Rainbow cycles for families of matchings

TL;DR

This paper investigates the minimum length of rainbow cycles in edge-colored graphs with matchings, showing that the rainbow girth is logarithmic in the number of matchings, which advances understanding of a generalized Caccetta-H"aggkvist conjecture.

Contribution

It proves that for graphs with matchings of size at least 2, the rainbow girth is bounded by a logarithmic function of the number of matchings, improving previous bounds.

Findings

Rainbow girth of n matchings is O(log n)

Supports conjecture that rainbow girth can be smaller with matching color classes

Extends understanding of rainbow cycles in edge-colored graphs

Abstract

Given a graph $G$ and a coloring of its edges, a subgraph of $G$ is called rainbow if its edges have distinct colors. The rainbow girth of an edge coloring of G is the minimum length of a rainbow cycle in G. A generalization of the famous Caccetta-H\"aggkvist conjecture, proposed by the first author, is that if in an coloring of the edge set of an $n$-vertex graph by $n$ colors, in which each color class is of size $k$, the rainbow girth is at most $\lceil \frac{n}{k} \rceil$. In the known examples for sharpness of this conjecture the color classes are stars, suggesting that when the color classes are matchings, the result may be improved. We show that the rainbow girth of $n$ matchings of size at least 2 is $O(\log n)$.