ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.70490 J. Algebra Comb. Discrete Appl. 3(3) • 187–194 Received: 06 November 2015 Accepted: 03 March 2016 Journal of Algebra Combinatorics Discrete Structures and Applications Group divisible designs of four groups and block size five with configuration (1,1,1,2) Research Article Ronald Mwesigwa, Dinesh G. Sarvate, Li Zhang Abstract: We present constructions and results about GDDs with four groups and block size five in which each block has Configuration (1,1,1,2), that is, each block has exactly one point from three of the four groups and two points from the fourth group. We provide the necessary conditions of the existence of a GDD(n,4,5;λ1,λ2) with Configuration (1,1,1,2), and show that the necessary conditions are sufficient for a GDD(n,4,5;λ1, λ2) with Configuration (1,1,1,2) if n 6≡ 0(mod 6), respectively. We also show that a GDD(n,4,5; 2n,6(n − 1)) with Configuration (1,1,1,2) exists, and provide constructions for a GDD(n = 2t,4,5;n,3(n− 1)) with Configuration (1,1,1,2) where n 6= 12, and a GDD(n = 6t,4,5; 4t,2(6t−1)) with Configuration (1,1,1,2) where n 6= 6 and 18, respectively. 2010 MSC: 05B05, 05B30 Keywords: Group divisible designs (GDDs), Latin squares, Block configurations, 1-factors, RGDDs, RBIBDs 1. Introduction Group divisible designs (GDDs) have been studied for their usefulness in statistics and for their universal application to constructions of new designs [13, 17, 18]. Certain difficulties are present especially when the number of groups is smaller than the block size. In [3, 4], the question of existence of GDDs D. G. Sarvate thanks the College of Charleston for granting sabbatical. R. Mwesigwa thanks Mbarara University of Science and Technology for its support. D. G. Sarvate and R. Mwesigwa also thank Council for International Exchange of Scholars and the U. S. Department of State’s Bureau of Educational and Cultural Affairs for granting D. Sarvate a Fulbright core fellowship which made this collaboration possible. L. Zhang thanks The Citadel Foundation for its support. Ronald Mwesigwa; Mbarara University of Science and Technology, Mbarara, Uganda (email: ronmwe- sigwa@yahoo.com). Dinesh G. Sarvate (Corresponding Author); College of Charleston, Department of Mathematics, Charleston, SC, 29424 (email: sarvated@cofc.edu). Li Zhang; The Citadel, Department of Mathematics and Computer Science, Charleston, SC, 29409 (email: li.zhang@citadel.edu). 187 R. Mwesigwa et. al. / J. Algebra Comb. Discrete Appl. 3(3) (2016) 187–194 for block size three was settled. There is a more technical proof given in the book “Triple System" [2]. Similar results were established for GDDs with block size four in [6, 8, 9, 14, 19]. In [7, 16], results about GDDs with two groups and block size five with fixed block configuration were presented. In [10], results about GDDs with block size six with fixed block configuration were established. A group divisible design GDD(n,m,k;λ1,λ2) is a collection of k-element subsets of a v-set V called blocks which satisfies the following properties: each point of V appears in r (called replication number) of the b blocks; the v = nm elements of V are partitioned into m subsets (called groups) of size n each; points within the same group are called first associates of each other and appear together in λ1 blocks; any two points not in the same group are called second associates