Strong edge-colouring and induced matchings

• Strong edge-colouring is NP-complete for subcubic planar graphs.
• Maximum induced matching problem is NP-complete for subcubic planar graphs.
• Tight upper bound for the strong chromatic index of outerplanar graphs.

A strong edge-colouring of a graph G is a proper edge-colouring such that every path of three edges uses three colours. An induced matching of a graph G   is a subset II of edges of G   such that the graph induced by the endpoints of II is a matching. In this paper, we prove the NP-completeness of strong 4-, 5-, and 6-edge-colouring and maximum induced matching in some subclasses of subcubic triangle-free planar graphs. We also obtain a tight upper bound for the minimum number of colours in a strong edge-colouring of outerplanar graphs as a function of the maximum degree.