Holey Generalized Balanced Tournament Designs

A group divisible design (GDD) of indexλand group size k is a triple (X,(?), B) where X is a set(of points), (?) is a partition of X into subsets (called groups), and B is a set of k-tuple subsets (called blocks) of X such that every pair of pointsfrom distinct groups occurs in exactlyλblocks, whilst no block contains a pairof points from the same group.If there are ai groups with size gi(1≤i≤u) in the (?), then we call ga11ga22…gauu the type of the GDD.Let (X, (?), B) be a (k, k-1)-GDD of type gu with (?)={G1, G2,...,Gu}. Suppose that k|g and the blocks of B can be arranged into a (gu|k)×(u-1)g array H whose rows and columns are indexed respectively with the elements 1, 2,..., gu/k and 1, 2,..., (u-1)g.Let Ri={t∈Z:((i-1)g／k)+1≤t≤ig／k} for i=1,2,...,u.suppose further that H satisfies the following two properties.(1) The blocks in each column of H form a parallel class over X.(2) Let x∈Ri(1≤i≤u). Then the blocks in row x form a partial k-parallel class over X\Gi.Then we refer to this GDD as a holey generalized balanced tournament designs(HGBTD), denoted by HGBTD(k, gu).The notion of an HGBTD(k, gu) was introduced recently by Yin, which can be regarded as an extention of a GBTD with Property C. HGBTD are of importance in the construction of GBTDs. This article investigates the existence of HGBTDs. We prove that for any integer u≥4, both an HGBTD(3, 3u) and an HGBTD(3, 6u) exist, with 6 possible exceptions. (see Theorem 1.1 and Theorem 1.2).