of each other and appear together in λ2 blocks. We note that in [13], the term GDD always refer to the case where λ1 = 0. When λ1 is not zero, the designs here are called group divisible PBIBDs [18]. In [6, 19], the necessary conditions are proved to be sufficient for the existence of a GDD(n,3,4;λ1,λ2) with Configuration (1,1,2) where each block has exactly one point from two of the three groups and two points from the third group. The purpose of this paper is to establish results for GDDs with block size five and four groups (i.e. GDD(n,4,5;λ1,λ2)) in which each block has Configuration (1,1,1,2), that is, each block has exactly one point from three of the four groups and two points from the fourth group. Unless otherwise stated, GDDs addressed in this paper all have the Configuration (1,1,1,2). First we find the relationship between λ2 and λ1. Theorem 1.1. The necessary conditions for the existence of a GDD(n,4,5; λ1,λ2) are n ≥ 2 and λ2 = 3(n−1)λ1 n . Proof. Suppose a GDD(n,4,5;λ1,λ2) exists, then the replication number r for an arbitrary point is λ1(n−1)+λ2(3n) 4 . Also, since vr = bk, we have b = n×[λ1(n−1)+λ2(3n)] 5 . On the other hand, since every block must contain exactly one first associate pair (with Configuration (1,1,1,2)), the group size n should be greater than or equal to 2, and the number of the first associates pairs 4n(n−1) 2 times λ1 must be equal to the number of blocks b. We have 2n(n−1)λ1 = n×[λ1(n−1)+λ2(3n)] 5 , that is, λ2 = 3(n−1)λ1 n . Corollary 1.2. A necessary condition for the existence of a GDD(3,4,5; λ1,λ2) is λ2 = 2λ1 and a necessary condition for the existence of a GDD(n,4,5;λ1,λ2) reduces to λ2 = (n − 1)t (for t ≥ 1) if n 6= 3. Proof. By Theorem 1.1, λ2 = 2λ1 if n = 3. If n 6= 3, then 3λ1 ≡ 0(mod n) = nt (t ≥ 1), thus λ1 = nt3 , and λ2 = (n−1)t for t ≥ 1. Corollary 1.3. For n 6≡ 0(mod 3), the minimum λ1 for the existence of a GDD(n,4,5;λ1,λ2) is n. For n ≡ 0(mod 3), the minimum λ1 is n3 . Proof. By Theorem 1.1, if n 6≡ 0(mod 3), then λ1 ≡ 0(mod n), thus the minimum λ1 for the existence of a GDD(n,4,5;λ1,λ2) is n. If n ≡ 0(mod 3), then λ1 ≡ 0(mod n3 ), thus the minimum λ1 is n 3 . Notice that if a GDD(n,4,5;λ1,λ2) exists, then a GDD(n,4,5;tλ1, tλ2) exists by taking t multiples of GDD(n,4,5;λ1,λ2). Therefore, we can reduce the problem to find a GDD(n,4,5;λ1,λ2) for the minimum value of λ1 (which are given in Corollary 1.3). Remark 1.4. If a GDD(n,4,5;λ1,λ2) for the minimum value of λ1 exists (it’s n for n 6≡ 0(mod 3) and n 3 for n ≡ 0(mod 3)), then a GDD(n,4,5;tλ1, tλ2) exists for t ≥ 1. 2. GDD(n,4,5;λ1,λ2) for n = 2,3,4 and n ≡ 1,5(mod 6) Theorem 2.1. Necessary conditions given in Theorem 1.1 are sufficient for the GDDs with n = 2,3 and 4. 188 R. Mwesigwa et. al. / J. Algebra Comb. Discrete Appl. 3(3) (2016) 187–194 Proof. By Theorem 1.1, the necessary condition for the existence of a GDD(2,4,5;λ1,λ2) is 2λ2 = 3λ1, that is, λ1 ≡ 0(mod 2) and λ2 ≡ 0(mod 3). The minimum values of λ1 and λ2 are 2 and 3, respec- tively. A GDD(2,4,5; 2,3) on the four groups {1,2}, {3,4}, {5,6} and {7,8} is as follows: {1,3,5,7,8}, {2,4,6,7,8}, {3,6,7,1,2}, {4,5,8,1,2}, {5,7,3,2,4}, {6,8,1,3,4}, {7,1,4,5, 6}, and {8,2,3,5,6}. By Remark 1.4, we have a GDD(2,4,5;λ1,λ2). By Corollary 1.2, the necessary condition for the existence of a GDD(3,4,5; λ1,λ2) is λ2 = 2λ1. The minimum values of λ1 and λ2 are 1 and 2, respectively. A construction for a