Generalizing Sufficient Conditions and Traceable Graphs

Abstract

In 2005, Rahman and Kaykobad proved that if G is a 2-connected graph with n vertices and d(u)+d(v)+((u,v)≥n+1 for each pair of distinct non-adjacent vertices u,v in G, then G is traceable [ Information Processing Letters, 94(2005), 1, 37-41]. In 2006, Li proved thatif G is a 2-connected graph with n vertices and d(u)+d(v)+((u,v)≥n+3 for each pair of distinct non-adjacent vertices u,v in G, then G is Hamiltonian-connected [Information Processing Letters, 98(2006), 4, 159-161]. In this present paper, we prove that if G is a 2-connected graph with n vertices and d(u)+d(v)+((u,v)≥n for each pair of distinct non-adjacent vertices u,v in G, then G has a Hamiltonian path or G belongs to a class of exceptional graphs. We also prove that if G is a 2-connected graph with n vertices and d(u)+d(v)+((u,v)≥n+2 for each pair of distinct non-adjacent vertices u,v in G, then G is Hamiltonian-connected or G belongs to a classes of exceptional graphs. Thus, our the two results generalize the above two results by Rahman et al. and Li, respectively.