A few more Kirkman squares and doubly near resolvable BIBDs with block size 3

Abstract

AbstractA 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. In a series of papers, Lamken established the existence of the following designs: KS3(v;1,2) with at most six possible exceptions [E.R. Lamken, The existence of doubly resolvable (v,3,2)-BIBDs, J. Combin. Theory Ser. A 72 (1995) 50–76], KS3(v;2,4) with two possible exceptions [E.R. Lamken, The existence of KS3(v;2,4)s, Discrete Math. 186 (1998) 195–216], and doubly near resolvable (v,3,2)-BIBDs with at most eight possible exceptions [E.R. Lamken, The existence of doubly near resolvable (v,3,2)-BIBDs, J. Combin. Designs 2 (1994) 427–440]. In this paper, we construct designs for all of the open cases and complete the spectrum for these three types of designs. In addition, Colbourn, Lamken, Ling, and Mills established the spectrum of KS3(v;1,1) in 2002 with 23 possible exceptions. We construct designs for 11 of the 23 open cases