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Balanced whist tournaments. (English) Zbl 1198.05078
Summary: Whist tournaments for \(v\) players, \(\text{Wh}(v)\) are known to exist for all \(v\equiv 0,1\pmod 4\). In this paper a new specialization of whist tournament design, namely a balanced whist tournament, is introduced. We establish that balanced whist tournaments on \(v\) players, \(\text{BWh}(u)\), exist for several infinite classes of \(v\). An adaptation of a classic construction due to R. C. Bose and J. M. Cameron [J. Res. Natl. Bur. Stand., Sect. B 69, 323–332 (1965; Zbl 0131.36101)] enables us to establish that \(\text{BWh}(4n+1)\) exist whenever \(4n+1\) is a prime or a prime power. It is also established that \(\text{BWh}(4n)\) exist for \(4n=2^kA\), with \(k\equiv 0\pmod {2,3\text{ or }5}\). We demonstrate that a \(\text{BWh}(4n+1)\) is equivalent to a conference matrix of order \(4n+2\). Consequently, a necessary condition for the existence of a \(\text{BWh}(4n+1)\) is that \(4n+1\) is a product of primes each of which is \(\equiv1\pmod 4\). Thus, in particular, BWh(21) and BWh(33) do not exist. Specific examples of \(\text{BWh}(v)\) are given for \(v = 4,8,9,20,24,32\). It is also shown that a BWh(12) does not exist.

05C20 Directed graphs (digraphs), tournaments