In 2017, Aharoni proposed the following generalization of the Caccetta-H\"{a}ggkvist conjecture: if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $\lceil n/r \rceil$. In this paper, we prove that, for fixed $r$, Aharoni's conjecture holds up to an additive constant. Specifically, we show that for each fixed $r \geq 1$, there exists a constant $c_r$ such that if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $n/r + c_r$.

翻译：2017年，Aharoni提出了以下对Caccetta–Häggkvist猜想的推广：若$G$是一个简单的$n$顶点边染色图，且其$n$个色类的大小均至少为$r$，则$G$包含一条长度至多为$\lceil n/r \rceil$的彩虹圈。本文证明，对于固定的$r$，Aharoni的猜想在加法常数意义下成立。具体而言，我们证明对每个固定的$r \geq 1$，存在常数$c_r$使得：若$G$是一个简单的$n$顶点边染色图，且其$n$个色类的大小均至少为$r$，则$G$包含一条长度至多为$n/r + c_r$的彩虹圈。