Panconnectivity of locally connected claw-free graphs

Abstract

Let G be a connected, locally connected, claw-free graph and x, y be two vertices of G. In this paper, we prove that if for any 2-cut S of G, S∩{ x, y}=∅, then G contains ( x, y)-paths of all possible lengths. As a corollary of the result, the following conjecture of Broersma and Veldman is proved: every locally connected, claw-free graph of order at least 4 is panconnected if and only if it is 3-connected.