# Fractional chromatic number, maximum degree and girth

## TL;DR

This paper develops new bounds for the fractional chromatic number of graphs with local constraints, especially focusing on triangle-free graphs and graphs with large girth, providing both upper and lower bounds in various regimes.

## Contribution

Introduces a novel method for bounding fractional chromatic number and independence number, deriving explicit bounds for graphs with given maximum degree and girth.

## Key findings

- Established a general upper bound for fractional chromatic number of triangle-free graphs.
- Derived explicit bounds for graphs with girth at least 7 and various maximum degrees.
- Provided new lower bounds on independence ratios for specific degree and girth combinations.

## Abstract

We introduce a new method for computing bounds on the independence number and fractional chromatic number of classes of graphs with local constraints, and apply this method in various scenarios. We establish a formula that generates a general upper bound for the fractional chromatic number of triangle-free graphs of maximum degree~$\Delta \ge 3$. This upper bound matches that deduced from the fractional version of Reed's bound for small values of~$\Delta$, and improves it when~$\Delta\ge 17$, transitioning smoothly to the best possible asymptotic regime, barring a breakthrough in Ramsey theory. Focusing on smaller values of~$\Delta$, we also demonstrate that every graph of girth at least~$7$ and maximum degree~$\Delta$ has fractional chromatic number at most~$1+ \min_{k \in \mathbb{N}} \frac{2\Delta + 2^{k-3}}{k}$. In particular, the fractional chromatic number of a graph of girth~$7$ and maximum degree~$\Delta$ is at most~$\frac{2\Delta+9}{5}$ when~$\Delta \in [3,8]$, at most~$\frac{\Delta+7}{3}$ when~$\Delta \in [8,20]$, at most~$\frac{2\Delta+23}{7}$ when~$\Delta \in [20,48]$, and at most~$\frac{\Delta}{4}+5$ when~$\Delta \in [48,112]$. In addition, we also obtain new lower bounds on the independence ratio of graphs of maximum degree~$\Delta \in \{3,4,5\}$ and girth~$g\in \{6,\dotsc,12\}$, notably~$1/3$ when~$(\Delta,g)=(4,10)$ and~$2/7$ when~$(\Delta,g)=(5,8)$.