α-Resolvable Group Divisible Designs With Block Size Four And Group Size Three

A group divisible design GD(k, λ, t; tn) is α-resolvable if its blocks can be partitioned into classes such that each point of the design occurs in precisely α blocks in each class. The necessary conditions for the existence of such a design are λt(n - 1) = r(k - 1),bk = rtn, k|αtn and α|r. α-resolvable group divisible designs have been studied by many researchers and found to have a number of applications. For α-resolvable GD(3, λ, t; tn), its existence has been solved by Jungnickel, Mullin and Vanstone [D. Jungnickel, R.C. Mullin and S.A. Vanstone, The spectrum of α-resolvable block designs with block size 3, Discrete Math. 97 (1991), 269-277] and Zhang and Du [Y. Zhang and B. Du, α-resolvable group divisible designs with block size three, J Combin Designs, 13 (2005), 139-151]. For α-resolvable GD(4,λ,t; tn), when t = 1, it is an α-resolvable {n,4,λ)-BIBD, whose existence has been solved by Furino and Mullin [S. Furino and R.C. Mullin, Block designs and large holes and α-resolvable BIBDs, J Combin Designs, 1 (1993), 101-112] and Vasiga, Furino and Ling [T.M.J. Vasiga, S. Furino and A.C.H. Ling, The spectrum of α-resolvable designs with block size four, J Combin Designs, 9 (2001), 1-16]; when α = 1 and A = 1, it is a resolvable GD(4, 1, t; tn), whose existence has been almost completely solved. It is shown in this paper that these conditions are also sufficient when k = 4 and t = 3, except for n = 4 and α = λ = 1.