Spanning Local Tournaments in Locally Semicomplete Digraphs

We investigate the existence of a spanning local tournament with possibly high connectivity in a highly connected locally semicomplete digraph. It is shown that every (3k 2)-connected locally semicomplete digraph contains a k-connected spanning local tournament. This improves the result of Bang-Jensen and Thomassen for semicomplete digraphs and of Bang-Jensen [I] for locally semicomplete digraphs. Kewords: Digraph; Connectivity 1. Terminology and preliminaries We denote by V(D) and E(D) the vertex set and the arc set of a digraph D, respectively. If .~y is an arc of D, then we say that x dominates y. More generally, if A and B are two disjoint subdigraphs of D such that every vertex of A dominates every vertex of B, then we say that A dominates B, denoted by A --f B. In addition, if A + B, but there is no arc from B to A, then we say that A strictly dominates B, denoted by A + B. The outset of a vertex x E V(D) is the set N+(x) = {y ( xy E E(D)}. Similarly, N-(x) = {,v 1 yx E E(D)} is th e inset of x. More generally, for a subdigraph A of D, we define its outset by N+(A) = UxEVcAj N+(x)-A and its inset by Np(A)=lJx~vcA,N-(x)-A (if necessary, we write N:(A) and N;(A) instead of N+(A) and N-(A), respectively). Every vertex of N+(A) is called an out-neighbour of A and every vertex of N-(A) is an in-neighbour of A. This paper forms a part of the author’s Ph.D. Thesis [2] written under the supervision of Professor L. Volkmann at RWTH Aachen. * E-mail: [email protected]. ’ The author is supported by a postdoctoral grant from “Deutsche Forschungsgemeinschaft” as a member of the “Graduiertenkolleg: Analyse und Konstruktion in der Mathematik” at RWTH Aachen. 0166-2 18X/97/$17.00