# Asymptotically optimal bound on the adjacent vertex distinguishing edge choice number

## TL;DR

This paper establishes an asymptotically optimal upper bound on the list size needed for adjacent vertex distinguishing edge colourings in graphs, using probabilistic methods to improve understanding of graph colourings.

## Contribution

It provides the first asymptotically tight bound on list sizes for adjacent vertex distinguishing edge colourings, extending to total colourings.

## Key findings

- List size bound is at least \u00b5()^4 for proper edge colourings.
- Bound applies to total colourings as well.
- Proof uses probabilistic techniques.

## Abstract

An adjacent vertex distinguishing edge colouring of a graph $G$ without isolated edges is its proper edge colouring such that no pair of adjacent vertices meets the same set of colours in $G$. We show that such colouring can be chosen from any set of lists associated to the edges of $G$ as long as the size of every list is at least $\Delta+C\Delta^{\frac{1}{2}}(\log\Delta)^4$, where $\Delta$ is the maximum degree of $G$ and $C$ is a constant. The proof is probabilistic. The same is true in the environment of total colourings.