Improved results on linkage problems

A digraph D is k-linked if |D|≥2k, and whenever x1,…,xk,y1,…,yk are 2k distinct vertices of D, there exist vertex-disjoint paths P1,…,Pk such that Pi is a path from xi to yi for each i∈[k]. A digraph is strongly connected if it has a directed path from x to y for every ordered pair of distinct vertices x,y and it is strongly k-connected (briefly k-connected) if it has at least k+1 vertices and remains strongly connected when we delete any set of at most k−1 vertices. In this paper, we prove that every (13k−6)-connected semicomplete digraph with minimum out-degree at least 19k−6 is k-linked. This result extends the result of Bang-Jensen and Johansen (2022) [3] to semicomplete digraphs. Moreover, we show the following holds. Let n,k,l∈N, and let D be a digraph of order n. Suppose that D is (39k+9l−28)-connected with minimum degree at least n−l and minimum out-degree at least 46k+15l−35. Then D is k-linked.