In this paper we study group divisible designs (GDDs) with block size 4 and two groups of different sizes when λ2 = 1. We obtain necessary conditions for the existence of such GDDs and prove that these necessary conditions are sufficient in several cases. Further, we present general constructions using resolvable designs.
A set system or design is a pair \((X, \mathcal{B})\) , where \(X\) is a non-empty finite set and \(\mathcal{B}\) is a collection of subsets of \(X\) , called blocks. Among all combinatorial designs, probably the most widely studied design is a Balanced Incomplete Block Design (BIBD) (See [ 15 ]).
Definition 1.1. A Balanced Incomplete Block Design BIBD \((v,k,\lambda)\) is an arrangement of elements of a \(v\) -set \(X\) into \(b\) blocks of size \((k < v)\) each, such that every element appears in exactly \(r\) blocks and every pair of distinct elements occurs together in exactly \(\lambda\) blocks.
The numbers \(v\) , \(b\) , \(r\) , \(k\) and \(\lambda\) are called parameters of a BIBD and satisfy the conditions or relationships \[bk=rv \quad \text{and}\quad r(k-1)=\lambda(v-1).\]
These conditions are called necessary conditions for the existence of a BIBD \((v , k, \lambda)\) .
Example 1.2. A famous BIBD is the Fano plane : a BIBD \((7,3,1)\) . The blocks are obtained from the triple \(\{1,3,4\}\) by adding the elements of \(X= \{0,1,2,3,4,5,6\}\) modulo \(7\) . So the blocks of the design are given in columns as follows:
| \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(0\) |
| \(3\) | \(4\) | \(5\) | \(6\) | \(0\) | \(1\) | \(2\) |
| \(4\) | \(5\) | \(6\) | \(0\) | \(1\) | \(2\) | \(3\) |
In 1961, Hanani [ 9 ] proved that the necessary conditions are sufficient for the existence of BIBDs with block size three as well as four. Specifically he proved:
Theorem 1.3. A BIBD \((v,4,\lambda)\) exists if and only if
\(\lambda\equiv1,5\pmod6\) and \(v\equiv1,4\pmod{12}\) ;
\(\lambda\equiv2,4\pmod6\) and \(v\equiv1\pmod3\) ;
\(\lambda\equiv3\pmod6\) and \(v\equiv0,1\pmod4\) ;
\(\lambda\equiv0\pmod6\) and \(v\ge 4\) .
Definition 1.4. A parallel class in a BIBD \((v,k,\lambda)\) is a set of disjoint blocks of the BIBD whose union is \(X\) . A partition of the collection of all blocks into \(r\) parallel classes is called a resolution.
A BIBD is said to be a resolvable BIBD, denoted RBIBD \((v,k,\lambda)\) , if it has a resolution.
A BIBD is called \(\alpha\) -resolvable BIBD if its blocks can be partitioned into classes in which each element occurs \(\alpha\) times.
Example 1.5. To construct a RBIBD \((9,3,1)\) , the blocks of a BIBD \((9,3,1)\) on \(\{1,2,3,4,\) \(5,6,7,8,9\}\) are partitioned into four parallel classes as demonstrated below:
| \(1\) | \(4\) | \(7\) | \(1\) | \(2\) | \(3\) | \(1\) | \(2\) | \(3\) | \(1\) | \(2\) | \(3\) |
| \(2\) | \(5\) | \(8\) | \(4\) | \(5\) | \(6\) | \(5\) | \(6\) | \(4\) | \(6\) | \(4\) | \(5\) |
| \(3\) | \(6\) | \(9\) | \(7\) | \(8\) | \(9\) | \(9\) | \(7\) | \(8\) | \(8\) | \(9\) | \(7\) |
Theorem 1.6. [ 3 ] A resolvable BIBD \((v,3,1)\) exists if and only if \(v\equiv3\pmod6\) .
A resolvable BIBD \((v,3,1)\) is called Kirkman Triple System of order \(v\) , denoted by KTS \((v)\) . There are \(\frac{v-1}{2}\) parallel classes in a KTS \((v)\) .
Let \(\alpha\) be a positive integer. An \(\alpha\) -parallel class or \(\alpha\) -resolution class in a design is a set of blocks containing every point of the design exactly \(\alpha\) times [ 1 ].
In [ 1 ] the following necessary conditions are given:
Theorem 1.7. Necessary conditions for the existence of an \(\alpha\) -resolvable BIBD \((v,k,\) \(\lambda)\) are (1) \(\lambda(v-1)\equiv0\pmod{(k-1)\alpha}\) ; (2) \(\lambda v(v-1)\equiv0\pmod{k(k-1)}\) and (3) \(\alpha v\equiv0\pmod k\) .
For block size \(3\) , following result is given in [ 1 ].
Theorem 1.8. The necessary conditions for the existence of an \(\alpha\) -resolvable BIBD \((v,3,\) \(\lambda)\) are sufficient except for \(v=6\) , \(\alpha=1\) , and \(\lambda\equiv2\pmod4\) .
From Theorem 1.8, we have the following corollary.
