A group divisible design GDDv=1+n+n,3, 3, λ 1, λ 2 is an ordered pair $V, \mathcal{B}$ where V is an 1+n+n-set of symbols and $\mathcal{B}$ is a collection of 3-subsets called blocks of V satisfying the following properties: the 1+n+n-set is divided into 3 groups of sizes 1, n and n; each pair of symbols from the same group occurs in exactly λ 1 blocks in $\mathcal{B}$ ; and each pair of symbols from different groups occurs in exactly λ 2 blocks in $\mathcal{B}$ . Let λ 1, λ 2 be positive integers. Then the spectrum of λ 1, λ 2, denoted by Specλ 1, λ 2, is defined by

Specλ 1, λ 2={n∈i¾?: a GDDv=1+n+n,3, 3, λ 1, λ 2i¾?exists}.

We found in [10] the spectrum Specλ 1, λ 2 provided that λ 1i¾?λ 2 in all situations. We find in this paper Specλ 1, λ 2 when λ 1<λ 2 in all situations.

/pdf/gdds-with-two-associate-classes-and-with-three-groups-of-2ynafoa2qo.pdf

GDDs with Two Associate Classes and with Three Groups of Sizes 1, n, n and λ1<λ2

Group divisible designs with two associate classes, and quadratic leaves of triple systems

A group divisible design GDD(m, n; 3, λ 1, λ 2) is an ordered triple (V,G,B), where V is a m + nset of symbols, G is a partition of V into 2 sets of sizes m, n, each set being called group, and B is a collection of 3subsets
(called blocks) of V , such that each pair of symbols from the same group occurs in exactly λ 1 blocks; and each pair of symbols from different groups occurs in exactly λ 2 blocks. In this paper, we find necessary and sufficient conditions for the existence of a GDD(m, n; 3, λ 1, λ 2) with λ 1 ≥ λ 2.

/pdf/group-divisible-designs-with-two-associate-classes-and-with-3sdcaaygwn.pdf

Group divisible designs with two associate classes and with two unequal groups

A group divisible design GDD(v =3+n+n, 3,3,λ1,λ2) is an ordered pair (V,B) whereV is an (3+n+n)-set of symbols and B is a collection of 3-subsets (called blocks) of V satisfying the following properties: the (3 + n + n)-set is divided into 3 groups of sizes 3, n and n; each pair of symbols from the same group occurs in exactly λ1 blocks in B; and each pair of symbols from diﬀerent groups occurs in exactly λ2 blocks in B. Let λ1,λ2 be positive integers. Then the spectrum of λ1,λ2, denoted by Spec(λ1,λ2), is deﬁned by Spec(λ1,λ2)={n ∈ N : a GDD(v =3+n + n,3,3,λ 1,λ2) exists }. We ﬁnd the spectrum Spec(λ1,λ2) for all λ1 ≥ λ2.

GDDs WITH TWO ASSOCIATE CLASSES AND WITH THREE GROUPS OF SIZES 3,nAND n

3-Cycle Systems and Structure within Graph Decompositions

A group divisible design GDD(v = v 1 + v 2 + · · · + v g,g, k, λ 1, λ 2) is an ordered triple (V, G, B), where V is a vset of symbols, G is a partition of V into g sets of size v 1, v 2,⋯ v g, each set being called group, and B is a collection of ksubsets (called blocks) of
V, such that each pair of symbols from the same group occurs in exactly λ 1 blocks; and each pair of symbols from different groups occurs in exactly λ 2 blocks. Here, we focus on an existence problem of GDDs with two associate classes or when g = 2, and with blocks of size 3, when the required designs have two groups of unequal sizes and λ 2 = 4. We obtain the necessary conditions and prove that these conditions are sufficient.

Group divisible designs with two associate classes and λ 2 = 4

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Group divisible designs with two associate classes and λ 2 = 4