Pairwise Balanced Designs and Sigma Clique Partitions

In this paper, we are interested in minimizing the sum of block sizes in a pairwise balanced design, where there are some constraints on the size of one block or the size of the largest block. For every positive integers n;m, where m ? n, let S(n;m) be the smallest integer s for which there exists a PBD on n points whose largest block has size m and the sum of its block sizes is equal to s. Also, let S0(n;m) be the smallest integers for which there exists a PBD on n points which has a block of size m and the sum of it block sizes is equal to s. We prove some lower bounds for S(n;m) and S0(n;m). Moreover, we apply these bounds to determine the asymptotic behaviour of the sigma clique partition number of the graph Kn-Km, Cocktail party graphs and complement of paths and cycles.