Room designs and one-factorizations

Abstract

The existence of a Room square of order 2 n is known to be equivalent to the existence of two orthogonal one-factorizations of the complete graph on 2 n vertices, where "orthogonal" means "any two one-factors involved have at most one edge in common." Define R(n) to be the maximal number of pairwise orthogonal one-factorizations of the complete graph on n vertices. The main results of this paper are bounds on the function R. If there is a strong starter of order 2 n-1 then R(2 n) ≥ 3. If 4 n-1 is a prime power, it is shown that R(4 n) ≥ 2 n-1. Also, the recursive construction for Room squares, to obtain, a Room design of side v(u - w) +w from a Room design of side v and a Room design of side u with a subdesign of side w, is generalized to sets of k pairwise orthogonal factorizations. It is further shown that R(2 n) ≤ 2 n-3. © 1981 Birkhäuser Verlag.