Corollary 1.9. For \(n\equiv0\pmod3\) , a resolvable BIBD \((n,3,n-2)\) exists.
Example 1.10. Let \(X=\{1,2,3,4,5,6\}\) . By Corollary 1.9, BIBD \((6,3,\) \(4)\) is a resolvable with \(10\) resolution classes; each class contains \(2\) mutually disjoint blocks which partition the point set \(X\) as follows. The blocks are written in columns.
| \(1\) | \(4\) | \(1\) | \(3\) | \(1\) | \(3\) | \(1\) | \(3\) | \(1\) | \(2\) | \(1\) | \(2\) | \(1\) | \(2\) | \(1\) | \(2\) | \(1\) | \(2\) | \(1\) | \(2\) |
| \(2\) | \(5\) | \(2\) | \(5\) | \(2\) | \(4\) | \(2\) | \(4\) | \(3\) | \(5\) | \(3\) | \(4\) | \(3\) | \(4\) | \(4\) | \(3\) | \(4\) | \(3\) | \(5\) | \(3\) |
| \(3\) | \(6\) | \(4\) | \(6\) | \(5\) | \(6\) | \(6\) | \(5\) | \(4\) | \(6\) | \(5\) | \(6\) | \(6\) | \(5\) | \(5\) | \(6\) | \(6\) | \(5\) | \(6\) | \(4\) |
Group divisible designs (GDDs) defined below play a role in the construction of BIBDs as well as other designs [ 10 ].
Definition 1.11.
A group divisible design, GDD
\((n_1, n_2,…,n_m, k; \lambda_{1}, \lambda_{2})\)
, is a triple
\((X, \mathcal{G}, \mathcal{B})\)
, where
\(X\)
is a
\(v\)
-set,
\(\mathcal{G}\)
is a partition of
\(X\)
into
\(m\)
subsets (called
groups
) of sizes
\(n_1, n_2, …, n_m\)
respectively
and
\(\mathcal{B}\)
is a collection of
\(k\)
-subsets of
\(X\)
(called
blocks
) such that each pair of points within the same group appear together in
\(\lambda_1\)
blocks, whereas each pair of points from different groups appear together in
\(\lambda_2\)
blocks. The points in the same group are called
first associate
of each other
and
points not in the same group are called
second associates
of each other.
Group divisible designs have been studied for their usefulness in statistics and for their universal applications in the constructions of new designs [ 16 , 17 ]. Most research has concentrated on those cases where the group size or group type was critical. When the number of groups is smaller than the block size, however, certain difficulties are present and little has been done for block size four. Fu, Rodger and Sarvate [ 5 , 6 ] obtained complete results on group divisible designs with block size \(3\) and \(m\) groups of size \(n\) . When all the groups are of the same size, the GDD is said to be uniform, otherwise it is said to be nonuniform. In 1992, Colbourn, Hoffman and Rees [ 4 ] proved the sufficiency of the necessary conditions for the existence of a GDD \((n,n,…,n,u,3;0,1)\) . When GDDs are used to construct other combinatorial designs, the groups are preferred to have different sizes to fit in various situations [ 4 ]. Unfortunately comparing with uniform GDDs where group sizes are equal, less is known for the construction of nonuniform GDDs. The construction of nonuniform GDDs is a challenging problem as no appropriate algebraic or geometric structures have been found which are useful for these constructions [ 7 ].
This paper is specially motivated by [
2
] , [
11
] and [
12
]. In 2011, Pabhapote and Punnim [
12
] studied all triples of positive integers
\((n_1,n_2,\lambda)\)
for which a GDD
\((n_1,n_2,3;\lambda,1)\)
exists. They proved that the necessary conditions are sufficient for the existence of a GDD
\((n_1,n_2,3;\lambda,1)\)
. Later in 2012, Pabhapote [
11
] proved the existence of a
GDD
\((n_1,n_2,3;\lambda_1,\lambda_2)\)
for all
\(n_1\ne 2\)
and
\(n_2\ne 2\)
in which
\(\lambda_1\ge \lambda_2\)
. In 2012, Chaiyasena, Hurd, Punim and Sarvate [
2
] investigated GDDs with two association classes with blocks of size
\(3\)
and groups of unequal sizes. They also obtained some general results for a GDD
\((n_1,n_2,n_3,3;\lambda_1,\lambda_2)\)
. However, much less is known about the existence of nonuniform GDDs when the block size is four. A few results are
available. For instance, in 2022, Sarvate and Woldemariam [
13
] studied the existence of a GDD
\((1,n,n,4;\lambda_{1},\lambda_{2})\)
and proved that the necessary conditions are sufficient for its existence when
\(n\equiv 0, 1, 4, 5, 8, 9 \pmod{12}\)
and
\(\lambda_{1} \ge \lambda_{2}\)
. In 2023, Sarvate, Woldemariam and Zhang [
14
] investigated the existence of nonuniform GDDs with three groups and block size
\(4\)
. Recently, Girma, Sarvate and Zhang [
8
] studied the existence of GDD
\((1,n,4;\lambda_{1},\lambda_{2})\)
and showed the necessary conditions are sufficient in several cases when
\(\lambda_{1}\ge \lambda_{2}\)
.
