On strong edge-colouring of subcubic graphs
Résumé
A strong edge-colouring of a graph $G$ is a proper edge-colouring such that every path of length 3 uses three different colours. In this paper we improve some previous results on the strong edge-colouring of subcubic graphs by showing that every subcubic graph with maximum average degree strictly less than $\frac{7}{3}$ (resp. $\frac{5}{2}$, $\frac{8}{3}$, $\frac{20}{7}$) can be strong edge-coloured with six (resp. seven, eight, nine) colours. These upper bounds are optimal except the one of $\frac{8}{3}$. Also, we prove that every subcubic planar graph without $4$-cycles and $5$-cycles can be strong edge-coloured with nine colours.
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