The Existence Of2X3Grid-block Packing And Coverings

Let Kv be a complete graph of v vertice, Kr×Kc be Cartersian product of Kr and Kc in which any two different vertices (a1,b1) and (a2,b2) are joined if and only if a1-a2or b1-b2. A Kr×Kc-packing (orKr×Kc-covering), also called rxc grid-block packing (or covering), is a pair (X,A), where X is the vertice set of Kv, A is a set of subgraphs which are isomorphism to Kr×Kc (called grid-blocks), satisfying each edge of Kv occurs at most (or at least) once in certain subgraph.A rxc grid-block packing (or covering) is called rxc grid-block design if each edge of Kv occurs exactly once in certain subgraph. F.Hwang et al, Taiwanese researchers, earlier defined grid-block designs and also illustrated their applications in DNA library screening. Since then, the existence of grid-block designs has absorbed much attention.The necessary condition of2×3grid-block design is v≡1(mod9), which was proved to be sufficient by J. Carter in1989. When v(?)1(mod9), the related packings and coverings appear naturally. Very recently, L.Wang et al almost established the existence of2×3grid-block packings, leaving two possible exceptions.In this thesis, we first remove two possible exceptions in the existence of2×3grid-block packings. In addition, we also almost establish the existence of2x3grid-block coverings, leaving three possible exceptions. In addition, we almost settle the existence of K2×K3-GDD with type gu...