GDD(3,4,5; 1,2) on the four groups {1,2,3}, {4,5,6}, {7,8,9} and {a,b,c} is as follows: {1,2,6,7,b}, {1,3,4,9,a}, {2,3,5,8,c}, {4,5,7,3,b}, {5,6,9,2,a}, {6,4,8,1, c}, {7,8,a,1,5}, {8,9,b, 2,4}, {9,7,c,3,6}, {c,b,1,5,9}, {b,a,3,6,8}, {c,a, 2,4,7}. Note that this construction is also listed in Clatworthy’s table (number 513 on page 902 in [1]). By Remark 1.4, we have a GDD(3, 4,5;λ1,λ2 = 2λ1). The necessary condition for the existence of a GDD(4,4,5; λ1,λ2) is λ1 ≡ 0(mod 4). The minimum values of λ1 and λ2 are 4 and 9, respectively. A construction for a GDD(4,4,5; 4,9) on the four groups {1,2,3,4}, {5,6,7,8}, {9,10,11,12} and {13,14,15,16} is as follows in Figure 1 (where each column represents a block). By Remark 1.4, we have a GDD(4,4,5;λ1,λ2). Figure 1. A GDD(4,4,5; 4,9) We use a different construction below for a GDD(n,4,5;λ1 = 2n,λ2 = 6(n − 1)) where λ1 is not of its minimum value (but twice of its minimum value if n 6≡ 0(mod 3) or six times of its minimum value if n ≡ 0(mod 3)). It’s an interesting construction as it uses a special kind of group divisible design GDD(n,k,k; 0,1). Such a GDD is called a transversal design, TD(k,n). The construction also uses a resolvable GDD (RGDD). A design is resolvable if the blocks of the design can be partitioned into parallel classes P1, . . . ,Ps, where every point of V occurs exactly once in each Pi. Similarly, one can define a resolvable transversal design, RTD(k,n). The following several theorems from the Handbook of Combinatorial Designs (2nd edition) [1], also see Rees [15] and Ge and Ling [5], are well-known theorems of RGDDs that we will use in our proof. Theorem 2.2. [1] (Theorem 5.35 on page 264) The necessary condition for the existence of a RGDD(n, m,k; 0,λ)) are (1) m ≥ k, (2) nm ≡ 0(mod k), and (3) λn(m−1) ≡ 0(mod (k −1)). Theorem 2.3. [1] (Theorem 5.43 on page 265) A RGDD(n,m,3; 0,λ) exists if and only if m ≥ 3, λn(m−1) is even, nm ≡ 0(mod 3), and (λ,n,m) 6∈ {(1,2,6),(1,6,3)}∪{(2j+1,2,3),(4j+2,1,6) : j ≥ 0}. Theorem 2.4. [1] (Theorem 5.44 on page 265) The necessary conditions for the existence of a RGDD(n, m,4; 0,1), namely, m ≥ 4, nm ≡ 0(mod 4) and n(m − 1) ≡ 0(mod 3), are also suffi- cient except for (n,m) ∈ {(2,4),(2,10),(3,4),(6,4)} and possibly excepting: n = 2 and m ∈ {34,46, 52,70,82,94,100,118,130,142,178,184, 202,214,238,250,334,346}; n = 10 and m ∈ {4,34,52,94}; n ∈ [14,454] ∪ {478,502,514,526,614,626,686} and m ∈ {10,70,82}; n = 6 and m ∈ {6,54,68}; n = 18 and m ∈ {18,38,62}; n = 9 and m = 44; n = 12 and m = 27; n = 24 and m = 23; and n = 36 and m ∈{11,14,15,18,23}. 189 R. Mwesigwa et. al. / J. Algebra Comb. Discrete Appl. 3(3) (2016) 187–194 A latin square L of side (or order) n is an n×n array in which each cell contains a single symbol from an n-set S, such that each symbol occurs exactly once in each row and exactly once in each column. Two latin squares L1 and L2 of the same order are orthogonal if L1(a,b) = L1(c,d) and L2(a,b) = L2(c,d), implies a = c and b = d. A set of latin squares L1, . . . ,Lm is mutually orthogonal, or a set of MOLS, if for every 1 ≤ i ≤ j ≤ m, Li and Lj are orthogonal. Theorem 2.5. [1] (Theorem 3.18 on page 161) The existence of k MOLS (n), the existence of a TD(k+ 2,n) and the existence of a RTD(k + 1,n) are equivalent where k ≥ 1. Theorem 2.4 implies the existence of a RGDD(n,4,4; 0,1) = RTD(4,n) except for n = 2,3,6 and 10. A construction of a TD(4,3) (it is