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arXiv 1209.3899math.CO
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图中Hamilton圈存在的两个充分条件

Two sufficient conditions for the existence of Hamilton cycles in graphs

Bo Ning, Bing Chen, Shenggui Zhang

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Withdraw this preprint since the results will be added to arxiv:1212.6466
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摘要：

设$G$是一个以$n\geq 3$为顶点的图，它是二分图$K_{1,3}$的一个分支，而$Z_i$是一个由三角形的一个顶点加上一条长为$i$的路而得到的图。如果$G$的每个诱导爪至少有一个端顶点的度至少为$n/2$，则称$G$为1-重的;如果$G$的每个诱导爪都有一对端顶点的度和至少为$n$，则称$G$为爪- \emph{o} -重的。本文证明了两个结果：（1）2-连通爪- $o$ -重图$G$是Hamilton图，如果具有$d_{H}(u,v)=2$的$G$的导出子图$Z_2$所包含的子图$H\cong Z_1$中的每对顶点$u,v$满足下列条件之一：（$a$）$|N(u)\cap N(v)|\geq 2$ ;（$b$）$\max(d(u),d(v))\geq n/2$ . (2)每个3-连通1-重图$G$是Hamilton图，如果具有$d_{H}(u,v)=2$的$G$的导出子图$H\cong Z_2$中的每对顶点$u,v$满足下列条件之一：（$a$）$|N(u)\cap N(v)|\geq 2$ ;（$b$）$\max(d(u),d(v))\geq n/2$ .我们的结果改进或推广了Broersma等人的定理，Chen等人，Fan，Goodman & Hedetniemi，Gould & Jacobson和Shi关于图中Hamilton圈的存在性.

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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