Fractional colorings of cubic graphs with large girth

TL;DR

This paper proves that large-girth cubic graphs have fractional chromatic numbers at most 2.2978, leading to improved bounds on independent set sizes and maximum cuts, with implications for random cubic graphs.

Contribution

It establishes a new upper bound on the fractional chromatic number of large-girth cubic graphs, enhancing understanding of their coloring and independence properties.

Findings

Fractional chromatic number at most 2.2978 for large-girth cubic graphs

Independent set size at least 0.4352n in such graphs

Improved lower bounds on maximum cut in large-girth cubic graphs

Abstract

We show that every (sub)cubic n-vertex graph with sufficiently large girth has fractional chromatic number at most 2.2978 which implies that it contains an independent set of size at least 0.4352n. Our bound on the independence number is valid to random cubic graphs as well as it improves existing lower bounds on the maximum cut in cubic graphs with large girth.