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On The Genus Of A Block Design, Joan Marie Rahn
On The Genus Of A Block Design, Joan Marie Rahn
Dissertations
The genus of a design (BIBD or PBIBD) is defined to be the genus of its corresponding hypergraph (objects as vertices, blocks as edges); that is, the genus of the bipartite graph associated with the hypergraph in a natural way. The Euler formula is used to establish a lower bound (gamma) for the genus of a block design. An imbedding of the design of the surface of genus (gamma) is then described by a voltage hypergraph or voltage graph. Use of the lower bound formula leads to a characterization of planar BIBDs. A connection between a block design derived from …
The Chromatic And Cochromatic Number Of A Graph, John Gordon Gimbel
The Chromatic And Cochromatic Number Of A Graph, John Gordon Gimbel
Dissertations
Clearly, there are many ways that one can partition the vertex sets of graphs. In the first chapter of this work I examine the problem of determining, for a given graph, the minimum order of a vertex partition having specified properties. In the remaining chapters I concentrate on partitions of two types--those in which each subset induces an empty graph and those in which each subset induces an empty or a complete graph.
The chromatic number of a graph G is the minimum number of subsets into which V(G) can be partitioned so that each subset induces an empty graph. …