A New Connectivity Bound for a Tournament to be Highly Linked

• Published in: Graphs and combinatorics Vol. 41; no. 5
• Main Authors: Chen, Bin, Hou, Xinmin, Yu, Gexin, Zhou, Xinyu
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.10.2025; Springer Nature B.V
• Subjects: Combinatorics; Engineering Design; Graph theory; Mathematics; Mathematics and Statistics; Original Paper; Upper bounds; Vertex sets; 05C20; Linkedness; Connectedness; Tournament; 05C38; 05C40
• ISSN: 0911-0119; 1435-5914
• DOI: 10.1007/s00373-025-02970-1

A digraph D is k -linked if for any pair of two disjoint vertex sets { x 1 , x 2 , … , x k } and { y 1 , y 2 , … , y k } in D , there exist vertex disjoint dipaths P 1 , P 2 , … , P k such that P i is a dipath from x i to y i for each i ∈ [ k ] . Pokrovskiy (JCTB, 2015) confirmed a conjecture of Kühn et al. (Proc. Lond. Math. Soc., 2014) by verifying that every 452 k -connected tournament is k -linked. Meng et al. (Eur. J. Comb., 2021) improved this upper bound by showing that any ( 40 k - 31 ) -connected tournament is k -linked. In this paper, we show a better upper bound by proving that every ⌈ 12.5 k - 6 ⌉ -connected tournament with minimum out-degree at least 21 k - 14 is k -linked. Furthermore, we improve a key lemma that was first introduced by Pokrovskiy (JCTB, 2015) and later enhanced by Meng et al. (Eur. J. Comb., 2021).