On the edge-connectivity of the square of a graph

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) <= delta(G), and graphs for which equality holds are said to be maximally edge-connected. The square G2 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 & LeftFloor;n+2 4 & RightFloor;, then G2 is maximally edge-connected, and this result is best possible. We also give lower bounds on lambda(G2) for the case that G2 is not maximally edge-connected: We prove that lambda(G2) >= 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(G2) >= 21 lambda(G)3/2 - 21 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. (c) 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).