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

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 $n=\mathrm{\mid }V(G)\mathrm{\mid }$ such that $\delta^+(v) \ge k$forall$v \in V(G)$,then$G$ contains a directed cycle of length at most $\lceil n\mathrm{/}k\rceil$. 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 $\lceil n\mathrm{/}k\rceil$. With Pelik'anov'a and Pokorn'a, we showed that this conjecture is true if each color class has size $\mathrm{\Omega }(k\log k)$. In this paper, we present a proof of the conjecture if each color class has size $\mathrm{\Omega }(k)$, 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.