also a RTD(4,3)) and a set of 2 MOLS(10) (which implies the existence of a TD(4,10) by Theorem 2.5) can be found in examples 6.5.1 and 6.5.10 in [11], respectively. Therefore, we have the following Lemma 2.6. Lemma 2.6. A TD(4,n) and a RTD(3,n) exist except for n = 2 and 6. Theorem 2.7. If a RGDD(n,3,3; 0,n− 1) exists (i.e. n 6= 2 by Theorem 2.3), then a GDD(n,4,5; 2n, 6(n−1)) also exists. Hence a GDD(n,4,5; 2n,6(n−1)) exists for all n > 2. Proof. A RGDD(n,3,3; 0,n−1) has n2(n−1) blocks and these are partitioned into n(n−1) parallel classes. First, we construct a RGDD(n,3,3; 0, n−1) on three groups G1, G2, and G3. There are n(n−1) parallel classes. Attach each pair of distinct points from G4 with blocks of two parallel classes to make blocks of size 5. In the same way, we construct RGDDs on G1, G2, and G4 and attach a pair from G3, and then construct RGDDs on G1, G3, and G4 and attach a pair from G2, and then construct RGDDs on G2, G3, and G4 and attach a pair from G1, on two parallel classes each. Now a parallel class has n triples and each pair from a Gi is attached to these triples of two parallel classes, λ1 is 2n. Now we show λ2 = 6(n−1). Let i ∈ Gi and j ∈ Gj. When we attach a pair from Gi containing i to two parallel classes from the RGDD that misses Gi, the pair {i,j} occurs in 2(n − 1) blocks. Likewise, when a pair from Gj containing j is attached to two parallel classes from the RGDD that misses Gj, the pair {i,j} occurs 2(n−1) times. The other two RGDDs intersect both Gi and Gj. Thus the pair {i,j} occurs n−1 times in each of these RGDDs. Hence the pair {i,j} occurs a total of 6(n−1) times. By using the same proof as in Theorem 2.7, we have the following corollary. Corollary 2.8. If a RGDD(n,3,3; 0, n−1 2 ) exists, then a GDD(n,4,5; n,3(n− 1)) also exists. Hence a GDD(n,4,5;n,3(n−1)) exists for all n ≡ 1(mod 2), i.e. odd numbers. Combine Corollary 1.3, Remark 1.4 and Corollary 2.8, we have the following result. Corollary 2.9. Necessary conditions are sufficient for a GDD(n,4,5;λ1, λ2) if n ≡ 1,5(mod 6), i.e. n ≡ 1(mod 2) and n 6≡ 0(mod 3). The following is a different proof of Theorem 2.7. Theorem 2.10. A GDD(n,4,5;λ1 = 2n,λ2 = 6(n−1)) exists for all n ≥ 2. Proof. In Theorem 2.1, we have proved that the necessary conditions are sufficient for n = 2. A TD(4,n) exists for every n except for n = 2 and 6 by Lemma 2.6. For any n > 2 and n 6= 6, a TD(4,n) has n2 blocks each of size 4 and replication number r = n. Any two elements from group Gi occur together 0 times and pairs from different Gi’s occur once. Take any block {a1,a2,a3,a4}, where aj ∈ Gj. and replace it by {x,a1,a2,a3,a4}, where x ∈ Gi −{ai}. We have 4n2(n − 1) blocks. It is easy to check the parameters λ1 = 2n, (because replication for TD(4,n) is n) and λ2 = 6(n− 1). Therefore, a GDD(n,4,5;λ1 = 2n,λ2 = 6(n−1)) always exists. The reason for λ2 = 6(n−1) is that suppose we have two elements a ∈ Gi, b ∈ Gj, and i 6= j. There are only three types of blocks which will involve a and/or b, that is, when both appear in one block, when a appears and b does not appear, and when a does not appear while b appears, in a block. In case 1, the number of pairs are 4(n− 1), while in cases 2 and 3, there are n−1 pairs in each, and this gives a total of 6(n−1). 