A new bound for parsimonious edge-colouring of graphs with maximum degree three

TL;DR

This paper establishes a new lower bound on the fraction of edges that can be 3-edge-colored in graphs with maximum degree three, improving previous bounds and providing specific results for graphs with certain odd girth values.

Contribution

The paper introduces a tighter bound for the maximum fraction of 3-edge-colorable edges in degree-3 graphs, extending prior results to include graphs with odd girth at least 5.

Findings

New bound: γ(G) ≥ 1 - 2 / (3 g_odd(G) + 2)

γ(G) ≥ 15/17 for graphs with odd girth ≥ 5

Improves previous bounds for triangle-free graphs

Abstract

In a graph $G$ of maximum degree 3, let $\gamma(G)$ denote the largest fraction of edges that can be 3 edge-coloured. Rizzi \cite{Riz09} showed that $\gamma(G) \geq 1-\frac{2\strut}{\strut 3 g_{odd}(G)}$ where $g_{odd}(G)$ is the odd girth of $G$, when $G$ is triangle-free. In \cite{FouVan10a} we extended that result to graph with maximum degree 3. We show here that $\gamma(G) \geq 1-\frac{2 \strut}{\strut 3 g_{odd}(G)+2}$, which leads to $\gamma(G) \geq 15/17$ when considering graphs with odd girth at least 5, distinct from the Petersen graph.