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Schema design is one of the fundamentals in database theory and practice as well. In this paper, we discuss the problem of locally valid dimensional attributes in a classification hierarchy of a typical OLAP scenario. In a first step, we show that the traditional star and snowflake schema approach is not feasible in this very natural case of a hierarchy. Therefore, we sketch two alternative modeling approaches resulting in practical solutions and a seamless extension of the traditional star and snowflake schema approach: In a pure relational approach, we replace each dimension table of a star / snowflake schema by a set of views directly reflecting the classification hierarchy. The second approach takes advantage of the object-relational extensions. Using object-relational techniques in the context for the relational representation of a multidimensional OLAP scenario is a novel approach and promises a clean and smooth schema design.

This thesis touches three different topics in graph theory, namely, rainbow colouring, product dimension and boxicity. Rainbow colouring An edge colouring of a graph is called a rainbow colouring, if every pair of vertices is connected by atleast one path in which no two edges are coloured the same. The rainbow connection number of a graph is the minimum number of colours required to rainbow colour it. In this thesis we give upper bounds on rainbow connection number based on graph invariants like minimum degree, vertex connectivity, and radius. We also give some computational complexity results for special graph classes. Product dimension The product dimension or Prague dimension of a graph G is the smallest natural number k such that G is an induced subgraph of a direct product of k complete graphs. In this thesis, we give upper bounds on the product dimension for forests, bounded tree width graphs and graphs of bounded degeneracy. Boxicity and cubicity The boxicity (cubicity of a graph G is the smallest natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes(axis-parallel unit cubes) in Rk .In this thesis, we study the boxicity and the cubicity of Cartesian, strong and direct products of graphs and give estimates on the boxicity and the cubicity of a product graph based on invariants of the component graphs. Separation dimension The separation dimension of a hypergraph H is the smallest natural number k for which the vertices of H can be embedded in Rk such that any two disjoint edges of H can be separated by a hyper plane normal to one of the axes. While studying the boxicity of line graphs, we noticed that a box representation of the line graph of a hypergraph has a nice geometric interpretation. Hence we introduced this new parameter and did an extensive study of the same.