Further approximations for Aharoni's rainbow generalization of the Caccetta-H\"{a}ggkvist conjecture

TL;DR

This paper proves a near-optimal approximation for Aharoni's rainbow generalization of the Caccetta-H"{a}ggkvist conjecture, showing that large enough color classes guarantee a short rainbow cycle, advancing understanding of rainbow cycle existence.

Contribution

The authors improve previous bounds by proving the conjecture holds when each color class has size proportional to k, reducing the gap to the original conjecture.

Findings

The conjecture is true if each color class has size proportional to k.

Improved bounds bring the result closer to Aharoni's original conjecture.

The paper explores the effect of relaxing the number of colors condition.

Abstract

For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge k$ for all $v \in V(G)$, then $G$ contains a directed cycle of length at most $\lceil n/k \rceil$. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. With Pelik\'anov\'a and Pokorn\'a, we showed that this conjecture is true if each color class has size ${\Omega}(k\log k)$. In this paper, we present a proof of the conjecture if each color class has size ${\Omega}(k)$, which improved the previous result and is only a constant factor away from Aharoni's conjecture. We also consider what happens when the condition on the number of colors is relaxed.