Rainbow Arborescence Conjecture

TL;DR

This paper introduces the Rainbow Arborescence Conjecture, a graph theory problem inspired by Latin square transversals, and provides partial results including NP-completeness, special case verifications, and solutions for cycle graphs.

Contribution

The paper formulates the Rainbow Arborescence Conjecture and proves several partial results, including NP-completeness and special case verifications, advancing understanding of arborescence structures.

Findings

NP-complete to test the conjecture with a fixed root

Verified the conjecture for cycle graphs

Established new results on systems of distinct representatives for cycle intervals

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.