@MISC{Wiedemann06hamminggeometry, author = {Douglas H. Wiedemann}, title = {HAMMING GEOMETRY}, year = {2006} }

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Abstract

This thesis deals with the geometry of the n-th Cartesian powers of the complete graphs K(b). Emphasis is placed on the n-th power of K(2), the graph of the n-cube. We inv estigate sets of vertices which behave like the convex sets of Euclidean geometry. A geometric characterization is given for the solution sets of 2SAT problems (systems of Boolean disjunctions of two literals). As a result, an algorithm is obtained for solving 2SAT problems with a limited number of additional parity constraints. Furthermore, an algorithm is obtained for computing the xed points of any contraction mapping of the graph of the n-cube to itself. The next chapter considers sets related to convexity in a more loose sense. For example, any convex set in real n-space meets some collection of closed orthants. Many properties are derived for the sets of vertices of the n-cube which represent these collections of orthants. The nal chapter looks at decomposition and covering problems of the n-cube. For example, it is shown that the vertices of the n-cube cannot be partitioned into