Fractional coloring of planar graphs of girth five

TL;DR

This paper investigates fractional coloring of planar graphs with girth five, establishing new bounds and coloring schemes that improve understanding of their fractional chromatic number.

Contribution

It introduces a novel fractional coloring method for triangle-free planar graphs and extends results to graphs with girth five and bounded degree.

Findings

Every triangle-free planar graph has a specific fractional coloring with subsets of size 6.

Every such graph on n vertices is (6n:2n+1)-colorable.

Planar graphs with girth at least five and bounded degree have fractional chromatic number at most 3-3/(2M_Delta+1).

Abstract

A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex x of G, the graph G has a set coloring phi by subsets of {1,...,6} such that |phi(v)|>=2 for each vertex v of G and |phi(x)|=3. As a corollary, every triangle-free planar graph on n vertices is (6n:2n+1)-colorable. We further use this result to prove that for every Delta, there exists a constant M_Delta such that every planar graph G of girth at least five and maximum degree Delta is (6M_Delta:2M_Delta+1)-colorable. Consequently, planar graphs of girth at least five with bounded maximum degree Delta have fractional chromatic number at most 3-3/(2M_Delta+1).