arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1602.01033math.CO
收藏

最小度大的图的谱半径与Hamilton性

Spectral radius and Hamiltonicity of graphs with large minimum degree

Vladimir Nikiforov

备注：
18 pages. This version gives tighter bounds
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

本文给出了图的Hamilton路和圈的充分条件。设$λ\left( G\right) $表示图$G,$的邻接矩阵的谱半径，本文的主要结果是：（1）设$n\geq k^{3}/2+k+4,$和$G$是一个阶为$n$，度最小的图，若\[ λ\left( G\right) \geq n-k-1, \]，则$G$有Hamilton圈，除非$G=K_{1}\vee(K_{n-k-1}+K_{k})$或$G=K_{k}\vee(K_{n-2k}+\overline{K}_{k})$ . (2)令$k\geq1,$ $n\geq k^{3}/2+k^{2}/2+k+5,$和令$G$是一个图的顺序$n$，与最小程度$δ\left( G\right) \geq k.$如果\[ λ\left( G\right) \geq n-k-2, \]然后$G$有一个哈密尔顿路径，除非$G=K_{k}\vee(K_{n-2k-1}+\overline {K}_{k+1})$或$G=K_{n-k-1}+K_{k+1}$此外，它表明，在上述声明中，在$n$上的边界是紧的加法项不超过$2$。

英文摘要：

This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $λ\left( G\right) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $k\geq1,$ $n\geq k^{3}/2+k+4,$ and let $G$ be a graph of order $n$, with minimum degree $δ\left( G\right) \geq k.$ If \[ λ\left( G\right) \geq n-k-1, \] then $G$ has a Hamiltonian cycle, unless $G=K_{1}\vee(K_{n-k-1}+K_{k})$ or $G=K_{k}\vee(K_{n-2k}+\overline{K}_{k})$. (2) Let $k\geq1,$ $n\geq k^{3}/2+k^{2}/2+k+5,$ and let $G$ be a graph of order $n$, with minimum degree $δ\left( G\right) \geq k.$ If \[ λ\left( G\right) \geq n-k-2, \] then $G$ has a Hamiltonian path, unless $G=K_{k}\vee(K_{n-2k-1}+\overline {K}_{k+1})$ or $G=K_{n-k-1}+K_{k+1}$ In addition, it is shown that in the above statements, the bounds on $n$ are tight within an additive term not exceeding $2$.

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。

↑