190 R. Mwesigwa et. al. / J. Algebra Comb. Discrete Appl. 3(3) (2016) 187–194 For n = 6, an RGDD(6,3,3; 0,5) exists by Theorem 2.3 since m ≥ 3 and (λ,n,m) = (5,6,3) does not belong to that set of exceptions in Theorem 2.3. Hence we get a GDD(6,4,5; 12,30) by Theorem 2.7, that is, a GDD(n,4,5; 2n,6(n−1)) with n = 6 exists. Thus, a GDD(n,4,5;λ1 = 2n,λ2 = 6(n−1)) exists for all n ≥ 2. 3. GDD(n,4,5;λ1,λ2) for n ≡ 2,4(mod 6) A balanced incomplete block design BIBD(v,k,λ) (λ ≥ 1) is a pair (V,B) where B is a collection of binary blocks of V such that every block contains exactly k < v points and every pair of distinct elements is contained in exactly λ blocks. A resolvable BIBD(v,k,λ) is denoted as RBIBD(v,k,λ). A 1-factor of a graph G is a set of pairwise disjoint edges which partition the vertex set. A 1- factorization of a graph G is the set of 1-factors which partition the edge set of the graph. A 1-factorization of a K2n (also a RBIBD(2n,2,1)) exists, and for all n ≥ 1 contains 2n−1 1-factors [12]. Theorem 3.1. Necessary conditions are sufficient for a GDD(n,4,5;λ1, λ2) if n ≡ 2,4(mod 6), i.e. a GDD(6t + 2,4,5; 6t + 2,3(6t + 1)) and a GDD(6t + 4,4,5; 6t + 4,3(6t + 3)) exist. Proof. The construction provided in this proof uses a TD(4, t), and it works for all n = 2t except for t = 2 and t = 6 (since a TD(4, t) does not exist for t = 2 or 6 from Lemma 2.6). Let n = 2t where t 6= 2 and 6. A 1-factorization of a Ki2t on 2t elements of Gi has (2t− 1) 1-factors. Each 1-factor has t edges. Let Fij be the jth 1-factor. We partition 2t elements of Gi according to the edges of F i j , that is, Fij1,F i j2, . . . ,F i jt. Construct a TD(4, t) on four groups Hi = {F i j1,F i j2, . . . ,F i jt}. From each block of the TD(4, t), which gives naturally four groups each of size 2, we construct a GDD(2,4,5; 2,3) with 8 blocks. We repeat this for each 1-factor of the 1-factorization. A detailed counting gives the required values for λ1 and λ2 (see Example 3.2 below for an illustration of the construction). For t = 2 (i.e., n = 4), a GDD(4,4,5;λ1,λ2) exists from Theorem 2.1. For t = 6 (i.e., n = 12), it is considered in the case of a GDD(n,4,5;λ1,λ2) for n ≡ 0(mod 6), a GDD(6t + 2,4,5; 6t + 2,3(6t + 1)) and a GDD(6t + 4,4,5; 6t + 4,3(6t + 3)) exist. Example 3.2. A GDD(6,4,5; 6,15) based on the construction procedure in Theorem 3.1 is as follows. Here t = 3 and we want to construct a GDD(6,4,5; 6,15). The number of blocks for the GDD is 360. We start with a TD(4,3). If we use the groups {A1,A2,A3}, {B1,B2,B3}, {C1,C2,C3}, and {D1,D2,D3} then the blocks of the TD are {{A1,B1,C1,D1}, {A1,B2,C2,D2}, {A1,B3,C3,D3}, {A2,B1,C2,D3}, {A2,B2,C3,D1}, {A2,B3,C1,D2}, {A3,B1,C3,D2}, {A3, B2,C1,D3}, {A3,B3,C2,D1}}. Take a RBIBD(6,2,1) on Gi and call it βi. Essentially this is a 1-factorization on Ki6, a com- plete graph on six vertices where the vertices are the elements of group Gi and we have 6 − 1 = 5 1-factors. The sets Gi are given as, say, G1 = {x1,x2,x3,x4,x5,x6}, G2 = {y1,y2,y3,y4,y5,y6}, G3 = {z1,z2,z3,z4,z5,z6}, and G4 = {w1,w2,w3,w4,w5,w6}. Taking, for example, the set Gi, with i = 1, the five 1-factors will appear as Fi1 = {(x1,x2),(x3,x4),(x5,x6)}, Fi2 = {(x1,x3),(x2,x5),(x4,x6)}, Fi3 = {(x1,x4),(x2,x6),(x3,x5)}, Fi4 = {(x1,x5),(x2,x4),(x3,x6)}, and Fi5 = {(x1,x6),(x2,x3),(x4,x5)}. In general, we write Fij , where i,j are the group number and one-factor position, respectively. Take for example, j = 3 and i = 1,2,3,4. This gives F13 = {(x1,x4),(x2,x6),(x3,x5)}, F23 = {(y1,y4),(y2,y6),(y3,y5)}, F33 = {(z1,z4),(z2,z6),(z3,z5)}, and