Spanning 3-strong tournaments in 5-strong semicomplete digraphs

A digraph is strong if it has a directed path from x to y for every ordered pair of distinct vertices x, y and it is k-strong if it has at least k + 1 vertices and remains strong when we delete any set of at most k - 1 vertices. A digraph D is a tournament (semicomplete digraph) if there is precisely (at least) one arc between any pair of distinct vertices of D . A subdigraph of a digraph D is called spanning if it contains all vertices of D . In this paper, we show that every 5-strong semicomplete digraph on at least 9 vertices contains a spanning 3-strong tournament. This shows that the conjecture of Bang-Jensen and Jord & aacute;n [Discrete Mathematics 310 (2010) 1424-1428] holds for k = 3. (c) 2023 Elsevier B.V. All rights reserved.