Esther R. Lamken

Decompositions of Edge-Colored Complete Graphs

We prove an asymptotic existence theorem for decompositions of edge-colored complete graphs into prespecified edge-colored subgraphs. Many combinatorial design problems fall within this framework. Applications of our main theorem require calculations involving the numbers of edges of each color and degrees of each color class of edges for the graphs allowed in the decomposition. We do these calculations to provide new proofs of the asymptotic existence of resolvable designs, near resolvable designs, group divisible designs, and grid designs. Two further applications are the asymptotic existence of skew Room d-cubes and the asymptotic existence of (v, k, 1)-BIBDs with any group of order k?1 as an automorphism group.

The Existence of Kirkman Squares—Doubly Resolvable ( v ,3,1)- BIBD s

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. 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

Constructions for generalized balanced tournament designs

v \equiv 3{\text{ (mod 6)}}

Existence results for doubly near resolvable ( v , 3, 2)-BIBDs

, v=3 and v≥27 with at most 23 possible exceptions for v.

Existence results for generalized balanced tournament designs with block size 3

Abstract A Kirkman square with index λ, latinicity μ, block size κ, and ν points, a KSκ(ν; μ, λ), is a t × t array ( t = λ(ν − 1) μ(κ − 1) ) defined on a ν-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 κ-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (ν, κ, λ)-BIBD. For μ = 1, the existence of a KSκ(ν; μ, λ) is equivalent to the existence of a doubly resolvable (ν, κ, λ)-BIBD. The spectrum of KS2(ν; 1, 1) or Room squares was completed by Mullin and Wallis in 1975. In this paper, we determine the spectrum of KS3(ν; 1, 2) or DR(ν, 3, 2)-BIBDs with at present six possible exceptions for ν.

Pooling, lattice square, and union jack designs

Abstract A generalized balanced tournament design, GBTD(n,k), defined on a kn-set V, is an arrangement of the blocks of a (kn,k,k−1)-BIBD defined on V into an n×(kn−1) array such that (1) every element of V is contained in precisely one cell of each column, and (2) every element of V is contained in at most k cells of each row. In this paper, we describe several constructions, both direct and recursive, for generalized balanced tournament designs. We include special cases of these constructions which can be used to produce factored and partitioned generalized balanced tournament designs.

The existence of KS 3 ( v ;2,4)

We construct pairwise balanced designs on 49, 57, 93, and 129 points of index unity, with block sizes 5, 9, 13, and 29. This completes the determination of the unique minimal finite basis for the PBD-closed set which consists of the integers congruent to 1 modulo 4. The design on 129 points has been used several times by a number of different authors but no correct version has previously appeared in print.

The Existence of Partitioned Generalized Balanced TournamentDesigns with Block Size 3

Abstract Let V be a set of υ elements. A (1, 2; 3, υ, 1)-frame F is a square array of side v which satisfies the following properties. We index the rows and columns of F with the elements of V , V ={ x 1 , x 2 ,…, x υ }. (1) Each cell is either empty or contains a 3-subset of V . (2) Cell ( x i , x i ) is empty for i =1, 2,…, υ . (3) Row x i of F contains each element of V −{ x i } once and column x i of F contains each element of V −{ x i } once. (4) The collection of blocks obtained from the nonempty cells of F is a (υ, 3, 2)-BIBD. A (1, 2; 3, υ, 1)-frame is a doubly near resolvable (υ, 3, 2)-BIBD. In this paper, we first present a survey of existence results on doubly near resolvable (υ, 3, 2)-BIBDs and (1, 2; 3, υ, 1)-frames. We then use frame constructions to provide a new infinite class of doubly near resolvable (υ, 3, 2)-BIBDs by constructing (1, 2; 3, υ, 1)-frames.