论文标题
森林的合作着色
Cooperative colorings of forests
论文作者
论文摘要
给定一个跨越公共顶点$ v $的家庭$ \ Mathcal g $，$ \ Mathcal g $的合作着色是一个独立集的集合，从$ \ Mathcal g $的每个图中，因此这些独立集合的结合等于$ v $。 We prove that when $d$ is large, there exists a family $\mathcal G$ of $(1+o(1)) \frac{\log d}{\log \log d}$ forests of maximum degree $d$ that admits no cooperative coloring, which significantly improves a result of Aharoni, Berger, Chudnovsky, Havet, and Jiang (Electronic Journal of Combinatorics, 2020）。我们的家人$ \ Mathcal g $完全由星森林组成，我们证明，对于$ | \ Mathcal g | $，在$ \ Mathcal g $的情况下，这种价值是渐进的，这是渐近的。
Given a family $\mathcal G$ of graphs spanning a common vertex $V$, a cooperative coloring of $\mathcal G$ is a collection of one independent set from each graph of $\mathcal G$ such that the union of these independent sets equals $V$. We prove that when $d$ is large, there exists a family $\mathcal G$ of $(1+o(1)) \frac{\log d}{\log \log d}$ forests of maximum degree $d$ that admits no cooperative coloring, which significantly improves a result of Aharoni, Berger, Chudnovsky, Havet, and Jiang (Electronic Journal of Combinatorics, 2020). Our family $\mathcal G$ consists entirely of star forests, and we show that this value for $|\mathcal G|$ is asymptotically best possible in the case that $\mathcal G$ is a family of star forests.
