arXiv 1606.050452016-06math.CO

奇可染$r$ -超图的性质及应用

Some properties and applications of odd-colorable $r$-hypergraphs

Xiying Yuan, Liqun Qi, Jiayu Shao, Chen Ouyang

备注：: 9pages; Some results are added, and some typos are corrected

摘要：

设$r\geq2$和$r$为偶数。一个$r$ -超图$G$在$n$顶点上称为奇可染的，如果存在一个映射$φ:[n]\rightarrow\lbrack r]$使得对于$G$的任意边$\{j_{1},j_{2},\cdots,j_{r}\}$，我们有$φ(j_{1})+φ(j_{2})+\cdot\cdot\cdot+φ(j_{r})\equiv r/2(\operatorname{mod}r).$本文首先确定了，如果$r=2^{q}(2t+1)$和$n\ge 2^{q}(2^{q}-1)r$，则$r$ -超图类中的最大色数是$2^q$，它回答了V. Nikiforov最近在[V. Nikiforov，Hypergraphs and hypermatrix with symmetric spectrum. Prinprint可在arXiv：1605.00709v2，10 May，2016中获得]。我们还研究了V. Nikiforov在同一篇文章中给出的奇可染$r$ -图的对称谱性质的一些应用。证明了$r$ -超图$G$的Laplacian谱与无符号Laplacian谱相等的充要条件是$G$是奇可染的，并进一步研究了这些谱性质的应用.

英文摘要：

Let $r\geq2$ and $r$ be even. An $r$-hypergraph $G$ on $n$ vertices is called odd-colorable if there exists a map $φ:[n]\rightarrow\lbrack r]$ such that for any edge $\{j_{1},j_{2},\cdots,j_{r}\}$ of $G$, we have $φ(j_{1})+φ(j_{2})+\cdot\cdot\cdot+φ(j_{r})\equiv r/2(\operatorname{mod}r).$ In this paper, we first determine that, if $r=2^{q}(2t+1)$ and $n\ge 2^{q}(2^{q}-1)r$, then the maximum chromatic number in the class of the odd-colorable $r$-hypergraphs on $n$ vertices is $2^q$, which answers a question raised by V. Nikiforov recently in [V. Nikiforov, Hypergraphs and hypermatrices with symmetric spectrum. Prinprint available in arXiv:1605.00709v2, 10 May, 2016]. We also study some applications of the symmetric spectral property of the odd-colorable $r$-graphs given in that same paper by V. Nikiforov. We show that the Laplacian spectrum and the signless Laplacian spectrum of an $r$-hypergraph $G$ are equal if and only if $G$ is odd-colorable, and then study some further applications of these spectral properties.

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。

摘要：

英文摘要：

相关论文