F43 = {(w1,w4),(w2,w6),(w3,w5)}. Construct a TD(4,3) where the groups are {F1j1,F 1 j2,F 1 j3}, {F 2 j1,F 2 j2, F 2 j3}, {F 3 j1,F 3 j2,F 3 j3}, and {F4j1,F 4 j2,F 4 j3}. Take each block of the transversal design, for example, the first block, {{x1,x4},{y1,y4},{z1,z4},{w1,w4}}. The elements of this block give the groups for a GDD(2,4,5; 2,3). The second block will have the groups {{x1,x4},{y2,y6},{z2,z6},{w2,w6}}, and so on, up to the ninth block with groups {{x3,x5},{y3,y5},{z2,z6},{w1,w4}}. From each of these nine blocks from a 1-factor we get 9 × 8 = 72 blocks. From 5 1-factors, we have constructed 360 required blocks for a GDD(6,4,5; 6,15). Now we show that λ1 = 6 and λ2 = 15. For example, observe that the pair {x1,x4} appears in 191 R. Mwesigwa et. al. / J. Algebra Comb. Discrete Appl. 3(3) (2016) 187–194 one 1-factor. Through that 1-factor, the element {x1,x4} appears in three blocks of the TD(4,3). In each block of the TD(4,3), the pair {x1,x4} appears two times. Thus, λ1 = 6. On the other hand, the pair {x1,y1} appears in 3 blocks of the GDD(2,4,5; 2,3). Since there are five 1-factors, we get λ2 = 15. Remark 3.3. Theorem 3.1 provides constructions for a GDD(n,4,5;n, 3(n−1)) for n = 2t (n 6= 4 and 12). 4. GDD(n,4,5;λ1,λ2) for n ≡ 0,3(mod 6) If n ≡ 0,3(mod 6), then n ≡ 0(mod 3) = 3s. The minimum value of λ1 is n3 = s by Corollary 1.3. Thus, if a GDD(n,4,5;s,3s− 1) exists, then a GDD(n,4,5;λ1,λ2) for n ≡ 0(mod 3) exists by Theorem 1.1 and Remark 1.4. Theorem 4.1. Necessary conditions are sufficient for a GDD(n,4,5;λ1, λ2) if n ≡ 3(mod 6), i.e., a GDD(6t + 3,4,5; 2t + 1,6t + 2) exists for t ≥ 0. Proof. We know that a TD(4,2t+1) exists (Lemma 2.6) and has a replication number 2t+1. We also know that a RBIBD(6t+3,3,1) exists and has 3t+1 parallel classes [1]. We also have a GDD(3,4,5; 1,2). We wish to construct a GDD(6t + 3,4,5; 2t + 1,6t + 2). Let the groups be G1 = {a1,a2, . . . ,a6t+3}, G2 = {b1,b2, . . . ,b6t+3}, G3 = {c1,c2, . . . ,c6t+3}, and G4 = {d1,d2, . . . ,d6t+3}. Let π1, π2, . . . , π3t+1 be parallel classes of a RBIBD(6t + 3,3,1) on {1,2, . . . ,6t + 3}. Use each πi to partition each of the four groups by relabelling the elements, i.e., if {j1,j2,j3} is the jth block of πi, then the jth partition set G1j of G1 is {aj1,aj2,aj3}. Similarly for other Gi, for i = 2,3,4. Use a TD(4,2t + 1) on groups {Gi1,Gi2, . . . ,Gi,2t+1}, i = 1,2,3,4. If a block of the TD(4,2t + 1) is {G1r,G2s,G3t,G4u}, construct a GDD(3,4,5; 1,2) on groups G1r, G2s, G3t, and G4u. The union of all the blocks of the GDDs thus constructed using all the πi’s is a GDD(6t + 3,4,5; 2t + 1,6t + 2). Clearly λ1 = 2t + 1 because in a TD(4,2t + 1) each element occurs 2t + 1 times. It means that Gij will be in 2t + 1 blocks of the TD and hence, when GDD(3,4,5; 1,2) is formed with Gij as one of the groups, pairs of elements within Gij will occur 2t + 1 times. Also, λ2 is 6t + 2 because pairs of elements between Gi and Gj (i 6= j) occur twice for each parallel class πi and there are 3t + 1 parallel classes. For n ≡ 0(mod 6) = 6t, we provide constructions for a GDD(6t,4,5; 4t, 2(6t− 1)) where λ1 = 4t is not of its minimum value (but twice of its minimum value which is 2t). Example 4.2. A construction of a GDD(12,4,5; 8,22) is as follows. First note that a TD(4,4) exists (by Lemma 2.6), and it has 16 blocks of size 4. Let G1, G2, G3, and G4 be the groups, each of size four for a