Pairwise Balanced Designs and Sigma Clique Partitions

TL;DR

This paper investigates minimizing the sum of block sizes in pairwise balanced designs with size constraints, establishing bounds and analyzing asymptotic behavior of related graph parameters.

Contribution

It introduces new bounds for the minimal sum of block sizes in constrained pairwise balanced designs and applies these to graph partition problems.

Findings

Established lower bounds for S(n;m) and S0(n;m).

Determined asymptotic behavior of sigma clique partition numbers for specific graphs.

Connected design theory bounds to graph partition problems.

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 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.