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In a graph $G$ of maximum degree $\Delta$ let $\gamma$ denote the largestfraction of edges that can be $\Delta$ edge-coloured. Albertson and Haas showedthat $\gamma \geq 13/15$ when $G$ is cubic . We show here that this result canbe extended to graphs with maximum degree 3 with the exception of a graph on 5vertices. Moreover, there are exactly two graphs with maximum degree 3 (onebeing obviously the Petersen graph) for which $\gamma = 13/15$. This extends aresult given by Steffen. These results are obtained by using structuralproperties of the so called $\delta$-minimum edge colourings for graphs withmaximum degree 3. Keywords : Cubic graph; Edge-colouring

On parsimonious edge-colouring of graphs with maximum degree three