Vertex-disjoint cycles in local tournaments

Abstract

Let r be a positive integer. The Bermond–Thomassen conjecture states that, a digraph of minimum out-degree at least 2r−1 contains r vertex-disjoint directed cycles. A digraph D is called a local tournament if for every vertex x of D, both the out-neighbours and the in-neighbours of x induce tournaments. Note that tournaments form the subclass of local tournaments. In this paper, we verify that the Bermond–Thomassen conjecture holds for local tournaments. In particular, we prove that every local tournament D with δ+(D)≥2r−1 contains r disjoint cycles C1,C2,…,Cr, satisfying that either Ci has the length at most 4 or is a shortest cycle of the original digraph of D−C1−…−Ci−1 for 1≤i≤r.