Existence of Z-Cyclic Triplewhist Tournaments for a Prime Number of Players

A Z-cyclic triplewhist tournament for 4n+1 players, or briefly a TWh(4n+1), is equivalent to a n-set {(ai, bi, ci, di) | i=1, …, n} of quadruples partitioning Z4n+1−{0} with the property that ∪ni=1 {±(ai−ci), ±(bi−di)}=∪ni=1 {±(ai−bi), ±(ci−di)}=∪ni=1 {±(ai−di), ±(bi−ci)}=Z4n+1−{0}. The existence problem for Z-cyclic TWh(p)ʹs with p a prime has been solved for p≢1 (mod 16). I. Anderson et al. (1995, Discrete Math.138, 31–41) treated the case of p≡5 (mod 8) while Y. S. Liaw (1996, J. Combin. Des.4, 219–233) and G. McNay (1996, Utilitas Math.49, 191–201) treated the case of p≡9 (mod 16). In this paper, besides giving easier proofs of these authorsʹ results, we solve the problem also for primes p≡1 (mod 16). The final result is the existence of a Z-cyclic TWh(v) for any v whose prime factors are all≡1 (mod 4) and distinct from 5, 13, and 17.