Lower Bounds for Group Covering Designs

Abstract

A group covering design (GCD) is a set of mn points in n disjoint groups of size m and a collection of b k-subsets, called blocks, such that every pairset not contained in the same group occurs in at least one block. For m = 1, a GCD is a covering design [5]. Particular cases of GCD’s, namely transversal covers, covering arrays, Sperner systems etc. have been extensively studied by Poljak and Tuza [22], Sloane [24], Stevens et al. [26] and others. Cohen et al. [8], [9] and Sloane [24] have also shown applications of these designs to software testing, switching networks etc.. Determining the group covering number, the minimum value of b, for given k,m and n, in general is a hard combinatorial problem. This paper determines a lower bound for b, analogous to Schönheim lower bound for covering designs [23]. It is shown that there exist two classes of GCD’s (Theorems 15 and 18) which meet these bound. Moreover, a construction of a minimum GCD from a covering design meeting the Schönheim lower bound is given. The lower bound is further improved by one for three different classes of GCD’s. In addition, construction of group divisible designs with consecutive block sizes (Theorems 20 and 21) using properties of GCD’s are given.