Solution to a Problem on Hamiltonicity of Graphs Under Ore- and Fan-Type Heavy Subgraph Conditions

A graph G is called claw-o-heavy if every induced claw ((Formula presented.)) of G has two end-vertices with degree sum at least |V(G)|. For a given graph S,  G is called S-f-heavy if for every induced subgraph H of G isomorphic to S and every pair of vertices (Formula presented.) with (Formula presented.) there holds (Formula presented.) In this paper, we prove that every 2-connected claw-o-heavy and (Formula presented.) -f-heavy graph is hamiltonian (with two exceptional graphs), where (Formula presented.) is the graph obtained by identifying one end-vertex of (Formula presented.) (a path with 4 vertices) with one vertex of a triangle. This result gives a positive answer to a problem proposed Ning and Zhang (Discrete Math 313:1715–1725, 2013), and also implies two previous theorems of Faudree et al. and Chen et al., respectively.