# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2019: 0.755

SCImago Journal Rank (SJR) 2019: 0.600

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

## 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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