Panconnected index of graphs

Abstract

For a connected graph G not isomorphic to a path, a cycle or a K1,3, let pc(G) denote the smallest integer n such that the nth iterated line graph Ln(G) is panconnected. A path P is a divalent path of G if the internal vertices of P are of degree 2 in G. If every edge of P is a cut edge of G, then P is a bridge divalent path of G; if the two ends of P are of degree s and t, respectively, then P is called a divalent(s,t)-path. Let ℓ(G)=max{m:G has a divalent path of length m that is not both of length 2 and in a K3}. We prove the following.(i) If G is a connected triangular graph, then L(G) is panconnected if and only if G is essentially 3-edge-connected.(ii) pc(G)≤ℓ(G)+2. Furthermore, if ℓ(G)≥2, then pc(G)=ℓ(G)+2 if and only if for some integer t≥3, G has a bridge divalent (3,t)-path of length ℓ(G).