Progress on the adjacent vertex distinguishing edge colouring conjecture

TL;DR

This paper improves the upper bound on the number of colours needed for adjacent vertex distinguishing edge colouring of graphs, reducing it from 0 to , bringing it closer to the conjectured .

Contribution

The authors significantly lowered the upper bound for the edge colouring problem using the Local Lemma, advancing towards the conjectured optimal number of colours.

Findings

Bound reduced from 0 to  colours.

Applicable to graphs with large maximum degree .

Supports the conjecture that  colours suffice for  graphs.

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 $\Delta$ and no isolated edge has an adjacent vertex distinguishing edge colouring with $\Delta + 300$ colours, provided $\Delta$ is large enough. We show that this bound can be reduced to $\Delta + 19$. This is motivated by the conjecture of Zhang, Liu, and Wang (2002) that $\Delta + 2$ colours are enough for $\Delta \geq 3$.