The Fractional Chromatic Number of Triangle-free Graphs with $Δ\leq 3$

Abstract

Let $G$ be any triangle-free graph with maximum degree $Δ\leq 3$. Staton proved that the independence number of $G$ is at least 5/14n. Heckman and Thomas conjectured that Staton's result can be strengthened into a bound on the fractional chromatic number of $G$, namely $χ_f(G)\leq 14/5. Recently, Hatami and Zhu proved $χ_f(G) \leq 3 -{3/64}$. In this paper, we prove $χ_f(G) \leq 3- 3/43$.

Second revision: typos fixed, 24 pages, 37 figures