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

Abstract

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, 2013), and also implies two previous theorems of Faudree et al. and Chen et al., respectively.