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A graph $G$ is called \emph{claw-o-heavy} if every induced claw ($K_{1,3}$)of $G$ has two end-vertices with degree sum at least $|V(G)|$ in $G$. For agiven graph $R$, $G$ is called \emph{$R$-f-heavy} if for every induced subgraph$H$ of $G$ isomorphic to $R$ and every pair of vertices $u,v\in V(H)$ with$d_H(u,v)=2$, there holds $\max\{d(u),d(v)\}\geq |V(G)|/2$. In this paper, weprove that every 2-connected claw-\emph{o}-heavy and $Z_3$-\emph{f}-heavy graphis hamiltonian (with two exceptional graphs), where $Z_3$ is the graph obtainedfrom identifying one end-vertex of $P_4$ (a path with 4 vertices) with onevertex of a triangle. This result gives a positive answer to a problem proposedin [B. Ning, S. Zhang, Ore- and Fan-type heavy subgraphs for Hamiltonicity of2-connected graphs, Discrete Math. 313 (2013) 1715--1725], and also implies twoprevious theorems of Faudree et al. and Chen et al., respectively.

Solution to a problem on hamiltonicity of graphs under Ore- and Fan-type heavy subgraph conditions