Short rainbow cycles in graphs and matroids

TL;DR

This paper proves that in any edge-colored graph with certain conditions, a short rainbow cycle exists, and extends the result to cographic matroids but not binary matroids, contributing to conjectures in graph theory.

Contribution

The paper establishes the existence of short rainbow cycles in edge-colored graphs and extends the result to cographic matroids, advancing understanding of related conjectures.

Findings

Existence of rainbow cycle of length at most ⌈n/2⌉ in certain colored graphs

Extension of the main result to cographic matroids

Failure of the result for binary matroids

Abstract

Let $G$ be a simple $n$-vertex graph and $c$ be a colouring of $E(G)$ with $n$ colours, where each colour class has size at least $2$. We prove that $(G,c)$ contains a rainbow cycle of length at most $\lceil \frac{n}{2} \rceil$, which is best possible. Our result settles a special case of a strengthening of the Caccetta-H\"aggkvist conjecture, due to Aharoni. We also show that the matroid generalization of our main result also holds for cographic matroids, but fails for binary matroids.