On the connectivity and restricted edge-connectivity of 3-arc graphs

Abstract

Let G↔ denote the symmetric digraph of a graph G. A 3-arc is a 4-tuple (y,a,b,x) of vertices such that both (y,a,b) and (a,b,x) are paths of length 2 in G. The 3-arc graphX(G) of a given graph G is defined to have vertices the arcs of G↔, and they are denoted as (uv). Two vertices (ay),(bx) are adjacent in X(G) if and only if (y,a,b,x) is a 3-arc of G. The purpose of this work is to study the edge-connectivity and restricted edge-connectivity of 3-arc graphs. We prove that the 3-arc graph X(G) of every connected graph G of minimum degree δ(G)≥3 has λ(X(G))≥(δ(G)−1)2. Furthermore, if G is a 2-connected graph, then X(G) has restricted edge-connectivity λ(2)(X(G))≥2(δ(G)−1)2−2. We also provide examples showing that all these bounds are sharp. Concerning the vertex-connectivity, we prove that κ(X(G))≥min{κ(G)(δ(G)−1),(δ(G)−1)2}. This result improves a previous one by [M. Knor, S. Zhou, Diameter and connectivity of 3-arc graphs, Discrete Math. 310 (2010) 37–42]. Finally, we obtain that X(G) is superconnected if G is maximally connected.