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 (K1,3​) 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.Comment: 21 pages, 6 figures, finial version for publication in Discussiones
Mathematicae Graph Theor