turkmath.org

A (simple, undirected) graph G is an ordered pair (V;E), where V is a set of elements called vertices (or points), and E is a set of 2-element subsets of V , called edges. A graph consisting of all possible edges on n vertices is called a complete graph, denoted by Kn. For a graph G, a decomposition of Kn into disjoint copies of G is called a G-design of order n. The spectrum for a graph G is the set of positive integers n such that there exists a G-design of order n. In the literature, the problem of determining the spectrum of a graph has been considered for several types of graphs, such as complete graphs, trees, cycles, matchings, paths, stars, cubes, graphs of geometric solids, even graphs, theta graphs, unions of graphs, all graphs with up to ve vertices, and all graphs with six vertices and up to nine edges. The spectrum problem has been completely solved for cycles, matchings, paths, stars, graphs with up to ve vertices, and graphs with six vertices and up to eight edges. In this talk, I will introduce some direct construction techniques for graph designs.