Rainbow Arborescence Conjecture

TL;DR

This paper investigates the Rainbow Arborescence Conjecture, a rainbow- transversal analogue for unions of spanning arborescences. It establishes NP-completeness for finding a fixed-root rainbow arborescence, proves the conjecture in several structural special cases, and analyzes relaxations of the problem. A central achievement is a nontrivial proof that the conjecture holds when the underlying undirected graph is a cycle, with additional corollaries for systems of distinct representatives on cycle intervals. The results advance understanding of rainbow structures in matroid-like settings and open avenues for further algorithmic and combinatorial exploration.

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.

Figures (13)

• Figure 1: Illustration of the definition of $X_\ell$ and $\overline{X_\ell}$.
• Figure 2: The three possible configurations of the first and last arcs of $X_i$ and $Y_{\ell}$ (in bold). In all cases, no spanning arborescence can be disjoint from all four of them.
• Figure 3: Possible relative positions of $\overline{Z_{\ell}}$ and $\overline{X_{\ell'}}$.
• Figure 4: Two easy cases where $u(X_i)$ and $u(Z_{\ell})$ have at most one edge in common.
• Figure 5: Possible configurations of $\overline{X_{\ell}}$ and $\overline{Z_i}$ in Case 1.

Theorems & Definitions (62)

• Conjecture 1.1: Rainbow Arborescence Conjecture
• Conjecture 1.2
• Conjecture 1.3
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Theorem 3.3