Despite these developments, the existence of GDDs with two unequal groups with block size \(4\) remains unknown when one of the indices \(\lambda_{1}\) or \(\lambda_{2}\) is fixed. In this paper, we study group divisible designs with block size 4 and two groups of sizes \(n_1\) and \(n_2\) when \(\lambda_{2}=1\) , namely a GDD \((n_1,n_2,4;\lambda,1 )\) .
Group divisible designs play a role in the construction of BIBDs, but the converse is also true, for example, an easy observation is the following result.
Theorem 1.12. If a BIBD \((n_1+n_2+ … +n_m,k,\lambda_2)\) and a BIBD \((n_i,k,\lambda_1)\) exist for \(i=1,2,…m\) , then a GDD \((n_1,n_2,…,n_m,k;\lambda_1+\lambda_2,\lambda_2)\) exists.
Corollary 1.13. If a BID \((n_1+n_2,4,\lambda_{2})\) and a BIBD \((n_i,4,\lambda_{1})\) for \(i=1,2\) exist, then a GDD \((n_1,n_2,4;\lambda_{1}+\lambda_{2},\lambda_{2})\) exists.
The converse of Theorem 1.12 is not always true, for example, a GDD \((9,24,4;\) \(6,1)\) exists which can be constructed by using Corollary 3.5 but a BIBD \((9,4,5)\) , a BIBD \((24,4,5)\) and a BIBD \((9+24,4,1)\) do not exist.
Notice that a GDD \((n_1,n_2,4;0,\lambda_2)\) does not exist as the number of groups is less than the block size.
Let \(r_i\) be the replication number of elements of the \(i\) – th group for \(i=1,2\) and \(b\) be the required number of blocks if a GDD \((n_1,n_2, k; \lambda_1, \lambda_2)\) exists. By combinatorial argument, \[\begin{aligned} r_1 = \frac{(n_1-1)\lambda_1 + n_2\lambda_2}{k-1}. \label{equation1} \end{aligned} \tag{1}\] \[\begin{aligned} r_2 = \frac{n_1\lambda_2 + (n_2-1)\lambda_1}{k-1}.\label{equation2} \end{aligned} \tag{2}\] \[\begin{aligned} b = \frac{[n^2_1 + n^2_2-(n_1+n_2{\tiny })]\lambda_1 + \lambda_2(2n_1n_2)}{k(k-1)}.\label{equation3} \end{aligned} \tag{3}\]
Theorem 2.1. A necessary condition for the existence of a GDD \((n_1,n_2,4;\lambda_1,\) \(\lambda_2)\) is \(b\ge max(2r_i-\lambda_1)\) .
Proof. Let ( \(x\) , \(y\) ) be a first associate pair from \(G_i\) for some \(i=1,2\) . Then as both of them come together \(\lambda_1\) times, there are \(r_i\) blocks containing \(x\) and \(r_i-\lambda_1\) blocks which contain \(y\) but do not contain \(x\) . So the number of blocks must be at least \(2r_i-\lambda_1\) to accommodate \(x\) and \(y\) \(r_i\) times in the design. ◻
Theorem 2.2. A necessary condition for the existence of a GDD \((n_1,n_2,4;\lambda_1,\) \(\lambda_2)\) is \(b\ge r_1+r_2-\lambda_2\) .
Proof. Let ( \(x\) , \(y\) ) be a second associate pair, say \(x\in G_1\) and \(y\in G_2\) . As both of them come together \(\lambda_2\) times, there are \(r_1+r_2-\lambda_2\) blocks containing \(x\) or \(y\) but do not contain both \(x\) and \(y\) . So the number of blocks must be at least \(r_1+r_2-\lambda_2\) . ◻
Suppose a GDD \((n_1,n_2, 4; \lambda_1, \lambda_2)\) exists. Then the number of first associate pairs equals \(\binom{n_1}{2}\lambda_1 + \binom{n_2}{2}\lambda_1 = \dfrac{n_1^2+n_2^2-(n_1+n_2)}{2}\lambda_1\) and the number of second associate pairs equals \((n_1n_2)\lambda_2\) . As the block size is \(4\) , there are three types of block configurations: \((4,0)\) , \((3,1)\) and \((2, 2)\) . Let \(x\) , \(y\) and \(z\) be the number of blocks of type \((4,0)\) , \((3,1)\) and \((2, 2)\) respectively. Note \(x+y+z=b\) and they contribute \(6x + 3y + 2z\) first associate pairs in the design. Hence, \[\begin{aligned} 6x + 3y + 2z = \frac{n_1^2+n_2^2-(n_1+n_2)}{2}\lambda_1. \label{equation4} \end{aligned} \tag{4}\]
Similarly, counting second associate pairs, \[\begin{aligned} 3y + 4z = n_1n_2\lambda_2. \label{equation5} \end{aligned} \tag{5}\]
From Eqs. (4) and (5), we have \(2z-6x=n_1n_2\lambda_2- \dfrac{n_1^2+n_2^2-(n_1+n_2)}{2}\lambda_1\le 2z\le 2b\) . Using Eq. (3), we obtain \(\lambda_2\le \dfrac{n_1^2+n_2^2-(n_1+n_2)}{n_1n_2}\lambda_1\) . Thus, we have the following theorem.
