On the edge-connectivity of the square of a graph

Abstract

Let $G$ be a connected graph. The edge-connectivity of $G$, denoted by $\lambda \left(G\right)$, is the minimum number of edges whose removal renders $G$ disconnected. Let $\delta \left(G\right)$ be the minimum degree of $G$. It is well-known that $\lambda \left(G\right)\le \delta \left(G\right)$, and graphs for which equality holds are said to be maximally edge-connected. The square $G^2$ of $G$ is the graph with the same vertex set as $G$, in which two vertices are adjacent if their distance is not more that 2.

In this paper we present results on the edge-connectivity of the square of a graph. We show that if the minimum degree of a connected graph $G$ of order $n$ is at least $\lfloor \frac{n+2}{4}\rfloor$, then $G^2$ is maximally edge-connected, and this result is best possible. We also give lower bounds on $\lambda \left(G^2\right)$ for the case that $G^2$ is not maximally edge-connected: We prove that $\lambda \left(G^2\right)\ge \kappa {\left(G\right)}^2+\kappa \left(G\right)$, where $\kappa \left(G\right)$ denotes the connectivity of $G$, i.e., the minimum number of vertices whose removal renders $G$ disconnected, and this bound is sharp. We further prove that $\lambda \left(G^2\right)\ge \frac{1}{2}\lambda {\left(G\right)}^{3/2}-\frac{1}{2}\lambda \left(G\right)$, and we construct an infinite family of graphs to show that the exponent $3/2$ of $\lambda \left(G\right)$ in this bound is best possible.