Partial steiner triple systems with equal-sized holes (original) (raw)

Group divisible designs with three groups and block size four

Discrete Mathematics, 2007

We present new constructions and results on GDDs with three groups and block size four and also obtain new GDDs with two groups of size nine. We say a GDD with three groups is even, odd, or mixed if the sizes of the non-empty intersections of any of its blocks with any of the three groups is always even, always odd, or always mixed. We give new necessary conditions for these families of GDDs and prove that they are sufficient for these three types and for all group sizes except for the minimal case of mixed designs for group size 5t(t>1)5t(t>1). In particular, we prove that mixed GDDs allow a maximum difference between indices. We apply the constructions given to show that the necessary conditions are sufficient for all GDDs with three groups and group sizes two, three, and five, and also for group size four with two possible exceptions, a GDD(4,3,4;5,9)GDD(4,3,4;5,9) and a GDD(4,3,4;7,12)GDD(4,3,4;7,12).

Covering triples by quadruples: An asymptotic solution

Journal of Combinatorial Theory, Series A, 1986

be the minimum number of four-element subsets (called blocks) of an n-element set, X, such that each three-element subset of X is contained in at least one block. Let L(3,4, n) = rn/4rn -1/3rn -2/2111. Schoenheim has shown that C(3,4, n) 2 L(3,4, n). The construction of Steiner quadruple systems of all orders n ~2 or 4 (mod 6) by Hanani (Canad. J. Math. 12 (1960), 145-157) can be used to show that C(3,4, n)= L(3,4, n) for all n=2, 3,4 or 5 (mod 6) and all n = 1 (mod 12). The case n = 7 (mod 12) is made more difficult by the fact that C(3,4,7) = L(3,4,7) + 1 and until recently no other value for C(3,4, n) with n=7 (mod 12) was known. In 1980 Mills showed by construction that C(3,4,499) = L(3,4,499).

Resolvable group divisible designs with block size 3

Discrete Mathematics, 1989

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, Г, β) where X is a set of points, is a partition of X, and β is a class of subsets of X with the following properties. (Members of Г are called groups and members of β are called blocks.) 1. The cardinality of X is v.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 inprecisely λ blocks.5. Every 2-subset {x, y} of X such that x and y belong to the same group is contained in noblock.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.

Odd and even group divisible designs with two groups and block size four

Discrete Mathematics, 2004

We show the necessary conditions are su cient for the existence of GDD(n; 2; 4; 1, 2) with two groups and block size four in which every block intersects each group exactly twice (even GDD's) or in which every block intersects each group in one or three points (odd GDD's). We give a construction for near 3-resolvable triple systems TS(n; 3; 6) for every n ¿ 4, and these are used to provide constructions for several families of GDDs.

3-GDDs with block size 4

Bull. ICA, 2018

We define a 3-GDD(n, 2, k, λ1, λ2) by extending the definitions of a group divisible design and a t-design and give some necessary conditions for its existence. We prove that these necessary conditions are sufficient for the existence of a 3-GDD(n, 2, 4, λ1, λ2) except possibly when n ≡ 1, 3 (mod 6), n 6= 3, 7, 13 and λ1 > λ2. It is known that a partition of all 3-subsets of a 7-set into 5 Steiner triple systems (a large set for 7) does not exist, but we show that the collection of all 3-sets of a 7-set along with a Steiner triple system on the 7-set can be partitioned into 6 Steiner triple systems. Such a partition is then used to prove the existence of all possible 3-GDDs for n = 7. ∗The authors thank the referees for their comments which improved the paper. They also thank Dr. Bob Mignone of the College of Charleston and Bishop Stuart University Administration, especially to the Vice Chancellor Dr. Maud Kamatenesi Mugisha, for their continued support and the Council for Intern...

Equitable Specialized Block-Colourings for Steiner Triple Systems

Graphs and Combinatorics, 2008

We continue the study of specialized block-colourings of Steiner triple systems initiated in in which the triples through any element are coloured according to a given partition π of the replication number. Such colourings are equitable if π is an equitable partition (i.e., the difference between any two parts of π is at most one). Our main results deal with colourings according to equitable partitions into two, and three parts, respectively.