DMGT

Discussiones Mathematicae Graph Theory 29(1) (2009) 51-69
DOI: https://doi.org/10.7151/dmgt.1432

CONVEX INDEPENDENCE AND THE STRUCTURE OF CLONE-FREE MULTIPARTITE TOURNAMENTS

Darren B. Parker

Department of Mathematics Grand Valley State University Allendale, MI 49401-6495, USA e-mail: parkerda@gvsu.edu

Randy F. Westhoff and Marty J. Wolf

Department of Mathematics & Computer Science Bemidji State University Bemidji, MN 56601, USA

e-mail: rwesthoff@bemidjistate.edu

e-mail: mjwolf@bemidjistate.edu

Abstract

We investigate the convex invariants associated with two-path convexity in clone-free multipartite tournaments. Specifically, we explore the relationship between the Helly number, Radon number and rank of such digraphs. The main result is a structural theorem that describes the arc relationships among certain vertices associated with vertices of a given convexly independent set. We use this to prove that the Helly number, Radon number, and rank coincide in any clone-free bipartite tournament. We then study the relationship between Helly independence and Radon independence in clone-free multipartite tournaments. We show that if the rank is at least 4 or the Helly number is at least 3, then the Helly number and the Radon number are equal.

Keywords: convex sets, rank, Helly number, Radon number, multipartite tournaments.

2000 Mathematics Subject Classification: 05C20, 06B99.

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Received 24 September 2007
Revised 27 June 2008
Accepted 14 October 2008