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 ( $$K_{1,3}$$ ) 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 $$u,v\in V(H)$$ with $$d_H(u,v)=2,$$ there holds $$\max \{d(u),d(v)\}\ge |V(G)|/2.$$ In this paper, we prove that every 2-connected claw- o-heavy and $$Z_3$$ - f-heavy graph is hamiltonian (with two exceptional graphs), where $$Z_3$$ is the graph obtained by identifying one end-vertex of $$P_4$$ (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, ), and also implies two previous theorems of Faudree et al. and Chen et al., respectively. [ABSTRACT FROM AUTHOR]