On parcimonious edge-colouring of graphs with maximum degree three
Résumé
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 showned that $\gamma \geq \frac{13}{15}$ when $G$ is cubic \cite{AlbHaa}. 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 Petersen's graph) for which $\gamma = \frac{13}{15}$. This extends a result given in \cite{Ste04}. These results are obtained in giving structural properties of the so called $\delta-$minimum edge colourings for graphs with maximum degree $3$
Domaines
Mathématique discrète [cs.DM]
Origine : Fichiers produits par l'(les) auteur(s)