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Abstract: Group divisible covering designs (GDCDs) were introduced by Heinrich and Yin as a natural generalization of both covering designs and group divisible designs. They have applications in software testing and universal data compression. The minimum number of blocks in a k‐GDCD of type g u is a covering number denoted by C ( k , g u ). When k = 3, the values of C ( 3 , g u ) have been determined completely for all possible pairs ( g , u ). When k = 4, Francetić et al. constructed many families of optimal GDCDs, but the determination remained far from complete. In this paper, two specific 4‐IGDDs are constructed, thereby completing the existence problem for 4‐IGDDs of type ( g , h ) u. Then, additional families of optimal 4‐GDCDs are constructed. Consequently the cases for ( g , u ) whose status remains undetermined arise when g ≡ 7 mod 12 and u ≡ 3 mod 6, when g ≡ 11 , 14 , 17 , 23 mod 24 and u ≡ 5 mod 6, and in several small families for which one of g and u is fixed. [ABSTRACT FROM AUTHOR]

The research on directed PBDs is motivated by the construction of t-deletion/insertion-correcting codes. Fuji-Hara, Miao, Wang, and Yin have determined the existence of directed PBDs with block sizes from the set [ABSTRACT FROM AUTHOR]

The construction of group divisible designs (GDDs) is a basic problem in design theory. While there have been some methods concerning the constructions of uniform GDDs, the construction of nonuniform GDDs remains a challenging problem. In this paper, we present a new approach to the construction of nonuniform GDDs with group type [ABSTRACT FROM AUTHOR]

A triple system is a collection of b blocks, each of size three, on a set of v points. It is j-balanced when every two j -sets of points appear in numbers of blocks that are as nearly equal as possible, and well balanced when it is j-balanced for each j ∈ {1, 2, 3}.Well-balanced systems arise in the minimization of variance in file availability in distributed file systems. It is shown that when a triple system that is 2-balanced and 3-balanced exists, so does one that is well balanced. Using known and new results on variants of group divisible designs, constructions for well-balanced triple systems are developed. Using these, the spectrum of pairs (v, b) for which such a well-balanced triple system exists is determined completely. [ABSTRACT FROM AUTHOR]

Grooming uniform all-to-all traffic in optical (SONET) rings with grooming ratio C requires the determination of a decomposition of the complete graph into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The determination of optimal C-groomings has been considered for [ABSTRACT FROM AUTHOR]

A G-design of order n is a decomposition of the complete graph on n vertices into edge-disjoint subgraphs isomorphic to G. Grooming uniform all-to-all traffic in optical ring networks with grooming ratio C requires the determination of graph decompositions of the complete graph on n vertices into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The existence spectrum problem of G-designs for five-vertex graphs is a long standing problem posed by Bermond, Huang, Rosa and Sotteau in 1980, which is closely related to traffic groomings in optical networks. Although considerable progress has been made over the past 30 years, the existence problems for such G-designs and their related traffic groomings in optical networks are far from complete. In this paper, we first give a complete solution to this spectrum problem for five-vertex graphs by eliminating all the undetermined possible exceptions. Then, we determine almost completely the minimum drop cost of 8-groomings for all orders n by reducing the 37 possible exceptions to 8. Finally, we show the minimum possible drop cost of 9-groomings for all orders n is realizable with 14 exceptions and 12 possible exceptions. [ABSTRACT FROM AUTHOR]

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]