The 3-restricted Edge-Connectivity of Strong Product Graphs

An edge subset $S \subseteq E(G)$ is called a 3-restricted edge-cut if $G-S$ is disconnected and each component of $G - S$ contains at least three vertices. The 3-restricted edge-connectivity of a graph $G$, denoted by $\lambda_3(G)$, is defined as the minimum cardinality among all 3-restricted edge-cuts if there are at least one; otherwise, $\lambda_3(G) = +\infty$. It is proved that $\lambda_3(G)\leq\xi_3(G)$ if $G$ has a 3-restricted edge-cut, where $\xi_3(G) = \min \{ |[X, V(G) \setminus X]_G||X \subseteq V(G),|X| = 3 \text{ and } G[X] \text{ is connected}\}.$ If $\lambda_3(G) = \xi_3(G)$, then $G$ is said to be maximally 3-restricted edge-connected. The strong product of graphs $G$ and $H$, denoted by $G \boxtimes H$, is the graph with the vertex set $V(G)\times V(H)$ and the edge set $\{(x_{1},y_{1})(x_{2},y_{2})|x_{1}=x_{2}\text{ and }y_{1}y_{2}\in E(H);\text{ or }y_{1}=y_{2}$ and $x_{1}x_{2}\in E(G)$; or $x_{1}x_{2}\in E(G)$ and $y_{1}y_{2}\in E(H)\}$. In this paper, we prove that $G \boxtimes C_{n}$ is maximally 3-restricted edge-connected, and determine the 3-restricted edge-connectivity of $G \boxtimes K_{n}$, where $G$ is a maximally edge-connected graph, $C_{n}$ and $K_{n}$ are the cycle and the complete graph of order $n$, respectively.