  Let claw be the graph $K_{1,3}$. A graph $G$ on $n\geq 3$ vertices is called
\emph{o}-heavy if each induced claw of $G$ has a pair of end-vertices with
degree sum at least $n$, and 1-heavy if at least one end-vertex of each induced
claw of $G$ has degree at least $n/2$. In this note, we show that every
2-connected $o$-heavy or 3-connected 1-heavy graph is Hamiltonian if we
restrict Fan-type degree condition or neighborhood intersection condition to
certain pairs of vertices in some small induced subgraphs of the graph. Our
results improve or extend previous results of Broersma et al., Chen et al.,
Fan, Goodman & Hedetniemi, Gould & Jacobson, and Shi on the existence of
Hamilton cycles in graphs.
