Fractional balanced chromatic number and arboricity of planar (signed) graphs

TL;DR

This paper introduces a counterexample to a conjecture by demonstrating a planar signed graph with a fractional balanced chromatic number exceeding 2, and explores bounds related to fractional arboricity.

Contribution

It provides the first known example of a planar signed graph with fractional balanced chromatic number greater than 2, refuting previous conjectures and establishing new bounds.

Findings

Counterexample with fractional balanced chromatic number > 2

Lower bound of 83/41 for planar signed graphs

Sequence of planar graphs with fractional arboricity approaching 2 + 2/25

Abstract

A fractional coloring of a signed graph $(G, {\sigma})$ is an assignment of nonnegative weights to the balanced sets (sets which do not induce a negative cycle) such that each vertex has an accumulated weight of at least 1. The minimum total wight among all such colorings is defined to be the fractional balanced chromatic number, denoted by $\chi-{fb}(G, {\sigma})$. This value is clearly upper bounded by the fractional arboricity of $G$, denoted $a_f (G)$, where weights are assigned to sets inducing no cycle rather than sets inducing no negative cycle. In this work we present an example of a planar signed simple graph of fractional balanced chromatic number larger than 2, thus in particular refuting a conjecture of Bonamy, Kardos, Kelly, and Postle suggesting that the fractional arboricity of planar graphs is bounded above by 2. By iterating the construction, we show that the supremum of the fractional balanced chromatic number of planar signed simple graphs is at least as $83/41 = 2 + 1/41$. With similar operations, we built a sequence of planar graphs whose limit of fractional arboricity is $a_f (G) = 2 + 2/25$.