Some new \(\mathbb{Z}\)-cyclic whist tournament designs
It is well known that a whist design on \(v\) players exists for all \(v \equiv 0, 1 \pmod{4}\). A whist design is called \(\mathbb Z\)-cyclic if it admits an automorphism consisting of a single cycle of length \(v\) if \(v \equiv 1 \pmod{4}\) and of a cycle of length \(v - 1\) and one fixed point if \(v \equiv 0 \pmod{4}\). In 1896, E. H. Moore constructed \(\mathbb Z\)-cyclic whist designs on \(3p+1\) players for every prime \(p \equiv 1 \pmod{4}\). The first new results on the problem were obtained in 1992, when the authors generalized Moore's construction to the case of \(3p^n+1\) players. Since then several additional constructions have been found. In particular, \textit{G. Ge} and \textit{L. Zhu} [Bull. Inst. Comb. Appl.\ 32, 53--62 (2001; Zbl 0978.05019)] constructed \(\mathbb Z\)-cyclic whist designs on \(3s+1\) players for a class of values of \(s\) with \(s \equiv 1 \pmod{4}\) and \textit{N. J. Finizio} [Discrete Math.\ 279, 203--213 (2004; Zbl 1039.05013)] constructed such designs on \(27s+1\) players for the same values of \(s\). In the present paper, the authors obtain a generalization of these two results. They construct \(\mathbb Z\)-cyclic whist designs on \(3^nt+1\) players for all odd integers \(n \geq 1\) and for a class of integers \(t \equiv 1 \pmod{4}\) that includes all previous values for \(s\). They also obtain \(\mathbb Z\)-cyclic whist designs on \(3^nw\) players for all odd integers \(n \geq 1\) and for a class of integers \(w \equiv 3 \pmod{4}\). They obtain several other infinite classes of \(\mathbb Z\)-cyclic whist designs. In particular, they construct such designs on \(3^n\) players for all even \(n \geq 4\) and on \(3^n+1\) players, for all odd integers \(n \geq 1\).
scientific article; zbMATH DE number 2173229