Ore- and Fan-type heavy subgraphs for Hamiltonicity of 2-connected graphs

Bedrossian characterized all pairs of forbidden subgraphs for a 2-connected graph to be Hamiltonian. Instead of forbidding some induced subgraphs, we relax the conditions for graphs to be Hamiltonian by restricting Ore- and Fan-type degree conditions on these induced subgraphs. Let G be a graph on n vertices and H be an induced subgraph of G . H is called o -heavy if there are two nonadjacent vertices in H with degree sum at least n , and is called f -heavy if for every two vertices u , v ∈ V ( H ) , d H ( u , v ) = 2 implies that max { d ( u ) , d ( v ) } ≥ n / 2 . We say that G is H - o -heavy ( H - f -heavy) if every induced subgraph of G isomorphic to H is o -heavy ( f -heavy). In this paper we characterize all connected graphs R and S other than P 3 such that every 2-connected R - f -heavy and S - f -heavy ( R - o -heavy and S - f -heavy, R - f -heavy and S -free) graph is Hamiltonian. Our results extend several previous theorems on forbidden subgraph conditions and heavy subgraph conditions for Hamiltonicity of 2-connected graphs.