Essential edge connectivity of line graphs

Abstract

Let G be a graph and L(G) be its line graph. In 1969, Chartrand and Stewart proved that κ′(L(G))≥2κ′(G)−2, where κ′(G) and κ′(L(G)) denote the edge connectivity of G and L(G) respectively. We show a similar relationship holds for the essential edge connectivity of G and L(G), written κe′(G) and κe′(L(G)), respectively. In this note, it is proved that if L(G) is not a complete graph and G does not have a vertex of degree two, then κe′(L(G))≥2κe′(G)−2. An immediate corollary is that κ(L2(G))≥2κ(L(G))−2 for such graphs G, where the vertex connectivity of the line graph L(G) and the second iterated line graph L2(G) are written as κ(L(G)) and κ(L2(G)) respectively.