  Let $G$ be a graph on $n\geq 3$ vertices, claw the bipartite graph $K_{1,3}$,
and $Z_i$ the graph obtained from a triangle by attaching a path of length $i$
to its one vertex. $G$ is called 1-heavy if at least one end vertex of each
induced claw of $G$ has degree at least $n/2$, and claw-\emph{o}-heavy if each
induced claw of it has a pair of end vertices with degree sum at least $n$. In
this paper we prove two results: (1) Every 2-connected claw-$o$-heavy graph $G$
is Hamiltonian if every pair of vertices $u,v$ in a subgraph $H\cong Z_1$
contained in an induced subgraph $Z_2$ of $G$ with $d_{H}(u,v)=2$ satisfies one
of the following conditions: ($a$) $|N(u)\cap N(v)|\geq 2$; ($b$)
$\max(d(u),d(v))\geq n/2$. (2) Every 3-connected 1-heavy graph $G$ is
Hamiltonian if every pair of vertices $u,v$ in an induced subgraph $H\cong Z_2$
of $G$ with $d_{H}(u,v)=2$ satisfies one of the following conditions: ($a$)
$|N(u)\cap N(v)|\geq 2$; ($b$) $\max(d(u),d(v))\geq n/2$. Our results improve
or extend previous theorems of Broersma et al., Chen et al., Fan, Goodman &
Hedetniemi, Gould & Jacobson and Shi on the existence of Hamilton cycles in
graphs.
