Group Divisible Designs with Block Size Four and Group Type gum1 with Minimum m (original) (raw)

Group divisible designs with block size 4 and group sizes 2 and 5

2021

In this paper we provide a 4-GDD of type 25, thereby solving the existence question for the last remaining feasible type for a 4-GDD with no more than 30 points. We then show that 4-GDDs of type 25 exist for all but a finite specified set of feasible pairs (t, s).

Group divisible designs with block size five

2022

We report some group divisible designs with block size five, including types 6 and 10. As a consequence we are able to extend significantly the known spectrum for 5-GDDs of type gu.

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).

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

arXiv (Cornell University), 2023

In this paper, we consider the existence of group divisible designs (GDDs) with block size 4 and group sizes 4 and 7. We show that there exists a 4-GDD of type 4 t 7 s for all but a finite specified set of feasible values for (t, s).

Group divisible designs with block-size four

Discrete Mathematics, 1977

It is proved that the obvious necessary conditions for the existence of a group divisible design with k = 4 are sufficient, except for the cases corresponding to the non-existing transversal designs T [4, 1; 2] and T [4, 1; 6].

Group divisible designs with block size four and two groups

Discrete Mathematics, 2008

We give some constructions of new infinite families of group divisible designs, GDD(n, 2, 4; 1 , 2 ), including one which uses the existence of Bhaskar Rao designs. We show the necessary conditions are sufficient for 3 n 8. For n = 10 there is one missing critical design. If 1 > 2 , then the necessary conditions are sufficient for n ≡ 4, 5, 8 (mod 12). For each of n=10, 15, 16, 17, 18, 19, and 20 we indicate a small minimal set of critical designs which, if they exist, would allow construction of all possible designs for that n. The indices of each of these designs are also among those critical indices for every n in the same congruence class mod 12. 1 = 2 (n − 3) + 3 = (2n − 3), 2 = 2 (n − 3) + 4 = (2n − 2).

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.