arXiv 2202.027702022-02-06math.CO

超图关联着色

Hypergraph incidence coloring

Weichan Liu, Guiying Yan

备注：: 17 pages

摘要：

超图$\mathcal{H}=(X,S)$的发生率是一对$(x,s)$与$x\in X$、$s\in S$和$x\in s$。如果（i）$x=x'$，或（ii）$\{x,x'\}\subseteq s$或$\{x,x'\}\subseteq s'$，则$(x,s)$和$(x',s')$的两个事件相邻。一个超图$\mathcal{H}$的恰当关联$k$ -着色是从$\mathcal{H}$的关联集到$\{1,2,\ldots,k\}$的映射$φ$，使得对于$\mathcal{H}$的任意两个相邻关联$(x,s)$和$(x',s')$，$φ(x,s)\neq φ(x',s')$. $\mathcal{H}$的关联色数$χ_I(\mathcal{H})$是使$\mathcal{H}$有恰当关联$k$ -着色的最小整数$k$.本文证明了对每一个具有$t<<r(\mathcal{H})$和充分大的$Δ(\mathcal{H})$的$t$ -拟线性超图，都有$χ_I(\mathcal{H})\leq (4/3+o(1))r(\mathcal{H})Δ(\mathcal{H})$，其中$r(\mathcal{H})$是$\mathcal{H}$中边的基数的最大值.证明了当$\mathcal{H}$是$α$ -无圈线性超图时，$χ_I(\mathcal{H})\leq Δ(\mathcal{H})+r(\mathcal{H})-1$是有界的，并且这个界是尖锐的.

英文摘要：

An incidence of a hypergraph $\mathcal{H}=(X,S)$ is a pair $(x,s)$ with $x\in X$, $s\in S$ and $x\in s$. Two incidences $(x,s)$ and $(x',s')$ are adjacent if (i) $x=x'$, or (ii) $\{x,x'\}\subseteq s$ or $\{x,x'\}\subseteq s'$. A proper incidence $k$-coloring of a hypergraph $\mathcal{H}$ is a mapping $φ$ from the set of incidences of $\mathcal{H}$ to $\{1,2,\ldots,k\}$ so that $φ(x,s)\neq φ(x',s')$ for any two adjacent incidences $(x,s)$ and $(x',s')$ of $\mathcal{H}$. The incidence chromatic number $χ_I(\mathcal{H})$ of $\mathcal{H}$ is the minimum integer $k$ such that $\mathcal{H}$ has a proper incidence $k$-coloring. In this paper we prove $χ_I(\mathcal{H})\leq (4/3+o(1))r(\mathcal{H})Δ(\mathcal{H})$ for every $t$-quasi-linear hypergraph with $t<<r(\mathcal{H})$ and sufficiently large $Δ(\mathcal{H})$, where $r(\mathcal{H})$ is the maximum of the cardinalities of the edges in $\mathcal{H}$. It is also proved that $χ_I(\mathcal{H})\leq Δ(\mathcal{H})+r(\mathcal{H})-1$ if $\mathcal{H}$ is an $α$-acyclic linear hypergraph, and this bound is sharp.

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：

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