Theorem 2.3. A necessary condition for the existence of a GDD \((n_1,n_2,4;\lambda_1,\) \(\lambda_2)\) is \[\lambda_2\le \dfrac{n_1^2+n_2^2-(n_1+n_2)}{n_1n_2}\lambda_1.\]
As every block of a GDD \((n_1,n_2,4;\lambda_1,\lambda_2)\) contains at least two first associate pairs, hence \(\left[\binom{n_1}{2}+\binom{n_2}{2}\right]\lambda_1\ge 2b\) . Thus, we have
Theorem 2.4. A necessary condition for the existence of a GDD \((n_1,n_2,4;\lambda_1,\) \(\lambda_2)\) is \[b\le \frac{n_1^2+n_2^2-(n_1+n_2)}{4}\lambda_1.\]
Example 2.5. A GDD \((7,10,4;1,3)\) does not exist as \(\dfrac{n_1^2+n_2^2-(n_1+n_2)}{4}\lambda_1=33\) which is less than the required number of blocks \(b=46\) .
Theorem 2.6. A necessary condition for the existence of a GDD \((n_1,n_2,4;\lambda_1,\) \(\lambda_2)\) is \(r_1\ge\lambda_1\) .
Proof. Let \(x\in G_1\) , as a first associate pair occurs \(\lambda_1\) times, \(r_1\ge \lambda_1\) for \(n_1>1\) . ◻
Example 2.7. A GDD \((2,5,4;4,1)\) does not exists as \(r_1=3<4=\lambda_1\) .
After a few observations on the necessary conditions for the existence of a GDD \((n_1,n_2,\) \(4; \lambda_{1},\lambda_{2})\) , next we will obtain the spectrum for the existence of a GDD \((n_1,n_2,4;\lambda,1)\) .
Suppose \(\lambda_1=\lambda\) and \(\lambda_2=1\) . Then from Eqs. (1), (2) and (3), we have \[\begin{aligned} r_1 = \frac{(n_1-1)\lambda + n_2}{3}. \label{equation6} \end{aligned} \tag{6}\] \[\begin{aligned} r_2 = \frac{n_1 + (n_2-1)\lambda}{3}.\label{equation7} \end{aligned} \tag{7}\] \[\begin{aligned} b = \frac{[n_1(n_1-1) + n_2(n_2-1)]\lambda + 2n_1n_2}{12}.\label{equation8} \end{aligned} \tag{8}\]
Since \(r_1\) and \(r_2\) must be integers, from Eqs. (6) and (7), we have the following:
\(\bullet\) If \(\lambda\equiv0\pmod3\) , then \(n_1\equiv0\pmod 3\) and \(n_2\equiv0\pmod 3\) .
\(\bullet\) If \(\lambda\equiv1\pmod3\) , \(n_1+n_2\equiv1\pmod3\) .
\(\bullet\) If \(\lambda\equiv2\pmod3\) , then there are no integer value for \(n_1\) and \(n_2\) that makes \(r_1\) and \(r_2\) integers.
Since \(b\) must be an integer, from Eq. (8), we have the following.
\(\bullet\) If \(\lambda\equiv0\pmod6\) , then \(n_1n_2\equiv0\pmod6\) .
\(\bullet\) If \(\lambda\equiv1\pmod6\) , then \(n_1+n_2\equiv0,1,4,9\pmod{12}\) .
\(\bullet\) If \(\lambda\equiv2\pmod6\) , then \((n_1+n_1)(n_1+n_2-1)-n_1n_2\equiv0\pmod6\) .
\(\bullet\) If \(\lambda\equiv3\pmod6\) , then \(n_1+n_2\equiv0,1\pmod{4}\) and \(n_1n_2\equiv0\pmod3\) .
\(\bullet\) If \(\lambda\equiv4\pmod6\) , then \(n_1+n_2\equiv0,1\pmod{3}\) and \(n_1n_2\equiv0\pmod2\) .
\(\bullet\) If \(\lambda\equiv5\pmod6)\) , then \(5(n_1+n_2)(n_1+n_2-1)-8n_1n_2\equiv0\pmod{12}\) .
Combining the cases above, we obtain the following theorem:
Theorem 2.8. Necessary conditions for the existence of a GDD \((n_1,n_2,4;\lambda,1)\) are
a. \(\lambda\equiv0\pmod6\) : \(n_1n_2\equiv0\pmod6\) , \(n_1\equiv0\pmod3\) and \(n_2\equiv0\pmod3\) .
b. \(\lambda\equiv1\pmod6\) : \(n_1+n_2\equiv1,4\pmod{12}\) .
c. \(\lambda\equiv3\pmod6\) : \(n_1+n_2\equiv0,9\pmod{12}\) and \(n_1n_2\equiv0\pmod3\) .
d. \(\lambda\equiv4\pmod6\) : \(n_1+n_2\equiv1\pmod3\) and \(n_1n_2\equiv0\pmod2\) .
