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

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

HOPs and COPs: Frames with partitionable transversals

In this paper, we construct Room frames with partitionable transversals. Direct and recursive constructions are used to find sets of disjoint complete ordered partitionable (COP) transversals and sets of disjoint holey ordered partitionable (HOP) transversals for Room frames. Our main results include upper and lower bounds on the number of disjoint COP transversals and the number of disjoint HOP transversals for Room frames of type 2 n . This work is motivated by the large number of applications of these designs. 1 Introduction and Definitions Let S be a set, and let fS 1 ; : : : ; S n g be a partition of S. An fS 1 ; : : : ; S n g\GammaRoom frame is an jSj \Theta jSj array, F , indexed by S, which satisfies the following properties: 1. every cell of F either is empty or contains an unordered pair of symbols of S, 2. the subarrays S i \Theta S i are empty, for 1 i n (these subarrays are referred to as holes), 3. each symbol x 62 S i occurs once in row (or column) s, for any s..