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Fractional coloring via entropy

Abhishek Dhawan

Abstract

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.

Fractional coloring via entropy

Abstract

In recent work, Martinsson and Steiner showed that every -free -degenerate graph has fractional chromatic number . 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 -colorable graphs , i.e., where for each vertex . We show that -degenerate locally -colorable graphs satisfy , strengthening a result of Alon (1996) on the independence number of such graphs. Second, we extend Martinsson and Steiner's result to -uniform -degenerate hypergraphs of girth at least . We show that such hypergraphs satisfy , 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 -degenerate linear hypergraphs. Our approach is constructive, yielding efficient algorithms to sample independent sets in each of the settings we consider.
Paper Structure (13 sections, 20 theorems, 61 equations, 1 figure)

This paper contains 13 sections, 20 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Notation and terminology
  4. Proof Overview
  5. Open problems
  6. Graphs with bounded local chromatic number: proof of Theorem \ref{['theo: r colorable ordinary coloring']}
  7. Algorithm description
  8. Expectation of $|S(v_k)|$: proof of Lemma \ref{['Lemma: exp size S(vk)']}
  9. Concentration of $|S(v_k)|$: proof of Lemma \ref{['Lemma: conc size S(vk)']}
  10. High-girth hypergraphs: proof of Theorem \ref{['theo: hypergraph free ordinary coloring']}
  11. Algorithm description
  12. Expectation of $|S(v_k)|$: proof of Lemma \ref{['Lemma: exp size S(vk) hypergraph']}
  13. Concentration of $|S(v_k)|$: proof of Lemma \ref{['Lemma: conc size S(vk) hypergraph']}

Key Result

Theorem 1.1

Let $G$ be an $n$-vertex $d$-degenerate $K_3$-free graph. Then, $\chi_f(G) \leq (4 + o(1))\frac{d}{\log d}$.

Figures (1)

  • Figure 1: An edge $e$ containing $v_i$ and $v_k$ such that $v_k$ is the right-most vertex of $e$. Here, $f$ consists of the vertices of $e$ appearing before $v_i$ in the degeneracy ordering.

Theorems & Definitions (33)

  • Theorem 1.1: martinsson2025random
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.6: MolloyReed
  • Theorem 1.7: bradavc2026coloring
  • Conjecture 1.9
  • Theorem : Restatement of Theorem \ref{['theo: r colorable ordinary coloring']}
  • Lemma 2.1
  • Lemma 2.2
  • ...and 23 more