A characterization of rich c-partite (c > 7) tournaments without (c + 2)-cycles

TL;DR

This paper characterizes the structure of rich c-partite tournaments with c > 7 that do not contain cycles of length c + 2, extending previous work on tournaments without cycles of length c + 1.

Contribution

It provides a structural characterization of rich c-partite tournaments without (c + 2)-cycles for c > 7, answering an open problem posed by Guo and Volkmann.

Findings

Characterization of rich c-partite tournaments without (c + 2)-cycles for c > 7

Extension of previous results on tournaments without (c + 1)-cycles

Addresses an open problem in tournament theory

Abstract

Let c be an integer. A c-partite tournament is an orientation of a complete c-partite graph. A c-partite tournament is rich if it is strong, and each partite set has at least two vertices. In 1996, Guo and Volkmann characterized the structure of all rich c-partite tournaments without (c + 1)-cycles, which solved a problem by Bondy. They also put forward a problem that what the structure of rich c-partite tournaments without (c + k)-cycles for some k>1 is. In this paper, we answer the question of Guo and Volkmann for k = 2.