Moreover, we have the following theorem from the divisiblity requirements of \(r_1\) and \(r_2\) .
Theorem 2.9. For \(\lambda\equiv2, 5\pmod6\) , a GDD \((n_1,n_2,4;\lambda,1)\) is not admissible.
Table 1 summarizes certain restrictions on \(\lambda\) where “None” means the design does not exist for any \(\lambda\) . Here \(n_1\) and \(n_2\) are given in modulo \(12\) .
| \(n_1 \backslash n_2\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | \(\lambda\equiv0,3\pmod6\) | \(\lambda\equiv1,4\pmod6\) | None | \(\lambda\equiv0\pmod6\) | \(\lambda\equiv1,4\pmod6\) | None | \(\lambda\equiv0\pmod6\) | \(\lambda\equiv4\pmod6\) | None | \(\lambda\equiv0,3\pmod6\) | \(\lambda\equiv4\pmod6\) | None |
| 1 | \(\lambda\equiv1,4\pmod6\) | None | None | \(\lambda\equiv1\pmod6\) | None | None | \(\lambda\equiv4\pmod6\) | None | None | None | None | None |
| 2 | None | None | \(\lambda\equiv1,4\pmod6\) | None | None | \(\lambda\equiv4\pmod6\) | None | None | \(\lambda\equiv4\pmod6\) | None | None | \(\lambda\equiv1,4\pmod6\) |
| 3 | \(\lambda\equiv0\pmod6\) | \(\lambda\equiv1\pmod6\) | None | None | \(\lambda\equiv4\pmod6\) | None | \(\lambda\equiv0,3\pmod6\) | None | None | \(\lambda\equiv3\pmod6\) | \(\lambda\equiv1,4\pmod6\) | None |
| 4 | \(\lambda\equiv1,4\pmod6\) | None | None | \(\lambda\equiv4\pmod6\) | None | None | \(\lambda\equiv4\pmod6\) | None | None | \(\lambda\equiv1,4\pmod6\) | None | None |
| 5 | None | None | \(\lambda\equiv4\pmod6\) | None | None | None | None | None | \(\lambda\equiv1,4\pmod6\) | None | None | \(\lambda\equiv1\pmod6\) |
| 6 | \(\lambda\equiv0\pmod6\) | \(\lambda\equiv4\pmod6\) | None | \(\lambda\equiv0,3\pmod6\) | \(\lambda\equiv4\pmod6\) | None | \(\lambda\equiv0,3\pmod6\) | \(\lambda\equiv1,4\pmod6\) | None | \(\lambda\equiv0\pmod6\) | \(\lambda\equiv1,4\pmod6\) | None |
| 7 | \(\lambda\equiv1,4\pmod6\) | None | None | None | None | None | \(\lambda\equiv1,4\pmod6\) | None | None | \(\lambda\equiv1\pmod6\) | None | None |
| 8 | None | None | \(\lambda\equiv4\pmod6\) | None | None | \(\lambda\equiv1,4\pmod6\) | None | None | \(\lambda\equiv1,4\pmod6\) | None | None | \(\lambda\equiv4\pmod6\) |
| 9 | \(\lambda\equiv0,3\pmod6\) | None | None | \(\lambda\equiv3\pmod6\) | \(\lambda\equiv1,4\pmod6\) | None | \(\lambda\equiv0\pmod6\) | \(\lambda\equiv1\pmod6\) | None | None | \(\lambda\equiv4\pmod6\) | None |
| 10 | \(\lambda\equiv4\pmod6\) | None | None | \(\lambda\equiv1,4\pmod6\) | None | None | \(\lambda\equiv1,4\pmod6\) | None | None | \(\lambda\equiv4\pmod6\) | None | None |
| 11 | None | None | \(\lambda\equiv1,4\pmod6\) | None | None | \(\lambda\equiv1\pmod6\) | None | None | \(\lambda\equiv4\pmod6\) | None | None | None |
Unless otherwise stated \(t\) is a nonnegative integer throughout this section.
Theorem 3.1. Necessary conditions are sufficient for the existence of a GDD \((n_1,\) \(n_2,4;\lambda,1)\) when \(\lambda\equiv1\pmod6\) and \(n_i\ge 4\) for \(i=1,2\) .
Proof. Let \(G_1\) and \(G_2\) be groups of sizes \(n_1\) and \(n_2\) respectively. When \(\lambda\equiv1\pmod6\) , from the necessary conditions, \(n_1+n_2\equiv1,4\pmod{12}\) . Then by Theorem 1.3, a BIBD \((n_1+n_2,4,1)\) exists. Also when \(n_i\ge 4\) for \(i=1,2\) , a BIBD \((n_i,4,6)\) exists. So the blocks of a BIBD \((n_1+n_2,4,1)\) on points of \(G_1\cup G_2\) together with \(t\) copies of the blocks of a BIBD \((n_i,4,6)\) on points of \(G_i\) provide the blocks of a GDD \((n_1,n_2,4;6t+1,1)\) . ◻
Theorem 3.2 Necessary conditions are sufficient for the existence of a GDD \((n_1,\) \(n_2,4;\) \(\lambda,1)\) for \(\lambda\equiv4\pmod6\) when
(i) \(n_1\equiv0\pmod{12}\) and \(n_2\equiv1,4\pmod{12}\) ,
(ii) \(n_1\equiv4\pmod{12}\) and \(n_2\equiv9\pmod{12}\) ,
(iii) \(n_1\equiv5,8\pmod{12}\) and \(n_2\equiv8\pmod{12}\) .
