Notes on Aharoni's rainbow cycle conjecture

TL;DR

This paper surveys Aharoni's rainbow cycle conjecture in edge-coloured graphs, discusses recent partial results, and introduces new bounds for the case where each colour class has size at least 3.

Contribution

It provides a comprehensive survey of the conjecture, reviews recent progress, and presents a new upper bound for rainbow cycles when each colour class has size at least 3.

Findings

Proved a new upper bound of (4n/9)+7 for rainbow cycles with colour classes of size at least 3.

Summarized recent partial results and related conjectures in the area.

Discussed potential generalizations to larger colour class sizes.

Abstract

In 2017, Ron Aharoni made the following conjecture about rainbow cycles in edge-coloured graphs: If $G$ is an $n$-vertex graph whose edges are coloured with $n$ colours and each colour class has size at least $r$, then $G$ contains a rainbow cycle of length at most $\lceil \frac{n}{r} \rceil$. One motivation for studying Aharoni's conjecture is that it is a strengthening of the Caccetta-H\"aggkvist conjecture on digraphs from 1978. In this article, we present a survey of Aharoni's conjecture, including many recent partial results and related conjectures. We also present two new results. Our main new result is for the $r=3$ case of Aharoni's conjecture. We prove that if $G$ is an $n$-vertex graph whose edges are coloured with $n$ colours and each colour class has size at least 3, then $G$ contains a rainbow cycle of length at most $\frac{4n}{9}+7$. We also discuss how our approach might generalise to larger values of $r$.