The Rainbow Arborescence Problem on Cycles

TL;DR

This paper proves the rainbow arborescence conjecture for the special case where the underlying undirected graph is a cycle, confirming the existence of a spanning arborescence with exactly one arc of each color.

Contribution

It establishes the conjecture's validity specifically for cycle graphs, a significant step in understanding the broader problem.

Findings

The conjecture holds for cycle graphs.

A spanning arborescence with one arc of each color exists in this case.

The proof confirms the conjecture's applicability to this class of graphs.

Abstract

The rainbow arborescence conjecture posits that if the arcs of a directed graph with $n$ vertices are colored by $n-1$ colors such that each color class forms a spanning arborescence, then there is a spanning arborescence that contains exactly one arc of every color. We prove that the conjecture is true if the underlying undirected graph is a cycle.