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Fractional hypergraph coloring

Margarita Akhmejanova, Sean Longbrake

TL;DR

This work extends hypergraph coloring to the fractional regime by analyzing $(a:b)$-colorings of $n$-uniform hypergraphs. It provides a sharp probabilistic upper bound: if $2\le b\le a-2\,\le n/(2\ln n)$ and $|E(H)|\, ext{is small enough}$, then $H$ is properly $(a:b)$-colorable; this is complemented by Erdős-style lower bounds showing non-colorability above certain edge thresholds. The results connect to Panchromatic colorings and leverage linear programming duality, as well as Cherkashin–Kozik’s chain methods, to derive both constructive coloring guarantees and non-colorability thresholds. Overall, the paper advances the understanding of fractional colorings in hypergraphs, providing tight-ish bounds and techniques that bridge combinatorics, probability, and LP theory. These findings have implications for scheduling, coding theory, and resource allocation where multi-color sharing is relevant.

Abstract

We investigate proper $(a:b)$-fractional colorings of $n$-uniform hypergraphs, which generalize traditional integer colorings of graphs. Each vertex is assigned $b$ distinct colors from a set of $a$ colors, and an edge is properly colored if no single color is shared by all vertices of the edge. A hypergraph is $(a:b)$-colorable if every edge is properly colored. We prove that for any $2\leq b\leq a-2\leq n/\ln n$, every $n$-uniform hypergraph $H$ with $ |E(H)| \leq (ab^3)^{-1/2}\left(\frac{n}{\log n}\right)^{1/2} \left(\frac{a}{b}\right)^{n-1} $ is proper $(a:b)$-colorable. We also address specific cases, including $(a:a-1)$-colorability.

Fractional hypergraph coloring

TL;DR

This work extends hypergraph coloring to the fractional regime by analyzing -colorings of -uniform hypergraphs. It provides a sharp probabilistic upper bound: if and , then is properly -colorable; this is complemented by Erdős-style lower bounds showing non-colorability above certain edge thresholds. The results connect to Panchromatic colorings and leverage linear programming duality, as well as Cherkashin–Kozik’s chain methods, to derive both constructive coloring guarantees and non-colorability thresholds. Overall, the paper advances the understanding of fractional colorings in hypergraphs, providing tight-ish bounds and techniques that bridge combinatorics, probability, and LP theory. These findings have implications for scheduling, coding theory, and resource allocation where multi-color sharing is relevant.

Abstract

We investigate proper -fractional colorings of -uniform hypergraphs, which generalize traditional integer colorings of graphs. Each vertex is assigned distinct colors from a set of colors, and an edge is properly colored if no single color is shared by all vertices of the edge. A hypergraph is -colorable if every edge is properly colored. We prove that for any , every -uniform hypergraph with is proper -colorable. We also address specific cases, including -colorability.

Paper Structure

This paper contains 9 sections, 4 theorems, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proper and Fractional Colorings
  3. Linear Programming Approach to Fractional Coloring
  4. Application
  5. Our results
  6. Proofs
  7. Proof of Theorem 1:
  8. Proof of Theorem 2:
  9. Proof of Proposition 2:

Key Result

Theorem 1

For any $2\leq b\leq a-2\leq n/(2\ln n)$, every $n$-uniform hypergraph $H$ with is $(a:b)$-colorable.

Figures (1)

  • Figure 1: An example of $3$-proper and proper $(5:2)$-fractional colorings of a pentagon graph $G$. Here, chromatic number $\chi(G)=3$, and fractional chromatic number $\chi_f(G)=2.5.$

Theorems & Definitions (4)

  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Proposition 2