The 3-restricted edge-connectivity of the direct 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 $\xi_3(G) = \min \left\{ |[X, V(G) \setminus X]_G|:|X| = 3 \text{ and } G[X] \text{ is connected} \right\}.$ If $\lambda_3(G) = \xi_3(G)$, then $G$ is said to be maximally 3-restricted edge-connected. The direct product of two graphs $G$ and $H$, denoted by $G \times H$, is defined as the graph with vertex set $V(G \times H) = V(G) \times V(H)$, where two vertices $(u_1, v_1)$ and $(u_2, v_2)$ are adjacent in $G \times H$ if and only if $u_1u_2 \in E(G)$ and $v_1v_2 \in E(H)$. In this paper, we determine, for a regular connected graph $G$, the 3-restricted edge-connectivity of $G \times C_n$, $G \times K_n$ and $G \times T_n$, where $C_n$, $K_n$ and $T_n$ are the cycle, the complete graph and the total graph with $n$ vertices, respectively. As corollaries, we establish sufficient conditions for the direct product graphs $G \times C_n$, $G \times K_n$ and $G \times T_n$ to be maximally 3-restricted edge-connected.