Every 4-connected line graph of a quasi claw-free graph is hamiltonian connected

Let G be a graph. For any two distinct vertices x and y in G, denote distG(x, y) the distance in G from x and y. For u, v ∈ V (G) with distG(u, v) = 2, denote JG(u, v) = {w ∈ NG(u)∩NG(v)|N(w) ⊆ N [u]∪ N [v]}. A graph G is claw-free if it contains no induced subgraph isomorphic to K1,3. A graph G is called quasi-claw-free if JG(u, v) 6= ∅ for any u, v ∈ V (G) with distG(u, v) = 2. Kriesell’s result in that every 4-connected line graph of a claw-free graph is hamiltonian connected. In this paper we show that every 4-connected line graph of a quasi claw-free graph is hamiltonian connected. This is joint work with Hong-Jian Lai and Yehong Shao.