An upper bound on the fractional chromatic number of triangle-free subcubic graphs

TL;DR

This paper establishes a new upper bound of approximately 2.867 on the fractional chromatic number for triangle-free subcubic graphs, improving previous bounds and supporting the conjecture of Heckman and Thomas.

Contribution

The paper proves a tighter upper bound of 43/15 on the fractional chromatic number for the class of triangle-free subcubic graphs, advancing understanding of their coloring properties.

Findings

Proved that the fractional chromatic number is at most 43/15 (~2.867).

Improved previous bounds from 32/11 (~2.909) to 43/15.

Supports the conjecture that the bound is at most 2.8 for these graphs.

Abstract

An $(a:b)$-coloring of a graph $G$ is a function $f$ which maps the vertices of $G$ into $b$-element subsets of some set of size $a$ in such a way that $f(u)$ is disjoint from $f(v)$ for every two adjacent vertices $u$ and $v$ in $G$. The fractional chromatic number $\chi_f(G)$ is the infimum of $a/b$ over all pairs of positive integers $a,b$ such that $G$ has an $(a:b)$-coloring. Heckman and Thomas conjectured that the fractional chromatic number of every triangle-free graph $G$ of maximum degree at most three is at most 2.8. Hatami and Zhu proved that $\chi_f(G) \leq 3-3/64 \approx 2.953$. Lu and Peng improved the bound to $\chi_f(G) \leq 3-3/43 \approx 2.930$. Recently, Ferguson, Kaiser and Kr\'{a}l' proved that $\chi_f(G) \leq 32/11 \approx 2.909$. In this paper, we prove that $\chi_f(G) \leq 43/15 \approx 2.867$.