Fractional coloring via entropy

TL;DR

This paper extends fractional coloring bounds to locally r-colorable graphs and certain hypergraphs using entropy analysis, providing new theoretical insights and efficient algorithms.

Contribution

It generalizes fractional chromatic number bounds to broader classes of graphs and hypergraphs via an entropy-based approach, with constructive algorithms.

Findings

Fractional chromatic number of locally r-colorable graphs is bounded by O(d log(2r)/log d).

Fractional chromatic number of r-uniform hypergraphs with girth at least 4 is bounded by c_r(d/log d)^{1/(r-1)}.

The approach yields efficient algorithms for sampling independent sets in these graph classes.

Abstract

In recent work, Martinsson and Steiner showed that every $K_3$-free $d$-degenerate graph $G$ has fractional chromatic number $\chi_f(G) = O\left(\frac{d}{\log d}\right)$. 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 $\chi(G[N(v)]) \leq r$ for each vertex $v$. We show that $d$-degenerate locally $r$-colorable graphs $G$ satisfy $\chi_f(G) = O\left(\frac{d\log (2r)}{\log d}\right)$, 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 $\chi_f(H) \leq c_r\left(\frac{d}{\log d}\right)^{\frac{1}{r-1}}$, implying a strict generalization of a seminal result of Ajtai, Koml\'os, Pintz, Spencer, and Szemer\'edi (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.