Highly linked tournaments.

*(English)*Zbl 1319.05063Summary: A (possibly directed) graph is \(k\)-linked if for any two disjoint sets of vertices \(\{x_1,\dots,x_k\}\) and \(\{y_1,\dots,y_k\}\) there are vertex disjoint paths \(P_1,\dots,P_k\) such that \(P_i\) goes from \(x_i\) to \(y_i\). A theorem of B. Bollobás and A. Thomason [Combinatorica 16, No. 3, 313–320 (1996; Zbl 0870.05044)] says that every 22\(k\)-connected (undirected) graph is \(k\)-linked. It is desirable to obtain analogues for directed graphs as well. Although C. Thomassen [“Note on highly connected non-2-linked digraphs”, Combinatorica 11, No. 3, 393–395 (1991)] showed that the Bollobás-Thomason theorem does not hold for general directed graphs, he [“Connectivity in tournaments”, in: B. Bollobás (ed.), Graph theory and combinatorics. Proceedings of the Cambridge combinatorial conference, in honour of Paul Erdős. London: Academic Press. 305–313 (1984)] proved an analogue of the theorem for tournaments – there is a function \(f(k)\) such that every strongly \(f(k)\)-connected tournament is \(k\)-linked. The bound on \(f(k)\) was reduced to \(O(k\log k)\) by D. Kühn et al. [Proc. Lond. Math. Soc. (3) 109, No. 3, 733–762 (2014; Zbl 1302.05069)], who also conjectured that a linear bound should hold. We prove this conjecture, by showing that every strongly \(452k\)-connected tournament is \(k\)-linked.

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