Abstract
A digraph D is k-linked if it satisfies that for every choice of disjoint sets {x1,…,xk} and {y1,…,yk} of vertices of D there are vertex disjoint paths P1,…,Pk such that Pi is an (xi,yi)-path. Confirming a conjecture by Kühn et al., Pokrovskiy proved in 2015 that every 452k-strong tournament is k-linked and asked for a better linear bound. Very recently Meng et al. proved that every (40k−31)-strong tournament is k-linked. In this note we use an important lemma from their paper to give a short proof that every (13k−6)-strong tournament of minimum out-degree at least 28k−13 is k-linked.
Highlights
A digraph D is strongly connected if it has a directed path from x to y (an (x, y)-path) for every ordered pair of distinct vertices x, y and it is k-strong if it has at least k + 1 vertices and remains strongly connected when we delete any set of at most k − 1 vertices
Agraph D is k-linked if it has vertex disjoint paths P1, . . . , Pk such that Pi is an (xi, yi)-path for every choice of disjoint sets {x1, . . . , xk} and {y1, . . . , yk} of vertices of D
It is natural to focus on special classes of digraphs and one such important class of digraphs is tournaments, that is, digraphs in which there is exactly one arc between every pair of distinct vertices
Summary
A digraph D is strongly connected if it has a directed path from x to y (an (x, y)-path) for every ordered pair of distinct vertices x, y and it is k-strong if it has at least k + 1 vertices and remains strongly connected when we delete any set of at most k − 1 vertices. A (di)graph D is k-linked if it has vertex disjoint paths P1, . This was improved to a polynomial in k by Kuhn, Lapinskas, Osthus, and Patel in [8] and they conjectured that a linear function would suffice Pokrovskiy confirmed this in [10] by showing that every 452k-strong tournament is k-linked. [10] For every narural number p every tournament T on at least 11p vertices contains disjoints sets X, Y both of size p such that X anchors Y in T. The proof in [9] that every (40k − 31)-strong tournament is k-linked uses an approach similar to that used by Pokrovskiy in [10] but is based on Lemma 2 instead of Lemma 1 and some matching arguments. If X is a subset of the vertices of D we use the notation D − X for the digraph D V (D) \ X
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