Leírás
Let G be a simple graph. A family of cliques of G is called an (edge) clique covering for G if every edge of G belongs to at least one member of the family. A clique covering in which each edge belongs to exactly one clique, is called a clique partition. The minimum size of a clique covering of G is called the clique cover number of G and is denoted by cc(G). Similarly, the clique partition number of G, denoted by cp(G), is defined as the minimum number of cliques in a clique partition of G. The subject of clique covering has been widely studied in recent decades. First time, Erd˝os et al. in 1966 presented a close relationship between the clique covering and the set intersection representation. Also, they proved that the clique partition number of a graph on n vertices cannot exceed n 2/4 (known as Erd˝os-Goodman-P´osa theorem). The connections of clique covering and other combinatorial objects have been explored in the literature. Also a number of different variants of the clique cover number have been proposed. In this talk, two clique covering parameters, namely sigma clique cover number and local clique cover number are considered. Among some results on theses parameters, we briefly touch the connection between clique partitions and pairwise balanced designs.