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 withdegree sum at least $n$, and 1-heavy if at least one end-vertex of each inducedclaw of $G$ has degree at least $n/2$. In this note, we show that every2-connected $o$-heavy or 3-connected 1-heavy graph is Hamiltonian if werestrict Fan-type degree condition or neighborhood intersection condition tocertain pairs of vertices in some small induced subgraphs of the graph. Ourresults improve or extend previous results of Broersma et al., Chen et al.,Fan, Goodman & Hedetniemi, Gould & Jacobson, and Shi on the existence ofHamilton cycles in graphs.
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机译：令claw为图$ K_ {1,3} $。 $ n \ geq 3 $顶点上的图形$ G $称为\ emph {o}-重，如果每个$ G $的诱导爪具有一对端点和，且其总和至少为$ n $，则为1-重$ G $的每个诱发爪的至少一个顶点的度数至少为$ n / 2 $。在此注释中，我们表明，如果严格的Fan型度条件或邻域相交条件确定图的某些小诱导子图中的成对顶点，则每2个连通的$ o $重或3个连通的1重图是哈密顿量。我们的结果改善或扩展了Broersma等，Chen等，Fan，Goodman＆Hedetniemi，Gould＆Jacobson和Shi关于图形中汉密尔顿循环的存在的先前结果。
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