Arc-Disjoint Cycles and Feedback Arc Sets

TL;DR

This paper investigates a problem in tournament graphs related to feedback arc sets and arc-disjoint cycles, providing a negative answer to a posed question and exploring cycle properties in oriented graphs.

Contribution

It disproves Isaak's conjecture about the equality of maximum arc-disjoint cycles and minimum feedback arc set size in certain tournaments.

Findings

The maximum number of arc-disjoint cycles can be less than the minimum feedback arc set size.

Vertices with maximum out-degree and adjacency to all others are contained in a specific number of arc-disjoint cycles.

Abstract

Isaak posed the following problem. Suppose $T$ is a tournament having a minimum feedback arc set which induces an acyclic digraph with a hamiltonian path. Is it true that the maximum number of arc-disjoint cycles in $T$ equals the cardinality of minimum feedback arc set of $T$? We prove that the answer to the problem is in the negative. Further, we study the number of arc-disjoint cycles through a vertex $v$ of the minimum out-degree in an oriented graph $D$. We prove that if $v$ is adjacent to all other vertices, then $v$ belongs to $\delta^+(D)$ arc-disjoint cycles.