  Let $G$ be a graph. Adopting the terminology of Broersma et al. and \v{C}ada,
respectively, we say that $G$ is 2-heavy if every induced claw ($K_{1,3}$) of
$G$ contains two end-vertices each one has degree at least $|V(G)|/2$; and $G$
is o-heavy if every induced claw of $G$ contains two end-vertices with degree
sum at least $|V(G)|$ in $G$. In this paper, we introduce a new concept, and
say that $G$ is \emph{$S$-c-heavy} if for a given graph $S$ and every induced
subgraph $G'$ of $G$ isomorphic to $S$ and every maximal clique $C$ of $G'$,
every non-trivial component of $G'-C$ contains a vertex of degree at least
$|V(G)|/2$ in $G$. In terms of this concept, our original motivation that a
theorem of Hu in 1999 can be stated as every 2-connected 2-heavy and
$N$-c-heavy graph is hamiltonian, where $N$ is the graph obtained from a
triangle by adding three disjoint pendant edges. In this paper, we will
characterize all connected graphs $S$ such that every 2-connected o-heavy and
$S$-c-heavy graph is hamiltonian. Our work results in a different proof of a
stronger version of Hu's theorem. Furthermore, our main result improves or
extends several previous results.
