The super restricted edge-connectedness of direct product graphs

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. An edge subset $F\subseteq E(G)$ is called a restricted edge-cut if $G-F$ is disconnected and has no isolated vertices. The restricted edge-connectivity $\lambda'(G)$ of $G$ is the cardinality of a minimum restricted edge-cut of $G$ if it has any; otherwise $\lambda'(G)=+\infty$. If $G$ is not a star and its order is at least four, then $\lambda'(G)\leq \xi(G)$, where $\xi(G)=$ min$\{d_G(u) + d_G(v)-2:\ uv \in E(G)\}$. The graph $G$ is said to be maximally restricted edge-connected if $\lambda'(G)= \xi(G)$; the graph $G$ is said to be super restricted edge-connected if every minimum restricted edge-cut isolates an edge from $G$. The direct product of graphs $G_1$ and $G_2$, denoted by $G_1\times G_2$, is the graph with vertex set $V(G_1\times G_2) = V(G_1)\times V(G_2)$, where two vertices $(u_{1} ,v_{1} )$ and $(u_{2} ,v_{2} )$ are adjacent in $G_1\times G_2$ if and only if $u_{1}u_{2} \in E(G_1)$ and $v_{1}v_{2} \in E(G_2)$. In this paper, we give a sufficient condition for $G\times K_{n}$ to be super restricted edge-connected, where $K_{n}$ is the complete graph on $n$ vertices.