The famous Ryser--Brualdi--Stein conjecture asserts that every $n \times n$ Latin square contains a partial transversal of size $n-1$. Since its appearance, the conjecture has attracted significant interest, leading to several generalizations. One of the most notable extensions is to matroid intersection given by Aharoni, Kotlar, and Ziv, focusing on the existence of a common independent transversal of the common independent sets of two matroids. In this paper, we study a special case of this setting, the Rainbow Arborescence Conjecture, which states that any graph on $n$ vertices formed by the union of $n-1$ spanning arborescences contains an arborescence using exactly one arc from each. We prove that the computational problem of testing the existence of such an arborescence with a fixed root is NP-complete, verify the conjecture in several cases, and explore relaxed versions of the problem.

翻译：著名的Ryser--Brualdi--Stein猜想断言：每个$n \times n$拉丁方都包含一个大小为$n-1$的部分横截。自该猜想提出以来，它引起了广泛关注，并催生了若干推广。其中最重要的扩展之一是由Aharoni、Kotlar和Ziv提出的拟阵交理论框架，该框架聚焦于两个拟阵的公共独立集是否存在公共独立横截。本文研究该框架下的一个特例——彩虹树状图猜想，该猜想断言：由$n-1$棵生成树状图构成的$n$顶点图中，必存在一棵树状图，其每条弧恰好取自各生成树状图的一条弧。我们证明了以固定根检测此类树状图存在性的计算问题是NP完全的，验证了该猜想在若干情形下的正确性，并探讨了该问题的松弛版本。