GDD(12,4,5; 8,22) which we wish to construct. Let Hi = {Gi1,Gi2,Gi3, Gi4}, i = 1,2,3,4, where each Gij has size 3. We construct TD(4,4)s on groups H = {H1,H2,H3,H4}. Any Gij is in four blocks. But a block of H gives four subsets each of size 3. So these groups can be used to get a GDD(3,4,5; 1,2). Do this for each block of H. The pair of elements within Gij occur four times and the pairs from Gij,Gst, i 6= s occur two times. Now, construct a RBIBD(12,3,2) with 11 parallel classes. Use each of the parallel classes and apply the construction. As pairs in BIBD appear twice we have any two elements (a,b) from Gi in eight blocks and (c,d) where c ∈ Gi and d ∈ Gj, i 6= j occur 22 times. Remark 4.3. A GDD(6t,4,5; 4t,2(6t−1)) where t 6= 1 and 3 can be constructed using a TD(4,2t) and a RBIBD(6t,3,2). By Lemma 2.6, a TD(4,n) exist except for n = 2 and 6. Use similar ideas as in Example 4.2, we construct a GDD(6t,4,5; 4t,2(6t−1)) except for t = 1 and 3 using a TD(4,n = 2t) and a GDD(3,4,5; 1,2). We use a partition of 6t elements according to the parallel classes πij of a RBIBD(6t,3,2) on Gi, i = 1,2,3,4 and j = 1,2, · · · ,6t − 1. Note for a RBIBD(6t,3,2), there are 6t − 1 parallel classes and the 192 R. Mwesigwa et. al. / J. Algebra Comb. Discrete Appl. 3(3) (2016) 187–194 number of blocks is b = 2t(6t − 1). We use a partition πs, to get groups, say Gij, i = 1,2,3,4 and j = 1,2, · · · ,2t. The pair of elements within Gij occur 2t times and pairs from Gij,Gst, i 6= s occur two times. This construction is repeated 6t− 1 times once for each of the 6t− 1 parallel classes. Therefore elements from the same group will occur 4t times and pairs of elements from two different groups will occur 2(6t−1) times and we have GDD(6t,4,5; 4t,2(6t−1)) where t 6= 1 and 3. To completely solve the case for n ≡ 0(mod 6) = 6t, one should construct a GDD(6t,4,5; 2t,6t−1). 5. Summary In this paper we studied constructions and results about GDD(n,4,5;λ1, λ2) with Configuration (1,1,1,2). We provide the necessary conditions of the existence of a GDD(n,4,5;λ1,λ2) with Config- uration (1,1,1,2), and show that the necessary conditions are sufficient for a GDD(n,4,5;λ1,λ2) with Configuration (1,1,1,2) if n 6≡ 0(mod 6), respectively. We also show that a GDD(n,4,5; 2n,6(n−1)) with Configuration (1,1,1,2) exists, and provide constructions for a GDD(n = 2t,4,5;n,3(n− 1)) with Con- figuration (1,1,1,2) where n 6= 12, and a GDD(n = 6t,4,5; 4t,2(6t − 1)) with Configuration (1,1,1,2) where n 6= 6 and 18, respectively. The remaining case of the problem is to show that the necessary conditions are sufficient for n ≡ 0(mod 6), i.e., to show the existence of a GDD(6t,4,5; 2t,6t − 1) with Configuration (1,1,1,2). Acknowledgment: We are thankful to both the referees for their useful comments. Our special thanks to one of the referees as we have used his/her wordings to count the value of λ1 in Theorem 2.7 and the values for λ1 and λ2 in Example 3.2 verbatim. References [1] C. J. Colbourn, D. H. Dinitz (Eds.), Handbook of Combinatorial Designs, Second Edition, Chapman and Hall, CRC Press, Boca Raton, FL, 2007. [2] C. J. Colbourn, A. Rosa, Triple System, Oxford Science Publications, Clarendon Press, Oxford, 1999. [3] H. L. Fu, C. A. Rodger, Group divisible designs with two associate classes: n = 2 or m = 2, J. Combin. Theory Ser. A 83(1) (1998) 94–117. [4] H. L. Fu, C. A. Rodger, D. G. Sarvate, The existence of group divisible designs with first and second associates, having block size three, Ars Combin. 