# On parsimonious edge-colouring of graphs with maximum degree three

Abstract : In a graph $G$ of maximum degree $\Delta$ let $\gamma$ denote the largest fraction of edges that can be $\Delta$ edge-coloured. Albertson and Haas showed that $\gamma \geq \frac{13}{15}$ when $G$ is cubic . We show here that this result can be extended to graphs with maximum degree $3$ with the exception of a graph on $5$ vertices. Moreover, there are exactly two graphs with maximum degree $3$ (one being obviously the Petersen graph) for which $\gamma = \frac{13}{15}$. This extends a result given by Steffen. These results are obtained by using structural properties of the so called $\delta$-minimum edge colourings for graphs with maximum degree $3$.\\ {\bf Keywords :} Cubic graph; Edge-colouring; \noindent {\bf Mathematics Subject Classification (2010) :} 05C15.
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Submitted on : Saturday, January 28, 2012 - 10:28:31 AM
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Jean-Luc Fouquet, Jean-Marie Vanherpe. On parsimonious edge-colouring of graphs with maximum degree three. Graphs and Combinatorics, Springer Verlag, 2013, 29 (3), pp.475-487. ⟨10.1007/s00373-012-1145-3⟩. ⟨hal-00325253v6⟩

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