Let T be an n-partite tournament and let k,(T) denote the number of r-kings of T. Gutin (1986) and Petrovic and Thomassen (1991) proved independently that if T contains at most one transmitter, then k4(T) >i 1, and found infinitely many bipartite tournaments T with at most one transmitter such that
The number of kings in a multipartite tournament
โ Scribed by K.M. Koh; B.P. Tan
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 401 KB
- Volume
- 167-168
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
โฆ Synopsis
We show that in any n-partite tournament, where n/> 3, with no transmitters and no 3-kings, the number of 4-kings is at least eight. All n-partite tournaments, where n/>3, having eight 4-kings and no 3-kings are completely characterized. This solves the problem proposed in Koh and Tan (accepted).
๐ SIMILAR VOLUMES
We show that in any bipartite tournament with no transmitters and no 3-kings, the number of 4-kings is at least eight. All such bipartite tournaments having exactly eight 4-kings are completely characterized.
A king in a tournament is a player who beats any other player directly or indirectly. According to the existence of a king in every tournament, Wu and Sheng [Inform. Process. Lett. 79 (2001) 297-299] recently presented an algorithm for finding a sorted sequence of kings in a tournament of size n, i.
Volkmann [L. Volkmann, A remark on cycles through an arc in strongly connected multipartite tournaments, Appl. Math. Lett. 20 (2007Lett. 20 ( ) 1148Lett. 20 ( -1150] ] conjectured that a strong c-partite tournament with c โฅ 3 contains three arcs that belong to a cycle of length m for each m โ {3, 4,