In recent work, Martinsson and Steiner showed that every $K_3$-free $d$-degenerate graph $G$ has fractional chromatic number $χ_f(G) = O\left(\frac{d}{\log d}\right)$. In this paper, we extend the result in two ways, employing an approach rooted in the analysis of the entropy of certain probability distributions. Our argument provides a template to tackle other problems, so it is of independent interest. First, we consider locally $r$-colorable graphs $G$, i.e., where $χ(G[N(v)]) \leq r$ for each vertex $v$. We show that $d$-degenerate locally $r$-colorable graphs $G$ satisfy $χ_f(G) = O\left(\frac{d\log (2r)}{\log d}\right)$, strengthening a result of Alon (1996) on the independence number of such graphs. Second, we extend Martinsson and Steiner's result to $r$-uniform $d$-degenerate hypergraphs $H$ of girth at least $4$. We show that such hypergraphs satisfy $χ_f(H) \leq c_r\left(\frac{d}{\log d}\right)^{\frac{1}{r-1}}$, implying a strict generalization of a seminal result of Ajtai, Komlós, Pintz, Spencer, and Szemerédi (1982) on the independence number of uncrowded hypergraphs. As a corollary, we obtain the same growth rate for the fractional chromatic number of $d$-degenerate linear hypergraphs. Our approach is constructive, yielding efficient algorithms to sample independent sets in each of the settings we consider.

翻译：在最近的工作中，Martinsson 和 Steiner 证明了每个不含 $K_3$ 的 $d$-退化图 $G$ 的分数色数满足 $χ_f(G) = O\left(\frac{d}{\log d}\right)$。本文采用一种基于某些概率分布熵分析的方法，从两个方向推广了这一结果。我们的论证为解决其他问题提供了模板，因此具有独立的意义。首先，我们考虑局部 $r$-可着色图 $G$，即对于每个顶点 $v$，有 $χ(G[N(v)]) \leq r$。我们证明 $d$-退化局部 $r$-可着色图 $G$ 满足 $χ_f(G) = O\left(\frac{d\log (2r)}{\log d}\right)$，这加强了 Alon (1996) 关于此类图独立数的一个结果。其次，我们将 Martinsson 和 Steiner 的结果推广到围长至少为 $4$ 的 $r$-一致 $d$-退化超图 $H$。我们证明此类超图满足 $χ_f(H) \leq c_r\left(\frac{d}{\log d}\right)^{\frac{1}{r-1}}$，这严格推广了 Ajtai、Komlós、Pintz、Spencer 和 Szemerédi (1982) 关于非拥挤超图独立数的开创性结果。作为推论，我们得到了 $d$-退化线性超图的分数色数相同的增长速率。我们的方法是构造性的，能够在所考虑的每种情形下生成高效算法来采样独立集。