Pairwise balanced designs and sigma clique partitions

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 K n − K m , Cocktail party graphs and complement of paths and cycles.

Clique coverings and claw-free graphs

“…It was conjectured by G. O. H. Katona and T. Tarján, and proved in the papers [4,13,11], that for every graph G on n vertices, scp(G) ≤ ⌊n 2 /2⌋ holds, with equality if and only if G is the complete bipartite graph K ⌊n/2⌋,⌈n/2⌉ . Also, this parameter relates to a number of other well-known problems (see [6]). The second author and R. Javadi proved the following Nordhaus-Gaddum type theorem for scp.…”

Sigma clique covering of graphs

“…Furthermore, the authors in [5], using the pairwise balanced designs, have proved that scp(K t (2)) ∼ (2t) For the lower bound, assume that {C 1 , . .…”

On clique coverings of complete multipartite graphs

“…. , S n of G. Similarly, the sigma clique partition number, denoted by scp(G), is the smallest weight of a clique partition (see for example [5]) of G. That is, scp(G) = min…”