(2, 6) GWhD\((v)\) -- existence results and some \(Z\)-cyclic solutions (Q2718333)

This paper is concerned with generalised whist tournament designs (GWhD). Let \(|S|= v\). A \((2,2e)\text{ GWhD}(v)\) on \(S\) is a collection of games, each involving \(e\) teams of two players from \(S\) competing against each other. If \(v\equiv 0\pmod{2e}\) there are \(v-1\) rounds, each player appearing in one game per round; if \(v\equiv 1\pmod{2e}\) there are \(v\) rounds, each player appearing in all but one of the rounds. Altogether, each pair of players partner each other once, and oppose each other \(2(e-1)\) times. The case \(e=2\) thus corresponds to the ``classical'' whist tournaments. In this present paper the authors consider \(e=3\). It is shown that a \((2,6)\text{ GWhD}(6n+ 1)\) exists for all \(n\), and a \((2,6)\text{ GWhD}(6n)\) exists for all \(n\) except possibly \(n=3,18,22,29,44\). As a byproduct, several new resolvable or near-resolvable \((v,6,5)\) designs are obtained.