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 graph X(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.
展开▼
