On k-intersection edge colourings

We propose the following problem. For some k ≥ 1, a graph G is to be properly edge coloured such that any two adjacent vertices share at most k colours. We call this the k-intersection edge colouring. The minimum number of colours sufficient to guarantee such a colouring is the k-intersection chromatic index and is denoted χk(G). Let fk be defined by fk(∆) = max G:∆(G)=∆ {χk(G)}. We show that fk(∆) = Θ( 2 k ). We also discuss some open problems.