2024/03/06 by Yandong Bai, Bai, Yandong, Wenpei Jia +1
Decision Sciences · Economics, Econometrics and Finance · #Combinatorics (math.CO) #FOS: Mathematics #Game Theory and Applications #Game Theory and Voting Systems #Sports Analytics and Performance
paper · pdf · doi:10.48550/arxiv.2403.03692
openalex publication_date 2024/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least 2k-1 contains k vertex-disjoint cycles,here k is a positive integer. Lichiardopol conjectured in 2014 that for every positive integer k there exists an integer g(k) such that every digraph with minimum outdegree at least g(k) contains k vertex-disjoint cycles of different lengths. Recently, Chen and Chang proved in [J. Graph Theory 105 (2) (2024) 297-314] that for k\geqslant 3 every tournament with minimum outdegree at least 2k-1 contains k vertex-disjoint cycles in which two of them have different lengths. Motivated by the above two conjectures and related results, we investigate vertex-disjoint cycles of different lengths in tournaments, and show that when k\geqslant 5 every tournament with minimum outdegree at least 2k-1 contains k vertex-disjoint cycles in which three of them have different lengths. In addition, we show that every tournament with minimum outdegree at least 6 contains three vertex-disjoint cycles of different lengths and the minimum outdegree condition is sharp. This answers a question proposed by Chen and Chang.