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

TL;DR

This paper proves a generalized rainbow cycle conjecture for edge-colored graphs with sufficiently large color classes, extending the classical Caccetta-H"{a}ggkvist conjecture to a rainbow setting.

Contribution

It establishes the conjecture for graphs where each color class has size proportional to k log k, a significant advancement in rainbow cycle theory.

Findings

Proves the rainbow cycle conjecture for color classes of size Ω(k log k)

Extends the classical Caccetta-H"{a}ggkvist conjecture to rainbow graphs

Provides bounds on color class size for guaranteed rainbow cycles

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 $\lceil n/k \rceil$. In [2], Aharoni proposes 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 $\lceil n/k \rceil$. In this paper, we prove this conjecture if each color class has size ${\Omega}(k \log k)$.