Vertex‐disjoint cycles of the same length in tournaments

Abstract

A tournament is a digraph which has exactly one arc between any two vertices. Let k and q be two positive integers with ≥ q 4. A q-cycle is a cycle of length q. We prove that for any real number ∕ α q > 2, there exists a constant k α such that for every ≥ k k α , any tournament T with minimum outdegree at least αk contains k disjoint q-cycles. This generalizes a result by Bang-Jensen, Bessy, and Thomassé for disjoint 3-cycles. The linear factor ∕ q 2 of minimum outdegree is the best possible.to denote the minimum outdegree of D. A number of digraphs are said to be vertex-disjoint (briefly disjoint) if no two of them have any common vertex. Similarly, a number of digraphs are called arc-disjoint if they have no common arc.It is NP-complete to decide whether a digraph has k disjoint cycles for an integer k in [2]. It turns out that we have to seek sufficient conditions to guarantee the existence of disjoint cycles in a digraph. The following beautiful conjecture of Bermond and Thomassen states that a digraph with large minimum outdegree must have many disjoint cycles.Conjecture 1 (Bermond and Thomassen [5]). For a given integerObserve that the complete digraph of order k 2 − 1 (with all possible arcs) shows that this minimum outdegree condition is the best possible. The conjecture is trivial for k = 1. It was