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提出对Caccetta-H\"{a}ggkvist 推测作如下概括:如果G$是一个简单的美元-外缘色色图,其彩虹周期以美元为单位,其彩虹周期以美元计,其数额至少为$/r\r\ceil$为单位。在本文中,我们证明,对于固定美元而言,Aharoni的猜想维持在一个添加值。具体地说,我们显示,对于每个固定的$\geq 1美元,存在一个固定的$_c_r$,如果G$是一个简单的美元-外缘色色图,其数额以美元为单位,其数额至少为$,那么$的彩虹周期以美元/r+c_r$为单位。