Strong edge-colouring of sparse planar graphs

Abstract

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 $Δ$ has a strong edge-colouring with at most $4Δ+4$ colours. We show that $3Δ+1$ colours suffice if the graph has girth 6, and $4Δ$ colours suffice if $Δ\geq 7$ 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.