Existence Of Group Divisible Designs Research Articles

Group divisible designs with block size 4 and group sizes 4 and 7

AbstractIn this paper, we consider the existence of group divisible designs (GDDs) with block size and group sizes and . We show that there exists a ‐GDD of type for all but a finite specified set of feasible values for .

Group divisible designs with block size four and type —II

AbstractWe show that the necessary conditions for the existence of group divisible designs with block size four (4‐GDDs) of type are sufficient for (mod ), = 39, 51, 57, 69, 87, 93, 111, 123 and 129, and for = 13, 17, 19, 23, 25, 29, 31 and 35. More generally, we show that for (mod 6), the possible exceptions occur only when , and there are no exceptions at all if has a divisor such that (mod 4) or is a prime not greater than 43. Hence, there are no exceptions when (mod 12). Consequently, we are able to extend the known spectrum for and 5 (mod 6). Also, we complete the spectrum for 4‐GDDs of type .

Group divisible designs with three unequal groups and larger first index

We show the necessary conditions are sufficient for the existence of group divisible designs (or PBIBDs) with block size k=3 with three groups of size (n,2,1) for any n≥2 and any two indices with λ1>λ2.

The asymptotic existence of group divisible designs of large order with index one

This paper gives the answer to a question of R.M. Wilson regarding the existence of group divisible designs of large order. Let k and u be positive integers such that 2 ⩽ k ⩽ u . Then there exists an integer m 0 = m 0 ( k , u ) such that there exists a group divisible design of group type m u with block size k and index one for any integer m ⩾ m 0 satisfying the necessary arithmetic conditions 1. m ( u − 1 ) ≡ 0 mod ( k − 1 ) , 2. m 2 u ( u − 1 ) ≡ 0 mod k ( k − 1 ) . This paper also presents a large-index asymptotic existence theorem for group divisible t-designs with a fixed number of groups, fixed group size and fixed block size.

A Complete Generalization of Clatworthy Group Divisible Designs

Partially balanced incomplete block designs (PBIBDs) have a long history and have been extensively used in agriculture and industrial experiments. Since the book of Clatworthy on two-associate-class partially balanced designs was published in 1973, little progress has been made on the construction of these designs. Group divisible designs (GDDs) are an important type of PBIBD with two associate classes. The existence of a GDD with block size $k=3$ was completely settled by Fu, Rodger, and Sarvate. In their works, the most difficult case to solve was when the number of groups, m, is less than the block size k. The existence of GDDs when $m<k$ is, in general, a difficult case to solve. Indeed, when $k=4$, very little is known about the existence of such GDDs. In this paper, we present two general construction methods for GDDs. The first one is a generalization of Wilson's fundamental construction in combinatorial design theory. The second is an extension of the traditional construction using double group divisible designs. As an application of our new construction methods, a complete solution is provided for GDDs that generalize all eleven designs in the old table of Clatworthy which have block size four, three groups, and replication number at most $10$. For these GDDs, no progress has been made until very recently.

Gregarious GDDs with Two Associate Classes

The existence of group divisible designs with two associate classes has been studied for over 50 years. Probably the most difficult cases to solve are those in which the number of groups is less than the size of the blocks. Recently, such an existence problem was solved in the case where the groups have the same size and the blocks have size 3. In this paper, we continue to focus on blocks of size 3, solving the existence problem when the required designs are gregarious (each block intersects each group). These designs are tight to construct in the sense that they satisfy equality in one of the bounds required for GDDs to exist.

Uniform Group Divisible Designs with Block Sizes Three and n

The existence of group divisible designs of type gt with block sizes three and n, 4≤ n≤10, is completely settled for all values of g and t.

An Existence Theorem for Group Divisible Designs of Large Order

The following result gives a partial answer to a question of R. M. Wilson regarding the existence of group divisible designs of large order. Let k and u be positive integers, 2⩽k⩽u. Then there exists an integer m0=m0(k, u) such that there exists a group divisible design of group type mu with block size k and index one for all integer m⩾m0 if and only if (i)u−1≡0mod(k−1), (ii)u(u−1)≡0modk(k−1). This is a generalization of the well-known result of Chowla, Erdős, and Straus on the existence of transversal designs of large order.

Partial steiner triple systems with equal-sized holes

The existence of group divisible designs of type u r 1 t with block size three is completely settled for all values of u, r, and t.