Let $G$ be a simple $n$-vertex graph and $c$ be a colouring of $E(G)$ with $n$ colours, where each colour class has size at least $2$. We prove that $(G,c)$ contains a rainbow cycle of length at most $\lceil \frac{n}{2} \rceil$, which is best possible. Our result settles a special case of a strengthening of the Caccetta-Haggkvist conjecture, due to Aharoni. We also show that the matroid generalization of our main result also holds for cographic matroids, but fails for binary matroids.

中文翻译：

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图和拟阵中的短彩虹周期

令$G$ 是一个简单的$n$-顶点图，$c$ 是$E(G)$ 的着色，具有$n$ 种颜色，其中每个颜色类别的大小至少为$2$。我们证明 $(G,c)$ 包含一个最长为 $\lceil \frac{n}{2} \rceil$ 的彩虹圈，这是最好的。由于 Aharoni，我们的结果解决了 Caccetta-Haggkvist 猜想加强的特殊情况。我们还表明，我们的主要结果的拟阵推广也适用于共形拟阵，但不适用于二元拟阵。