Dudeney's round table problem

Abstract A set of Hamilton cycles in the complete graph on n vertices is called a Dudeney set, and denoted D ( n ), if every path of length two lies on exactly one of the cycles. In this paper it is shown that: 1. (a) There is a Dudeney set D ( p + 2) if p is prime and 2 is a generator of the multiplicative subgroup of GF( p ). (b) If there is a Dudeney set D ( n + 1), then there is a Dudeney set D (2 n ). (c) For n ⩽ 50, the only n for which the existence of a Dudeney set D ( n ) remains in doubt are n ϵ {27, 29, 35, 37, 41, 47}.