A proper edge-colouring of a graph G is an assignment of colours to the edges of G such that two distinct edges sharing a vertex are coloured with distinct colours. The strong edge-colouring of G is a proper edge-colouring of G such that two edges joined by another edge are coloured with distinct colours. A k-intersection edge-colouring of G is a proper edge-colouring such that every pair of adjacent vertices u,v share at most k colours. Observe that a 1-intersection edge-colouring is exactly the strong edge-colouring.

The first part of the talk will be focused on an overview of the notion of strong edge-colouring. The complexity, few conjectures and some results related to this notion will be presented. In the second part we discuss the k-intersection edge-colouring. We show that the associated decision problem is NP-complete for every k, within a very restricted setting. We also prove upper and lower bounds for minimum number of colours needed for a k-intersection edge-colouring, for various classes of graphs. This is a part of a series of joint works with various sets of authours from Bordeaux, Montpellier, Orsay, India and Japan.