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

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 $\gamma(G) \geq \frac{15}{17}$ when considering graphs with odd girth at least $5$, distinct from the Petersen graph.
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https://hal.archives-ouvertes.fr/hal-00516702
Contributor : Jean-Marie Vanherpe <>
Submitted on : Sunday, February 27, 2011 - 11:46:56 AM
Last modification on : Thursday, February 7, 2019 - 4:46:16 PM
Long-term archiving on : Saturday, May 28, 2011 - 2:24:24 AM

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• HAL Id : hal-00516702, version 2
• ARXIV : 1102.5523

### Citation

Jean-Luc Fouquet, Jean-Marie Vanherpe. A new bound for parsimonious edge-colouring of graphs with maximum degree three.. 2011. ⟨hal-00516702v2⟩

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