Strong edge-colouring of sparse planar graphs

Julien Bensmail (1) , Ararat Harutyunyan (2) , Hervé Hocquard (1) , Petru Valicov (3, 4)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 MI - Mathematical Institute [Oxford]
3 LIP - Laboratoire de l'Informatique du Parallélisme
4 MC2 - Modèles de calcul, Complexité, Combinatoire

A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching. It is known that every planar graph with maximum degree $\Delta$ has a strong edge-colouring with at most $4\Delta+4$ colours. We show that $3\Delta+1$ colours suffice if the graph has girth 6, and $4\Delta$ colours suffice if $\Delta\geq 7$ or the girth is at least 5. In the last part of the paper, we raise some questions related to a long-standing conjecture of Vizing on proper edge-colouring of planar graphs.

Mots clés: planar graphs; girth; proper edge-colouring; strong edge-colouring

Domaines: Mathématique discrète [cs.DM]

Julien Bensmail, Ararat Harutyunyan, Hervé Hocquard, Petru Valicov. Strong edge-colouring of sparse planar graphs. Discrete Applied Mathematics, 2014, ⟨10.1016/j.dam.2014.07.006⟩. ⟨hal-00932945v3⟩