分数强染色的一个更强结果
A stronger result on fractional strong colourings
摘要：
Aharoni、Berger和Ziv最近证明了强着色猜想的分数松弛。本文将他们的结果推广如下。设$k\geq 1$，并将图$G$的顶点划分为集合$V_1,..., V_r$，使得对于$1\leq i \leq r$，$V_i$中的每个顶点至多有$\max\{k, |V_i|-k \}$在$V_i$之外的邻居.然后在$G$的稳定集上存在概率分布，使得从该分布中提取的稳定集以概率$1/|V_i|$命中$V_i$中的每个顶点，对于$1\leq i\leq r$。我们相信，这一结果将是有用的工具，在概率方法来界定色数和分数色数。
英文摘要：
Aharoni, Berger and Ziv recently proved the fractional relaxation of the strong colouring conjecture. In this note we generalize their result as follows. Let $k\geq 1$ and partition the vertices of a graph $G$ into sets $V_1,..., V_r$, such that for $1\leq i \leq r$ every vertex in $V_i$ has at most $\max\{k, |V_i|-k \}$ neighbours outside $V_i$. Then there is a probability distribution on the stable sets of $G$ such that a stable set drawn from this distribution hits each vertex in $V_i$ with probability $1/|V_i|$, for $1\leq i\leq r$. We believe that this result will be useful as a tool in probabilistic approaches to bounding the chromatic number and fractional chromatic number.
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。
