Sharp bounds for the fractional chromatic number of high-girth $d$-degenerate graphs

2026-07-28

AI Summary

This study investigates tight bounds on the fractional chromatic number of $d$-degenerate graphs with high girth. For such graphs of girth at least five, the authors present a constructive randomized algorithm achieving an upper bound of $(1+o(1))\frac{d}{\log d}$ on the fractional chromatic number. Complementarily, by analyzing the uniform attachment random graph model, they construct a family of graphs whose fractional chromatic number reaches $(1-o(1))\frac{d}{\log d}$, thereby establishing a matching lower bound. This work not only confirms a conjectured upper bound but also strengthens the known lower bound, while demonstrating that the uniform attachment model circumvents the algorithmic complexity barriers commonly encountered in Erdős–Rényi random graphs, highlighting its distinctive structural advantages in extremal graph theory.

Abstract

Martinsson and Steiner recently proved that the fractional chromatic number of any $d$-degenerate triangle-free graph $G$ satisfies $χ_f(G) = O\left(\frac{d}{\log d}\right)$. They further conjectured a sharp leading constant $1 + o(1)$. In this paper, we confirm their upper bound conjecture for graphs having girth at least $5$. Our proof is constructive: it gives an efficient randomized algorithm that, with high probability, computes a fractional coloring of weight at most $(1 + o(1))\frac{d}{\log d}$ in such graphs. Furthermore, we establish their conjectured lower bound in a stronger form: for any constant $g \ge 4$, there exist $d$-degenerate graphs having girth at least $g$ with $χ_f(G) \ge (1 - o(1))\frac{d}{\log d}$. This lower bound is achieved by analyzing a random graph based on the uniform attachment model. Notably, our results reveal that this model lacks the typical computational complexity barriers found in Erdős-Rényi graphs, where there is a conjectured factor-$2$ algorithmic gap for this problem.

Problem

• fractional chromatic number
• d-degenerate graphs
• high girth
• sharp bounds
• random graph models

Innovation

• fractional chromatic number
• d-degenerate graphs
• high girth
• randomized algorithm
• uniform attachment model