Intersection Problems For Some Classes Of Block Designs

Let Kv be the complete graph with v vertices and λKv denote the graph Kv with each of its edges replicated λ times. Given a family (?) of graphs each of which is simple and connected, a λ-fold (?)-design of order v, denoted by (λKV,(?))-design, is a pair (X, B) where X is the vertex set of Kv and B is a collection of subgraphs (called blocks) of λKV, such that each block is isomorphic to a graph in (?), and each edge of λKv belongs to exactly λ blocks of B. If in the definition of (?)-designs we replace the term "exactly" with "at most"(or "at least"), we have a (λKv,(?))-packing (or covering). When λ=1, a (Kv,(?))-packing (or covering) is called a (?)-packing (or covering) of order v. When (?) contains a single graph G, i.e.,(?)={G}, a (λKv,{G})-design (packing or covering) is simply written as a (λKv, G)-design (packing or covering). If G is the complete graph Kk, a (Kv, Kk)-design is called a Steiner system S(2, k, v).The intersection problem for S(2, k, v)’s was first introduced by Kramer and Mes-ner in1974. This initial work was later extended to many other combinatorial structures, such as intersection problem for Latin squares and intersection problem for G-designs. In this thesis, we investigate the the fine triangle intersections for maximum kite pack-ings, the fine triangle intersection problem for minimum kite coverings, the flower in-tersection problem for S(2,4, v)s, and the intersection problem for PBD(4,7*).This thesis is organized as follows.Chapter1gives a brief introduction on the background of intersection problem for combinatorial designs.In Chapter2, a complete solution to the fine triangle intersection problem for maximum kite packings of small orders v is made by direct constructions, where v∈{4,5,6,7,10}. Using recursive constructions and fine triangle intersection numbers of (Kv\Khv, G)-design (or maximum packing), the fine triangle intersection problems for maximum kite packings are completed.In Chapter3, a complete solution to the fine triangle intersection problem for min-imum kite coverings of small orders v is made by direct constructions, where v∈{4,5,6,7,10}. Applying the recursive constructions and known results of Chapter2, the fine triangle intersection problems for minimum kite coverings are completely solved.In Chapter4, the flower intersection problem for S(2,4, v)s is investigated. Flower intersection numbers of S(2,4, v)s of small orders v are established by direct construc-tions, where v∈{13,16,25,28,37}. Using the recursive constructions, the flower intersection problem for S(2,4, v)s is dealt with, apart from three undecided values for v=25,28,37.In Chapter5, the intersection problem for PBD(4,7*)s is investigated. Some inter-section numbers of PBD(4,7*) of small orders v are established by direct constructions, where v∈{22,31,34,46,58,70}. Applying the recursive constructions and known re-sults of Chapter4, the intersection numbers of PBD(4,7*) are determined, apart from six undecided values for v=22,31,34,46,58,70.