The famous Ryser--Brualdi--Stein conjecture asserts that every $k \times k$ Latin square contains a partial transversal of size $k-1$. Since its appearance, the conjecture has attracted significant interest, leading to several proposed generalizations. One of the most notable of these, by Aharoni, Kotlar, and Ziv, conjectures that $k$ disjoint common bases of two matroids of rank $k$ have a common independent partial transversal of size $k-1$. Although simple counterexamples show that the size $k-1$ above cannot be improved to $k$ (i.e., a transversal instead of a partial transversal), it is remarkable that no such counterexample is known for the special case of spanning arborescences. This motivated the formulation of the Rainbow Arborescence Conjecture: 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 several partial results on this conjecture. We show that the computational problem of testing the existence of such an arborescence with a fixed root is NP-complete, verify the conjecture in several special cases, and study relaxations of the problem. In particular, we establish the validity of the conjecture when the underlying undirected graph is a cycle; this also yields a new result on systems of distinct representatives for intervals on a cycle.

翻译：著名的Ryser--Brualdi--Stein猜想断言，每个$k \times k$拉丁方均包含一个大小为$k-1$的部分横截。自该猜想提出以来，它引起了广泛关注，并催生了若干推广形式。其中最为显著的推广之一由Aharoni、Kotlar和Ziv提出，他们猜想两个秩为$k$的拟阵的$k$个不相交公共基集，存在一个大小为$k-1$的公共独立部分横截。尽管简单的反例表明上述$k-1$的规模无法提升至$k$（即无法保证存在完整横截而非部分横截），但值得注意的是，在生成树状图的特殊情形中尚未发现此类反例。这促使了彩虹树状图猜想的提出：由$n-1$个生成树状图构成的$n$顶点图中，必存在一个树状图，其每条弧恰好取自其中一个生成树状图。我们针对该猜想证明了若干部分结果。我们证明了测试固定根节点下此类树状图存在性的计算问题是NP完全的，验证了猜想在多种特殊情形下的成立性，并研究了该问题的松弛形式。特别地，我们证明了当底层无向图为环时猜想成立；这也为环上区间系统的相异代表系问题提供了新结论。