54 (2000) 33–50. [5] G. Ge, A. C. H. Ling, Asymptotic results on the existence of 4-RGDDs and uniform 5-GDDs, J. Combin. Des. 13(3) (2005) 222–237. [6] D. Henson, D. G. Sarvate, S. P. Hurd, Group divisible designs with three groups and block size four, Discrete Math. 307(14) (2007) 1693–1706. [7] S. P. Hurd, N. Mishra, D. G. Sarvate, Group divisible designs with two groups and block size five with fixed block configuration, J. Combin. Math. Comput. 70 (2009) 15–31. [8] S. P. Hurd, D. G. Sarvate, Odd and even group divisible designs with two groups and block size four, Discrete Math. 284(1-3) (2004) 189–196. [9] S. P. Hurd, D. G. Sarvate, Group divisible designs with block size four and two groups, Discrete Math. 308(13) (2008) 2663–2673. [10] M. S. Keranen, M. R. Laffin, Fixed block configuration group divisible designs with block size six, Discrete Math. 308(4) (2012) 745–756. [11] C. C. Lindner, C. A. Rodger, Design Theory, Second edition, CRC Press, Boca Raton, 2008. [12] E. Lucas, Récréations Mathématiques, Vol. 2, Gauthier-Villars, Paris, 1883. [13] R. C. Mullin, H. O. F. Gronau, PBDs and GDDs: the basics, C. J. Colbourn J. H. Dinitz (Eds.), 193 http://dx.doi.org/http://dx.doi.org/10.1006/jcta.1998.2868 http://dx.doi.org/http://dx.doi.org/10.1006/jcta.1998.2868 http://www.ams.org/mathscinet-getitem?mr=1742405 http://www.ams.org/mathscinet-getitem?mr=1742405 http://dx.doi.org/10.1002/10.1002/jcd.20020 http://dx.doi.org/10.1002/10.1002/jcd.20020 http://dx.doi.org/10.1016/j.disc.2006.09.017 http://dx.doi.org/10.1016/j.disc.2006.09.017 http://www.ams.org/mathscinet-getitem?mr=2542659 http://www.ams.org/mathscinet-getitem?mr=2542659 http://dx.doi.org/10.1016/j.disc.2004.01.010 http://dx.doi.org/10.1016/j.disc.2004.01.010 http://dx.doi.org/10.1016/j.disc.2005.02.024 http://dx.doi.org/10.1016/j.disc.2005.02.024 http://dx.doi.org/10.1016/j.disc.2011.11.002 http://dx.doi.org/10.1016/j.disc.2011.11.002 R. Mwesigwa et. al. / J. Algebra Comb. Discrete Appl. 3(3) (2016) 187–194 The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL (1996), 185–193. [14] N. Punnim, D. G. Sarvate, A construction for group divisible designs with two groups, Congr. Numer. 185 (2007) 57–60. [15] R. S. Rees, Two new direct product-type constructions for resolvable group-divisible designs, J. Combin. Des. 1(1) (1993) 15–26. [16] D. G. Sarvate, S. P. Hurd, Group divisible designs with two groups and block configuration (1,4), J. Combin. Inform. System Sci. 32 (2007) 297–306. [17] A. P. Street, D. J. Street, Combinatorics of Experimental Design, Clarendon Press, Oxford, 1987. [18] A. P. Street, D. J. Street, Partially balanced incomplete block designs, C. J. Colbourn J. H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL (1996), 419–423. [19] M. Zhu, G. Ge, Mixed group divisible designs with three groups and block size four, Discrete Math. 310(17-18) (2010) 2323–2326. 194 http://www.ams.org/mathscinet-getitem?mr=2408798 http://www.ams.org/mathscinet-getitem?mr=2408798 http://dx.doi.org/10.1002/jcd.3180010104 http://dx.doi.org/10.1002/jcd.3180010104 http://dx.doi.org/10.1016/j.disc.2010.05.014 http://dx.doi.org/10.1016/j.disc.2010.05.014 Introduction GDD(n, 4, 5; 1, 2) for n = 2, 3, 4 and n 1, 5 (mod 6) GDD(n, 4, 5; 1, 2) for n 2, 4 (mod 6) GDD(n, 4, 5; 1, 2) for n 0, 3 (mod 6) Summary References