On the Edge-Connectivity of the Square of a Graph

TL;DR

This paper investigates the edge-connectivity of the square of a graph, establishing conditions under which it is maximally edge-connected and providing bounds for cases when it is not, with proofs of optimality.

Contribution

It presents new bounds on the edge-connectivity of graph squares, including conditions for maximal edge-connectivity and sharp bounds involving graph connectivity and edge-connectivity.

Findings

If minimum degree ≥ ⌊(n+2)/4⌋, then G² is maximally edge-connected.

Provides lower bounds on λ(G²) based on κ(G) and λ(G).

Shows the exponent 3/2 in the bounds is optimal.

Abstract

Let $G$ be a connected graph. The edge-connectivity of $G$, denoted by $\lambda(G)$, is the minimum number of edges whose removal renders $G$ disconnected. Let $\delta(G)$ be the minimum degree of $G$. It is well-known that $\lambda(G) \leq \delta(G)$, 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 $\lambda(G^2)$ for the case that $G^2$ is not maximally edge-connected: We prove that $\lambda(G^2) \geq \kappa(G)^2 + \kappa(G)$, where $\kappa(G)$ 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 $\lambda(G^2) \geq \frac{1}{2}\lambda(G)^{3/2} - \frac{1}{2} \lambda(G)$, and we construct an infinite family of graphs to show that the exponent $3/2$ of $\lambda(G)$ in this bound is best possible.