Progress on the adjacent vertex distinguishing edge colouring conjecture

Abstract

A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree $\hat{I}”$ and no isolated edge has an adjacent vertex distinguishing edge colouring with $\hat{I}”+300$ colours, provided $\hat{I}”$ is large enough. We show that this bound can be reduced to $\hat{I}”+19$. This is motivated by the conjecture of Zhang, Liu, and Wang (2002) that $\hat{I}”+2$ colours are enough for $\hat{I}”\ge 3$.

v2: Revised following referees' comments