TY - JOUR

T1 - Resolvable Group Divisible Designs with Block Size 3

AU - Assaf, Ahmed M.

AU - Hartman, Alan

PY - 1989/1/1

Y1 - 1989/1/1

N2 - Let v be a non negative integer, let λ be a positive integer, and let K and M be sets of positive integers. Agroup divisible design, denoted by GD[K, λ, M. ν] is a triple (α γ β) where X is a set of points, γ = {G1, G2…} is a partition of α, and β is a class of subsets of X with the following properties. (Members of Γ are called groups and members of β are called blocks 1. The cardinality of X is ν. 2. The cardinality of each group is a member of M. 3. The cardinality of each block is a member of K. 4. Every 2–subset {x, y} of X such that x and y belong to distinct groups is contained in precisely λ blocks. 5. Every 2–subset {x, y) of X such that x and y belong to the same group is contained in no block. A group divisible design is resolvable if there exists a partition II= P1, P2, of β such that each part P1, is itself a partition of X. In this paper we investigate the existence of resolvable group divisible designs with K = {3}, M a singleton set, and all λ The case where M = {1} has been solved by Ray-Chaudhuri and Wilson for λ = 1, and by Hanani for all λ > The case where M is a singleton set, and λ= 1 has recently been investigated by Rees and Stinson. We give some small improvements to Rees and Stinson's results, and give new results for the cases where λ> We also investigate a class of designs, introduced by Hanani, which we call frame resolvable group divisible designs and prove necessary and sufficient conditions for their existence.

AB - Let v be a non negative integer, let λ be a positive integer, and let K and M be sets of positive integers. Agroup divisible design, denoted by GD[K, λ, M. ν] is a triple (α γ β) where X is a set of points, γ = {G1, G2…} is a partition of α, and β is a class of subsets of X with the following properties. (Members of Γ are called groups and members of β are called blocks 1. The cardinality of X is ν. 2. The cardinality of each group is a member of M. 3. The cardinality of each block is a member of K. 4. Every 2–subset {x, y} of X such that x and y belong to distinct groups is contained in precisely λ blocks. 5. Every 2–subset {x, y) of X such that x and y belong to the same group is contained in no block. A group divisible design is resolvable if there exists a partition II= P1, P2, of β such that each part P1, is itself a partition of X. In this paper we investigate the existence of resolvable group divisible designs with K = {3}, M a singleton set, and all λ The case where M = {1} has been solved by Ray-Chaudhuri and Wilson for λ = 1, and by Hanani for all λ > The case where M is a singleton set, and λ= 1 has recently been investigated by Rees and Stinson. We give some small improvements to Rees and Stinson's results, and give new results for the cases where λ> We also investigate a class of designs, introduced by Hanani, which we call frame resolvable group divisible designs and prove necessary and sufficient conditions for their existence.

UR - http://www.scopus.com/inward/record.url?scp=77957023427&partnerID=8YFLogxK

U2 - 10.1016/S0167-5060(08)70092-X

DO - 10.1016/S0167-5060(08)70092-X

M3 - Article

AN - SCOPUS:77957023427

VL - 42

SP - 5

EP - 20

JO - Annals of Discrete Mathematics

JF - Annals of Discrete Mathematics

SN - 0167-5060

IS - C

ER -