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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.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments
##### Keywords:
whist tournament; conference matrix