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

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.