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

Abstract

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.

Group divisible designs with block size four and two groups

“…When 2 > 1, a GDD(6, 2, 4; 3j, 2 ) is obtained from a GDD(6, 2, 4; 3(j − 1), 2 − x) and a suitable (3, x) with 1 x 5. We now have: For n = 18, the critical designs are (3, 1), (3, 2), (3,4), (3,5), (6, 1), and (6,11). The case n = 7: In this case, 2 must be a multiple of six, but there are no other restrictions from Table 1 on 1 .…”

“…One example: the design GDD(17, 2, 4; 39, 72) should exist but is not constructible using the "nearby" even design GDD(17, 2, 4; 34, 64). The needed critical designs for n = 17 are the ( 1 , 2 ) pairs: (5,8), (8,14), (11,20) and (14,26).…”

“…Table 1 does not give the whole story, however. There are several other known necessary conditions [11].…”

“…There are two special types of GDD(n, 2, 4; 1 , 2 ) called odd and even which we investigated in [11].A GDD(n, 2, 4, Lemma 2 (Hurd and Sarvate [11]). (a) For even GDDs, n 2 = 2 1 (n − 1).…”

A Family of Group Divisible Designs with Arbitrary Block Sizes

“…Since GDD has been widely applied to graphs [7] and matrices [13], many authors proposed different constructions of a GDD. One can see [7,13,8] (X, G, B), where X is a finite set of cardinality v, G is a partition of X into groups, and B is a family of subsets (blocks) of X that satisfy…”