Note on disjoint cycles in multipartite tournaments

In 1981, Bermond and Thomassen conjectured that for any positive integer k, every digraph with minimum out-degree at least 2k - 1 admits k vertex-disjoint directed cycles. In this short paper, we verify the Bermond-Thomassen conjecture for trianglefree multipartite tournaments and 3-partite tournaments. Furthermore, we characterize 3-partite tournaments with minimum out-degree at least 2k - 1 (k >= 2) such that in each set of k vertex-disjoint directed cycles, every cycle has the same length. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.