Acyclic edge colourings of graphs with large girth

TL;DR

This paper proves that graphs with sufficiently large girth can be acyclically edge coloured using nearly optimal number of colours proportional to their maximum degree, improving understanding of colourings in sparse graphs.

Contribution

It establishes a bound on acyclic edge colourings for graphs with large girth, showing they can be coloured with just over the maximum degree in colours.

Findings

Acyclic edge colouring with (1+ε)Δ colours for large girth graphs.

Existence of a girth threshold g(ε) for such colourings.

Extension of colouring theory to sparse graphs with large cycles.

Abstract

An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\varepsilon>0$, there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then $G$ admits an acyclic edge colouring with at most $(1+\varepsilon)\Delta$ colours.