arXiv 0903.45672009-03-26math.CO

Hamilton图的泛圈性与高连通图

Pancyclicity of Hamiltonian and highly connected graphs

Peter Keevash, Benny Sudakov

备注：: 15 pages, 1 figure

摘要：

一个n阶图G是哈密尔顿图，如果它包含一个长度为n的圈;如果它对所有$3 \le \ell \le n$都包含长度为$\ell$的圈，则G是泛圈图.写$α(G)$为$G$的独立数，即不包含边的顶点集的最大子集的大小，和$κ(G)$为（顶点）连通性，即可以删除以获得不连通图的顶点集的最小子集的大小。Chvátal和Erdés的一个著名定理说，如果$κ(G) \ge α(G)$，则$G$是哈密顿的。此外，Bondy提出，几乎任何非平凡的图的哈密尔顿性条件也应该暗示泛圈性。受此启发，我们证明了如果$κ(G) \ge 600α(G)$，那么G是泛环的。这建立了一个猜想杰克逊和Ordaz到一个常数的因素。此外，我们还得到了更一般的结果：如果G是最小度为$δ(G) \ge 600α(G)$的Hamilton算子，则G是泛圈的.改进了Erdens的一个结果，证明了G是泛圈的，如果它是Hamilton的，并且是$n \ge 150α(G)^3$ .我们的论点使用以下定理的独立兴趣的周期长度的图：如果$δ(G) \ge 300α(G)$，那么G包含一个周期的长度$\ell$的所有$3 \le \ell \le δ(G)/81$。

英文摘要：

A graph G on n vertices is Hamiltonian if it contains a cycle of length n and pancyclic if it contains cycles of length $\ell$ for all $3 \le \ell \le n$. Write $α(G)$ for the independence number of $G$, i.e. the size of the largest subset of the vertex set that does not contain an edge, and $κ(G)$ for the (vertex) connectivity, i.e. the size of the smallest subset of the vertex set that can be deleted to obtain a disconnected graph. A celebrated theorem of Chvátal and Erdős says that $G$ is Hamiltonian if $κ(G) \ge α(G)$. Moreover, Bondy suggested that almost any non-trivial conditions for Hamiltonicity of a graph should also imply pancyclicity. Motivated by this, we prove that if $κ(G) \ge 600α(G)$ then G is pancyclic. This establishes a conjecture of Jackson and Ordaz up to a constant factor. Moreover, we obtain the more general result that if G is Hamiltonian with minimum degree $δ(G) \ge 600α(G)$ then G is pancyclic. Improving an old result of Erdős, we also show that G is pancyclic if it is Hamiltonian and $n \ge 150α(G)^3$. Our arguments use the following theorem of independent interest on cycle lengths in graphs: if $δ(G) \ge 300α(G)$ then G contains a cycle of length $\ell$ for all $3 \le \ell \le δ(G)/81$.

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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英文摘要：

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