Connectivity of Kronecker products by K2

Abstract: Let $\kappa(G)$ be the connectivity of $G$. The Kronecker product $G_1\times G_2$ of graphs $G_1$ and $G_2$ has vertex set $V(G_1\times G_2)=V(G_1)\times V(G_2)$ and edge set $E(G_1\times G_2)={(u_1,v_1)(u_2,v_2):u_1u_2\in E(G_1),v_1v_2\in E(G_2)}$. In this paper, we prove that $\kappa(G\times K_2)=\textup{min}{2\kappa(G), \textup{min}{|X|+2|Y|}}$, where the second minimum is taken over all disjoint sets $X,Y\subseteq V(G)$ satisfying (1)$G-(X\cup Y)$ has a bipartite component $C$, and (2) $G[V(C)\cup {x}]$ is also bipartite for each $x\in X$.