Room designs and one-factorizations

Abstract

The existence of a Room square of order 2n is known to be equivalent to the existence of two orthogonal one-factorizations of the complete graph on 2n vertices, where “orthogonal” means “any two one-factors involved have at most one edge in common.” DefineR(n) to be the maximal number of pairwise orthogonal one-factorizations of the complete graph onn vertices.

The main results of this paper are bounds on the functionR. If there is a strong starter of order 2n−1 thenR(2n) ≥ 3. If 4n−1 is a prime power, it is shown thatR(4n) ≥ 2n−1. Also, the recursive construction for Room squares, to obtain, a Room design of sidev(u − w) +w from a Room design of sidev and a Room design of sideu with a subdesign of sidew, is generalized to sets ofk pairwise orthogonal factorizations. It is further shown thatR(2n) ≤ 2n−3.