Edge connectivity and restricted edge connectivity of cartesian product of graphs

Abstract

An edge cutset $E\subset E(G)$ of a graph $G$ is called a restricted edge cutset if every component of $G-E$ has order at least $2$. We let ${\lambda }^{\prime }(G)$ denote the minimum cardinality of a restricted edge cutset of $G$, and let ${\delta }^{\prime }(G)$ denote the minimum of ${\mathrm{deg}}_G(x)+{\mathrm{deg}}_G(y)-2$ as $x$ and $y$ range over all adjacent vertices of $G$. We let $\lambda (G)$ and $\delta (G)$ denote the edge connectivity and the minimum degree of $G$, respectively. Among other results, we show that if $G_1$ and $G_2$ are graphs such that $\lambda (G_i)=\delta (G_i)\ge 2$ and ${\lambda }^{\prime }(G_i)={\delta }^{\prime }(G_i)\ge 2$ for each $i=1,2$, then ${\lambda }^{\prime }(G_1\otimes G_2)={\delta }^{\prime }(G_1\otimes G_2)=\min \{{\delta }^{\prime }(G_1)+2\delta (G_2),{\delta }^{\prime }(G_2)+2\delta (G_1)\}$, where $G_1\otimes G_2$ denotes the cartesian product of $G_1$ and $G_2$.