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

TL;DR

This paper improves the upper bound on the fractional chromatic number of triangle-free graphs with maximum degree at most 3, advancing understanding of graph coloring properties in this class.

Contribution

The paper establishes a tighter upper bound of 3 - 3/43 on the fractional chromatic number for these graphs, improving previous bounds.

Findings

Proved $oxed{ ext{ } oxed{rac{14}{5}}}$ as an upper bound for fractional chromatic number.

Enhanced the bound from 3 - 3/64 to 3 - 3/43.

Contributed to the theory of graph coloring in triangle-free graphs with bounded degree.

Abstract

Let $G$ be any triangle-free graph with maximum degree $\Delta\leq 3$. 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 $\chi_f(G)\leq 14/5. Recently, Hatami and Zhu proved $\chi_f(G) \leq 3 -{3/64}$. In this paper, we prove $\chi_f(G) \leq 3- 3/43$.