Some hamiltonian properties of L1-graphs Original Research Article

A graph G is called an L1-graph if, for each triple of vertices u, v, and w with d(u,v)=2 and w∈N(u)∩N(v), d(u)+d(v)⩾|N(u)∪N(v)∪N(w)|−1. Let G be a 2-connected L1-graph of order n. If δ(G)⩾(n−2)/3 or σ3(G)⩾n, then G is hamiltonian or G∈K, where σ3(G)=min{d(u)+d(v)+d(w): {u,v,w} is an independent set in G}, K={G: Kp,p+1⊆G⊆Kp+(p+1)K1 for some p ⩾2}. Some results on the traceability of L1-graphs are also obtained.