Short rainbow cycles for families of small edge sets

TL;DR

This paper extends results on rainbow cycles in edge-colored graphs, showing that even a small fraction of non-star color classes guarantees a logarithmic rainbow girth, and identifies the threshold for this transition.

Contribution

It proves that a small fraction of non-star color classes suffices for logarithmic rainbow girth and determines the transition threshold from linear to logarithmic girth.

Findings

Small fraction of non-star classes ensures logarithmic rainbow girth

Logarithmic bound is tight and optimal

Threshold fraction for girth transition identified

Abstract

In 2019, Aharoni proposed a conjecture generalizing the Caceetta-H\"aggkvist conjecture: if an $n$-vertex graph $G$ admits an edge coloring (not necessarily proper) with $n$ colors such that each color class has size at least $r$, then $G$ contains a rainbow cycle of length at most $\lceil n/r\rceil$. Recent works \cite{AG2023,ABCGZ2023,G2025} have shown that if a constant fraction of the color classes are non-star, then the rainbow girth is $O(\log n)$. In this note, we extend these results, and we show that even a small fraction of non-star color classes suffices to ensure logarithmic rainbow girth. We also prove that the logarithmic bound is of the right order of magnitude. Moreover, we determine the threshold fraction between the types of color classes at which the rainbow girth transitions from linear to logarithmic.