Triangle-free $d$-degenerate graphs have small fractional chromatic number

Abstract

A well-known conjecture by Harris states that any triangle-free $d$-degenerate graph has fractional chromatic number at most $O\left(\frac{d}{\ln d}\right)$. This conjecture has gained much attention in recent years, and is known to have many interesting implications, including a conjecture by Esperet, Kang and Thomass\'e that any triangle-free graph with minimum degree $d$ contains a bipartite induced subgraph of minimum degree $\Omega(\log d)$. Despite this attention, Harris' conjecture has remained wide open with no known improvement on the trivial upper bound, until now. In this article, we give an elegant proof of Harris' conjecture. In particular, we show that any triangle-free $d$-degenerate graph has fractional chromatic number at most $(4+o(1))\frac{d}{\ln d}.$ The conjecture of Esperet et al. follows as a direct consequence. We also prove a more general result, showing that for any triangle-free graph $G$, there exists a random independent set in which each vertex $v$ is included with probability $\Omega(p(v))$, where $p:V(G)\rightarrow [0,1]$ is any function that satisfies a natural condition.