Arc-Disjoint Cycles and Feedback Arc Sets

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 δ(D) arc-disjoint cycles. Mathematics Subject Classification: 05C20, 05C38.