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Rainbow Arborescence Conjecture

Year: 2024
Type: preprint
Abstract: 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.
DOI: 10.48550/arxiv.2412.15457
Language: English
Open Access status: green
Awards: マン・マシン・システムとしてのCAI研究, Research on Integrated Techniques of Enumeration and Optimization Based on Discrete Structure Manipulation Systems, Exploration into Matroid Common Base Packing Problem, G4171720266, G4183452178, Integrated Techniques and Applications of Enumeration, Optimization, and SAT-Solving Based on Discrete Structure Manipulation, G660829768, G721023497, G8058665615, G925228835
OpenAlex ID: W4405714359
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