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：在近期工作中，Martinsson与Steiner证明了每个不含$K_3$子图的$d$-退化图$G$的分数色数满足$χ_f(G) = O\left(\frac{d}{\log d}\right)$。本文通过分析特定概率分布熵的方法，从两个方向拓展该结果。我们的论证为处理其他问题提供了模板，因此具有独立研究价值。首先，我们考虑局部$r$-可着色图$G$，即每个顶点$v$满足$χ(G[N(v)]) \leq r$。我们证明$d$-退化局部$r$-可着色图$G$满足$χ_f(G) = O\left(\frac{d\log (2r)}{\log d}\right)$，强化了Alon（1996）关于此类图独立数的一个结果。其次，我们将Martinsson与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与Szemerédi（1982）关于非拥挤超图独立数的开创性成果。作为推论，我们得到$d$-退化线性超图的分数色数具有相同的增长阶。我们的方法具有构造性，能为所考虑的各种场景提供采样独立集的有效算法。