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Non-uniform group divisible designs have been studied by numerous researchers in the past two decades due to their vital applications in the constructions for other types of designs. Much progress has been made for the existence of $$\{4\}$$ -GDDs of type $$g^um^1$$ , especially when $$gu$$ is even. The corresponding problem for block size three had been solved by Colbourn et al. (J Comb Theory Ser A 59:73-89, ). In this paper, we consider the entire existence problem for such $$\{4\}$$ -GDDs. We show that, for each given g, up to a small number of undetermined cases of $$u$$ , the necessary conditions on $$(u,m)$$ for the existence of a $$\{4\}$$ -GDD of type $$g^um^1$$ are also sufficient. [ABSTRACT FROM AUTHOR]

In this paper, we continue to investigate the existence of 5 -GDDs, 4-frames and 4-RGDDs, which have been studied by numerous researchers in the past two decades due to their vital applications in the constructions for other types of designs. Much progress has been made for the existence problems of these designs, while the problems are still open. Our new results mainly lie in the cases with relatively large group size. We also construct some smaller designs, including 4 -RGDDs of types 2 184 , 9 44 , 14 10 , 18 18 , 22 10 and 36 11 . [ABSTRACT FROM AUTHOR]

Constructing non-uniform designs has become a topic of considerable interest over the last two decades due to the vital role of such designs in the constructions for other types of designs. Ge, Rees and Shalaby started the investigation into the spectrum of non-uniform Kirkman frames of type $$h^{u}m^{1}$$. They determined the spectrum for the cases where $$h\in \{2,4,6,8,10,12\}$$, leaving 24 cases of $$(h,u,m)$$. In this paper, we continue to investigate this spectrum problem by removing all of these 24 possible exceptions. Using this result, we show that for each value of $$h \equiv 0 {\,\,\mathrm{mod}\, 12}\,$$ the obvious necessary conditions on $$u, m$$ are also sufficient. [ABSTRACT FROM AUTHOR]

Abstract: Non-uniform group divisible designs (GDDs) and non-uniform Kirkman frames are useful in the constructions for other types of designs. In this paper, we consider the existence problems for -GDDs of type with and Kirkman frames of type . First, we determine completely the spectrum for uniform -GDDs of type . Then, we consider the entire existence problem for non-uniform -GDDs of type with . We show that, for each given , up to a small number of undetermined cases of , the necessary conditions on for the existence of a -GDD of type are also sufficient, except possibly when for and when for . Finally, a similar result for Kirkman frames of type is obtained. We show that, for each given , up to a small number of undetermined cases of , the necessary conditions on for the existence of a Kirkman frame of type are also sufficient, except possibly when for and when for . [Copyright &y& Elsevier]

Nonuniform group divisible designs (GDDs) have been studied by numerous researchers for the past two decades due to their essential role in the constructions for other types of designs. In this paper, we investigate the existence problem of [ABSTRACT FROM AUTHOR]

Abstract: Non-uniform group divisible designs are instrumental in the constructions for other types of designs. Most of the progress for the existence of -GDDs of type is on the case when is even, where the existence for small has played a key role. In order to determine the spectrum for -GDDs of type with being odd, we continue to investigate the small cases with in this paper. We show that, for each , the necessary conditions for the existence of a -GDD of type are also sufficient. As the applications of these GDDs, we obtain a few pairwise balanced designs with minimum block size 4. Meanwhile, we also improve the existence result for frame self-orthogonal Mendelsohn triple systems of type by reducing an infinite class of possible exceptions, namely and , to eight undetermined cases. [Copyright &y& Elsevier]