On the Path Covering Number of Given Subdigraphs of Regular Multipartite Tournaments

Lutz Volkmann 1, Stefan Winzen1
1Lehrstuhl II fiir Mathematik, RWTH Aachen, 52056 Aachen, Germany

Abstract

A tournament is an orientation of a complete graph, and a multipartite or \( c \)-partite tournament is an orientation of a complete \( c \)-partite graph. If we speak of a path, then we mean a directed path.

Let \( D \) be a regular \( c \)-partite tournament with \( r \) vertices in each partite set, and let \( X \subseteq V(D) \) be an arbitrary set with exactly \( 2 \) vertices from each partite set. For all \( c \geq 4 \), the authors determined in a recent article the minimal value \( g(c) \) such that \( D – X \) is Hamiltonian for every regular multipartite tournament with \( r \geq g(c) \). In this paper, we will supplement this result by postulating a given path covering number instead of the Hamiltonicity of the digraph \( D – X \). This means, for all \( c \geq 4 \) and \( k \geq 1 \), we will determine the minimal value \( h(k, c) \) such that \( D – X \) can be covered by at most \( k \) paths for every regular \( c \)-partite tournament with \( r \geq h(k, c) \). Moreover, we will present the minimal path covering number of \( D – X \), if \( D \) is a regular \( 3 \)-partite tournament and \( X \) contains exactly \( s \) vertices (\( s \geq 2 \)) from every partite set.

Keywords: Multipartite tournaments; Regular multipartite tourna- ments; Path covering number