Proof. Let \(G_1\) and \(G_2\) be groups of sizes \(n_1\) and \(n_2\) respectively. When \(\lambda\equiv4\pmod6\) , from the necessary conditions, \(n_1+n_2\equiv1\pmod3\) and \(n_1n_2\equiv0\pmod2\) . But \(n_1+n_2\equiv1\pmod3\) implies \(n_1+n_2\equiv1,4,7,10\pmod{12}\) . When \(n_1+n_2\equiv1,4\pmod{12}\) , by Theorem 1.3, a BIBD \((n_1+n_2,4,1)\) exists. As \(n_1+n_2\equiv1,4\pmod{12}\) implies \(n_1\equiv0\pmod{12}\) and \(n_2\equiv1,4\pmod{12}\) ; \(n_1\equiv9\pmod{12}\) and \(n_2\equiv4\pmod{12}\) ; and \(n_1\equiv8\pmod{12}\) and \(n_2\equiv5,8\pmod{12}\) . Also by Theorem 1.3 a BIBD \((n_i,4,3)\) for \(i=1,2\) exists. Since a BIBD \((n_i,4,6)\) for \(n_i\ge4\) exists, the blocks of a BIBD \((n_1+n_2,4,1)\) on points of \(G_1\cup G_2\) together with the blocks of a BIBD \((n_i,4,3)\) and \(t\) copies of the blocks of a BIBD \((n_i,4,6)\) on points of \(G_i\) provide the blocks of a GDD \((n_1,n_2,4;6t+4,1)\) . ◻
Theorem 3.3. The necessary conditions are sufficient for the existence a GDD \((n_1,n_2,4;\) \(\lambda,1)\) , except possibly for the following cases:
\(\bullet\) \(\lambda\equiv0\pmod{6}\) , \(n_1\equiv0\pmod{12}\) and \(n_2\equiv0\pmod{3}\) .
\(\bullet\) \(\lambda\equiv0\pmod{6}\) , \(n_1\equiv0\pmod{6}\) and \(n_2\equiv3,9\pmod{12}\) .
\(\bullet\) \(\lambda\equiv3\pmod6\) , \(n_1\equiv0\pmod{12}\) and \(n_2\equiv0,9\pmod{12}\) .
\(\bullet\) \(\lambda\equiv3\pmod6\) , \(n_1\equiv3\pmod{12}\) and \(n_2\equiv6,9\pmod{12}\) .
\(\bullet\) \(\lambda\equiv4\pmod6\) , \(n_1\equiv0,6\pmod{12}\) and \(n_2\equiv7,10\pmod{12}\) .
\(\bullet\) \(\lambda\equiv4\pmod6\) , \(n_1\equiv1,4\pmod{12}\) and \(n_2\equiv6\pmod{12}\) .
\(\bullet\) \(\lambda\equiv4\pmod6\) , \(n_2\equiv2\pmod{12}\) and \(n_2\equiv2,5,8,11\pmod{12}\) .
\(\bullet\) \(\lambda\equiv4\pmod6\) , \(n_1\equiv3,9\pmod{12}\) and \(n_2\equiv10\pmod{12}\) .
\(\bullet\) \(\lambda\equiv4\pmod6\) , \(n_1\equiv8\pmod{12}\) and \(n_2\equiv11\pmod{12}\) .
Though the sufficiency of the above cases is not known, in the next subsection we establish the existence of certain infinite families of a GDD \((n_1,n_2,4;\lambda,1)\) when \(\lambda\equiv0,3,4\pmod6\) using resolvable designes.
Theorem 3.4. If an \(\alpha\) -resolvable BIBD \((n_1,k,\lambda)\) with replication number \(r\) and a BIBD \((\frac{r}{\alpha},\\ k+1,\lambda)\) exist, then a GDD \((n_1,n_2=\frac{r}{\alpha},k+1;\lambda,\alpha)\) exists.
Proof. Let \(G_1\) and \(G_2\) be groups of size \(n_1\) and \(n_2=\frac{r}{\alpha}\) respectively. Suppose the blocks of the \(\alpha\) -resolvable BIBD \((n_1,k,\lambda)\) are partitioned into \(\frac{r}{\alpha}\) classes namely: \(\pi_1\) , \(\pi_2\) , …, \(\pi_{\frac{r}{\alpha}}\) each of size \(\frac{\alpha n_1}{k}\) .
