TY - JOUR

T1 - Resolvable group divisible designs with block size 3

AU - Assaf, Ahmed M.

AU - Hartman, Alan

PY - 1989

Y1 - 1989

N2 - Let v be a non negative integer, let λ be a positive integer, and let K and M be sets of positive integers. A group divisible design, denoted by GD[K, λ, M, v], is a triple (X, G{cyrillic}, β) where X is a set of points, G{cyrillic} = {G1, G2,...} is a partition of X, and β is a class of subsets of X with the following properties. (Members of G{cyrillic} are called groups and members of β are called blocks.) 1. 1. The cardinality of X is v. 2. 2. The cardinality of each group is a member of M. 3. 3. The cardinality of each block is a member of K. 4. 4. Every 2-subset {x, y} of X such that x and y belong to distinct groups is contained in 5. precisely λ blocks. 6. 5. Every 2-subset {x, y} of X such that x and y belong to the same group is contained in no 7. block. A group divisible design is resolvable if there exists a partition Π = {P1, P2,...} of β such that each part Pi 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 > 1. 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 > 1. 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. A group divisible design, denoted by GD[K, λ, M, v], is a triple (X, G{cyrillic}, β) where X is a set of points, G{cyrillic} = {G1, G2,...} is a partition of X, and β is a class of subsets of X with the following properties. (Members of G{cyrillic} are called groups and members of β are called blocks.) 1. 1. The cardinality of X is v. 2. 2. The cardinality of each group is a member of M. 3. 3. The cardinality of each block is a member of K. 4. 4. Every 2-subset {x, y} of X such that x and y belong to distinct groups is contained in 5. precisely λ blocks. 6. 5. Every 2-subset {x, y} of X such that x and y belong to the same group is contained in no 7. block. A group divisible design is resolvable if there exists a partition Π = {P1, P2,...} of β such that each part Pi 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 > 1. 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 > 1. 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=38249024380&partnerID=8YFLogxK

U2 - 10.1016/0012-365X(89)90346-4

DO - 10.1016/0012-365X(89)90346-4

M3 - Article

AN - SCOPUS:38249024380

VL - 77

SP - 5

EP - 20

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -