Neighbour-Distinguishing Edge Colourings of Random Regular Graphs

A proper edge colouring of a graph is neighbour-distinguishing if for all pairs of adjacent vertices v, w the set of colours appearing on the edges incident with v is not equal to the set of colours appearing on the edges incident with w. Let ndi(G) be the least number of colours required for a proper neighbour-distinguishing edge colouring of G. We prove that for d ≥ 4, a random d-regular graph G on n vertices asymptotically almost surely satisfies ndi(G) ≤ d3d/2e. This verifies a conjecture of Zhang, Liu and Wang for almost all 4-regular graphs.