First, we construct blocks of the required GDD by taking union of the \(i^{th}\) element of \(G_2\) with each block of the \(i^{th}\) class for \(i=1,2,…,\frac{r}{\alpha}\) . These blocks provide blocks of size \(k+1\) , where pairs from \(G_1\) occur \(\lambda\) times and pairs from different groups occur \(\alpha\) times. Then, these blocks together with the blocks of a BIBD \((n_2=\frac{r}{\alpha},k+1,\lambda)\) on the points of \(G_2\) provide the blocks for GDD \((n_1,n_2=\frac{r}{\alpha},k+1;\lambda,\alpha)\) . ◻
Corollary 3.5. Suppose a KTS \((n_1)\) with replication number \(r\) and a BIBD \((\) \(n_2,4,\frac{n_2}{r})\) exist. Then there exists a GDD \((n_1,n_2,4;\frac{n_2}{r},1)\) .
Proof. Let \(G_1\) and \(G_2\) be groups of size \(n_1\) and \(n_2\) respectively such that a KTS \((n_1)\) with replication number \(r\) and a BIBD \((n_2,4,\frac{n_2}{r})\) exist. Then a KTS \((n_1)\) has \(r\) parallel classes namely: \(\pi_1, \pi_2, …, \pi_r\) each of size \(\frac{n_1}{3}\) and \(r\mid n_2\) . Let \(T_1, T_2, …, T_{\frac{n_2}{r}}\) be partitions of \(G_2\) into parts of size \(r\) each.
First we construct the blocks of the required GDD by taking union of the \(i^{th}\) element of \(T_j\) with each block of \(\pi_i\) for \(i=1,2,…,r\) and \(j=1,2,…,\frac{n_2}{r}\) . These blocks provide blocks of size \(4\) , where pairs from \(G_1\) occur \(\frac{n_2}{r}\) times and pairs from different groups occur only once. Then these blocks together with the blocks of a BIBD \((n_2,4,\frac{n_2}{r})\) on points \(G_2\) provides the blocks for GDD \((n_1,n_2,4;\frac{n_2}{r},1)\) . ◻
Theorem 3.6. A GDD \((n_1,n_2,4;\) \(\lambda,1)\) exists for \(\lambda\equiv3\pmod6\) when
(i) \(n_1=24t+9\) and \(n_2=36t+12\) , and
(ii) \(n_1=24t+15\) and \(n_2=36t+21\) .
Proof. Let \(G_1\) and \(G_2\) be groups of sizes \(n_1\) and \(n_2\) respectively. Let \(n_1=24t+9\) for a nonnegative integer \(t\) . Then a KTS \((n_1,3,1)\) with replication number \(r=12t+4\) exists. As a BIBD \((36t+12,4,\frac{36t+12}{r}=3)\) exists, by Corollary 3.5, a GDD \((n_1,n_2,4;3,1)\) exists. Since a BIBD \((n_i,4,6)\) for \(i=1,2\) exists, the blocks of a GDD \((n_1,n_2,4;3,1)\) on points of \(G_1\cup G_2\) together with \(s\) copies of the blocks of a BIBD \((n_i,4,6)\) on points of \(G_i\) provide the blocks of a GDD \((n_1,n_2,4;3+6s,1)\) for a nonnegative integer \(s\) . Similarily, by Corollary 3.5 and Theorem 1.3 we can construct a GDD \((n_1,n_2,4;6s+3,1)\) for \(n_1=24t+15\) and \(n_2=36t+21\) . ◻
Theorem 3.7. A GDD \((n_1,n_2,4;\lambda,1)\) exists for \(\lambda\equiv4\pmod6\) when \(n_1=6t+3\) and \(n_2=12t+4\) .
Proof. Let \(G_1\) and \(G_2\) be groups of sizes \(n_1\) and \(n_2\) respectively. Let \(n_1=6t+3\) . Then a KTS \((n_1,3,1)\) with replication number \(r=3t+1\) exists. As a BIBD \((12t+4,4,\frac{12t+4}{r}=4)\) exists, by Corollary 3.5, a GDD \((n_1,n_2,4;4,1)\) exists. Since a BIBD \((n_i,4,6)\) for \(i=1,2\) and \(n_1>3\) exists, the blocks of a GDD \((n_1,n_2,4;4,1)\) on points of \(G_1\cup G_2\) together with \(s\) copies of the blocks of a BIBD \((n_i,4,6)\) on points of \(G_i\) provide the blocks of a GDD \((n_1,n_2,4;4+6s,1)\) for a nonnegative integer \(s\) . ◻
Theorem 3.8. A GDD \((n_1,n_2,4;\) \(\lambda,1)\) exists for \(\lambda\equiv0\pmod6\) when
(i) \(n_1=6t+3\) and \(n_2=18t+6\) , and
(ii) \(n_1=12t+9\) and \(n_2=36t+24\) .
