Abstract
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 S′(n,m) be the smallest integer s 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 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.
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