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In a graph $G$ of maximum degree 3, let $\gamma(G)$ denote the largestfraction 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 $g_{odd}(G)$ isthe odd girth of $G$, when $G$ is triangle-free. In \cite{FouVan10a} weextended 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.

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