Proof. Let \(G_1\) and \(G_2\) be groups of sizes \(n_1\) and \(n_2\) respectively. Let \(n_1=6t+3\) . Then a KTS \((n_1,3,1)\) with replication number \(r=3t+1\) exists. As a BIBD \((18t+6,4,\frac{18t+6}{r}=6)\) exists, by Corollary 3.5, a GDD \((n_1,n_2,4;6,1)\) exists. Since a BIBD \((n_i,4,6)\) for \(i=1,2\) and \(n_1>3\) exists, the blocks of a GDD \((n_1,n_2,4;6,1)\) on points of \(G_1\cup G_2\) together with \(s\) copies of the blocks of a BIBD \((n_i,4,6)\) on points of \(G_i\) provide the blocks of a GDD \((n_1,n_2,4;6+6s,1)\) for a nonnegative integer \(s\) . Similarily, by Corollary 3.5 and Theorem 1.3 we can construct a GDD \((n_1,n_2,4;6s+6,1)\) when \(n_1=12t+9\) and \(n_2=36t+24\) . ◻
Example 3.9. As a KTS \((9)\) with replication number \(4\) and a BIBD \((12,\) \(4,12/4=3)\) exist, we have a GDD \((9,12,4;3,1)\) .
Example 3.10. As a KTS \((15)\) with repliation number \(7\) and a BIBD \((21,4,\) \(21/7=3)\) exist, we have a GDD \((15,21,4;3,1)\)
Theorem 3.11. If a BIBD \((\frac{(n-1)(n-2)}{2},4,n-2)\) exists, then a GDD \(\left(n, \frac{(n-1)(n-2)}{2},4;n-2,1\right)\) exists for \(n\equiv0\pmod3\) .
Proof. Let \(G_1\) be a set with \(n\) elements. For \(n\equiv0\pmod3\) , a BIBD \((n,3,n-2)\) is resolvable. By Corollary 1.9, a BIBD \((n,3,n-2)\) has \(N=\frac{(n-1)(n-2)}{2}\) parallel classes; each class consists of \(\frac{n}{3}\) mutually disjoint blocks which partitions \(G_1\) . Let \(\pi_1, \pi_2, …, \pi_{N}\) be the resolution classes. Let \(G_2\) be a set with \(N\) elements. Then the union of each block of \(\pi_i\) with the \(i^{th}\) element of \(G_2\) provides \(\frac{n(n-1)(n-2)}{6}\) blocks of size \(4\) in which pairs from \(G_1\) occur \(n-2\) times while pairs from \(G_1\) and \(G_2\) occur only once. If a BIBD \(\left(\frac{(n-1)(n-2)}{2},4,n-2\right)\) exists, then the blocks of a BIBD \(\left(\frac{(n-1)(n-2)}{2},4,n-2\right)\) on \(G_2\) along with the above blocks just constructed provide the required GDD. ◻
Example 3.12. A GDD \((6,10,4;4,1)\) exists.
Proof. Let \(G_1=\{1,2,3,4,5,6\}\) and \(G_2=\{a,b,c,d,e,f,g,h,i,j\}\) be two groups of sizes \(6\) and \(10\) respectively. By Example 1.10 a BIBD(6,3,4) on \(G_1\) has \(10\) resolution classes, each with \(2\) pairwise disjoint blocks which partition \(G_1\) . Let \(\pi_1, \pi_2, …, \pi_{10}\) be the parallel classes. Then by taking the union of each block of \(\pi_l\) with the \(l^{th}\) element of \(G_2\) for \(l=1,2,…,10\) , we have the following blocks of size \(4\) in which pairs of elements from \(G_1\) occur \(4\) times while pairs from both groups occur only once.
| \(1\) | \(4\) | \(1\) | \(3\) | \(1\) | \(3\) | \(1\) | \(3\) | \(1\) | \(2\) | \(1\) | \(2\) | \(1\) | \(2\) | \(1\) | \(2\) | \(1\) | \(2\) | \(1\) | \(2\) |
| \(2\) | \(5\) | \(2\) | \(5\) | \(2\) | \(4\) | \(2\) | \(4\) | \(3\) | \(5\) | \(3\) | \(4\) | \(3\) | \(4\) | \(4\) | \(3\) | \(4\) | \(3\) | \(5\) | \(3\) |
| \(3\) | \(6\) | \(4\) | \(6\) | \(5\) | \(6\) | \(6\) | \(5\) | \(4\) | \(6\) | \(5\) | \(6\) | \(6\) | \(5\) | \(5\) | \(6\) | \(6\) | \(5\) | \(6\) | \(4\) |
| \(a\) | \(a\) | \(b\) | \(b\) | \(c\) | \(c\) | \(d\) | \(d\) | \(e\) | \(e\) | \(f\) | \(f\) | \(g\) | \(g\) | \(h\) | \(h\) | \(i\) | \(i\) | \(j\) | \(j\) |
Since a BIBD \((10,4,4)\) exists by Theorem 1.3, hence the blocks of BIBD \((10,4,4)\) on \(G_2\) together with the above blocks just constructed provide the blocks of a GDD \((6,10,4;4,\) \(1)\) . ◻
We obtained necessary conditions for the existence of a GDD \((n_1, n_2, 4; \lambda, 1)\) and proved that the necessary conditions were sufficient in several cases. Further, we presented several new general constructions for GDDs with two unequal groups of sizes \(n_1\) and \(n_2\) with block size \(k\) as well as block size \(4\) and \(\lambda_{2}=1\) .
The authors thank the referees for their valuable comments, which improved the paper. They also express their gratitude to Professor Dinesh G. Sarvate for his critical insights, constructive feedback, and unwavering support throughout the preparation and refinement of this work.