Pairwise balanced designs and sigma clique partitions

In this paper, we are interested inminimizing 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, wherem ≤ n, letS (n,m) be the smallest integer s for which there exists a PBD on n points whose largest block has sizem and the sum of its block sizes is equal to s. Also, letS (n,m) be the smallest integer s for which there exists a PBD on n points which has a block of sizem and the sum of it block sizes is equal to s. We prove some lower bounds for S (n,m) and S (n,m). Moreover, we apply these bounds to determine the asymptotic behaviour of the sigma clique partition number of the graph Kn − Km, the Cocktail party graphs and complement of paths and cycles. © 2015 Elsevier B.V. All rights reserved.