2014/11/02 by Akbar Davoodi, Davoodi, Akbar, Ramin Javadi +3 · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO
paper · pdf · doi:10.48550/arxiv.1411.0266
15 pages, 1 figures
arxiv created 2014/11/02 · arxiv updated 2014/11/04
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.