhou, Discrete Math., 113856, (2024)] and proves the factor-loopless case of [Conjecture~1.3, Wang, Qin and Xia, arXiv:2509.26170]. We also give affirmative answers to [Questions~3.5, 3.6, Gan, Liu and Xia, J. Combin. Theory Ser. B, 140--164, (2025)] and a negative answer to [Question~3.7, Gan, Liu and Xia, J. Combin. Theory Ser. B, 140--164, (2025)].