A $Z$-cyclic triplewhist tournament for $4n+1$ players is equivalent to a set of $n$ quadruples $(a_i,b_i,c_i,d_i)$ such that $\bigcup_i\{a_i,b_i,c_i,d_i\}=\bigcup_i\{\pm(a_i-b_i),\pm(c_i-d_i)\}=\bigcup_i\{\pm(a_i-c_i),\pm(b_i-d_i)\}=\bigcup_i\{\pm(a_i-d_i),\pm(b_i-c_i)\}=Z_{4n+1}-\{0\}$. The case where $4n+1$ is a prime $p$ is considered. The existence of such tournaments for all $p\equiv 5\pmod 8,\ p\geq 29$, was established by I. Anderson, S. D. Cohen and N. J. Finizio [Discrete Math. 138 (1995), no. 1-3, 31--41], and the case $p\equiv 9\pmod{16}$ was fully dealt with by Y. S. Liaw [J. Combin. Des. 4 (1996), no. 4, 219--233] and G. McNay [Utilitas Math. 49 (1996), 191--201]. In this paper, besides giving easier proofs of these authors' results, we solve the problem also for primes $p \equiv 1$ (mod 16). The final result is the existence of a Z-cyclic TWh(v) for any v whose prime factors are all $\equiv 1$ (mod 4) and distinct from 5, 13, and 17.

### Existence of Z-cyclic triplewhist tournaments for a prime number of players

#### Abstract

A $Z$-cyclic triplewhist tournament for $4n+1$ players is equivalent to a set of $n$ quadruples $(a_i,b_i,c_i,d_i)$ such that $\bigcup_i\{a_i,b_i,c_i,d_i\}=\bigcup_i\{\pm(a_i-b_i),\pm(c_i-d_i)\}=\bigcup_i\{\pm(a_i-c_i),\pm(b_i-d_i)\}=\bigcup_i\{\pm(a_i-d_i),\pm(b_i-c_i)\}=Z_{4n+1}-\{0\}$. The case where $4n+1$ is a prime $p$ is considered. The existence of such tournaments for all $p\equiv 5\pmod 8,\ p\geq 29$, was established by I. Anderson, S. D. Cohen and N. J. Finizio [Discrete Math. 138 (1995), no. 1-3, 31--41], and the case $p\equiv 9\pmod{16}$ was fully dealt with by Y. S. Liaw [J. Combin. Des. 4 (1996), no. 4, 219--233] and G. McNay [Utilitas Math. 49 (1996), 191--201]. In this paper, besides giving easier proofs of these authors' results, we solve the problem also for primes $p \equiv 1$ (mod 16). The final result is the existence of a Z-cyclic TWh(v) for any v whose prime factors are all $\equiv 1$ (mod 4) and distinct from 5, 13, and 17.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/22771
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