An existence theorem for cyclic triplewhist tournaments
type: Item
instance of: scholarly article
title: An existence theorem for cyclic triplewhist tournaments
zbMATH Open document ID: 0818.05027
DOI: 10.1016/0012-365X(94)00186-M
author: Ian Anderson
author: Stephen D. Cohen
author: Norman J. Finizio
published in: Discrete Mathematics
publication date: 1995-04-17T00:00:00Z
review text: A triplewhist tournament on \(v= 4m+ 1\) players is a schedule of games \((a,b,c,d)\) involving two players \(a\) and \(c\) opposing two other players \(b\) and \(d\). The games are arranged into \(4m+ 1\) rounds such that each player plays in exactly one game in all but one round, partners each other player exactly once and opposes each other player exactly once in each of the possible two positions (as \(b\) or \(d\)-player, respectively). Such a tournament is called cyclic if the players are taken as the residues modulo \(v\) and the rounds are obtained from an initial round by developing modulo \(v\). The authors provide the first general existence theorem for this type of structure by proving the existence of a cyclic triplewhist tournament for \(v\) players whenever \(v\) is a product of powers of primes \(\equiv 5\pmod 8\). The interesting method of proof rests on three constructions together with a proof of the following result on primitive roots modulo \(p\) which is of independent interest: Given any prime \(p\equiv 5\pmod 8\) with \(p\geq 29\), \(p\neq 61\), there exists a primitive root \(\omega\text{ mod }p\) such that both \(\omega^ 2+ \omega+ 1\) and \(\omega^ 2- \omega+ 1\) are squares. Naturally enough, this result is proved by using character sums.
Mathematics Subject Classification ID: 05B30
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 91A46
zbMATH DE Number: 743994
zbMATH Keywords: triplewhist tournament
zbMATH Keywords: game
zbMATH Keywords: player
zbMATH Keywords: existence theorem
zbMATH Keywords: cyclic triplewhist tournament
zbMATH Keywords: primitive roots
zbMATH Keywords: character sums
reviewed by: Dieter Jungnickel
MaRDI profile type: Publication
cites work: Q3139840
cites work: Cyclically resolvable designs and triple whist tournaments
cites work: Q4865136
cites work: Q4188358
cites work: Primitive roots and powers among values of polynomials over finite fields.
cites work: Q3136939
cites work: Q4028460
cites work: Z‐cyclic triplewhist tournaments when the number of players involves primes of the form 8u + 5
cites work: Q4695307
cites work: Q5690468
Recommended article: Q3139840
Recommended article: Existence of \(Z\)-cyclic triplewhist tournaments for a prime number of players
Recommended article: Q4221870
Recommended article: Q4389772
Recommended article: Q4378178
Recommended article: Q4393367
Recommended article: Q4489238
Recommended article: Q4222117
Recommended article: Q4267202
description: scientific article; zbMATH DE number 743994