Some new sufficient conditions and Hamiltonian connected graphs

In 2005, Rahman and Kaykobad introduced the Rahman-Kaykobad condition for the research of Hamiltonian path graphs and proved that if G is a 2-connected graph with n vertices and d(u)+d(v)+δ(u,v)≥n+1 for each pair of distinct non-adjacent vertices u,v in G, then G has a Hamiltonian path. In 2006 Li proved that if G is a 3-connected graph with n vertices and d(u)+d(v)+δ(u,v)≥n+3 for each pair of distinct non-adjacent vertices u,v in G, then G is Hamiltonian-connected graphs. In this present paper, we consider some better conditions for the research of Hamiltonian-connected graphs and prove that if G is a 2-connected graph with n vertices and d(u)+d(v)+δ(u,v)≥n+2 for each pair of distinct non-adjacent vertices u,v in G, then G is Hamiltonian-connected graphs or G belongs to a class of well-structured graphs.