The Existence of Kirkman Squares-Doubly Resolvable (v, 3, 1)-BIBDs

A Kirkman square with index λ, latinicity μ, block size k, and v points, KSk(v; μ, λ), is a t × t array (t = λ(v − 1)/μ(k − 1)) defined on a v-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V , and (3) the collection of blocks obtained from the non-empty cells of the array is a (v, k, λ)-BIBD. For μ = 1, the existence of a KSk(v; μ, λ) is equivalent to the existence of a doubly resolvable (v, k, λ)-BIBD. The spectrum of KS2(v; 1, 1) or Room squares was completed by Mullin and Wallis in 1975. In this paper, we determine the spectrum for a second class of doubly resolvable designs with λ = 1. We show that there exist KS3(v; 1, 1) for v ≡ 3 (mod 6), v = 3 and v ≥ 27 with at most 23 possible exceptions for v.