Browsing by Author "Carla D. Savage, Committee Chair"

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    Hamilton Cycle Heuristics in Hard Graphs
    (2004-03-23) Shields, Ian Beaumont; Jon Doyle, Committee Member; Matthias F. Stallmann, Committee Member; Robert E. Hartwig, Committee Member; Carla D. Savage, Committee Chair
    In this thesis, we use computer methods to investigate Hamilton cycles and paths in several families of graphs where general results are incomplete, including Kneser graphs, cubic Cayley graphs and the middle two levels graph. We describe a novel heuristic which has proven useful in finding Hamilton cycles in these families and compare its performance to that of other algorithms and heuristics. We describe methods for handling very large graphs on personal computers. We also explore issues in reducing the possible number of generating sets for cubic Cayley graphs generated by three involutions.
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    Symmetric chain decompositions and independent families of curves.
    (2003-07-08) Jiang, Zongliang; Carla D. Savage, Committee Chair; George N. Rouskas, Committee Member; Matthias Stallmann, Committee Member
    This thesis shows that symmetric independent families of n curves with the minimum possible number of regions exist for all n less than or equal to 16. Recent research has shown that such an independent family of curves exists for all prime n [Griggs, Killian, and Savage 2002]. For composite n, before this thesis, such a symmetric independent family of curves was known to exist only for n = 2,4,6,8,9, and 10. An independent family of curves is a collection of simple closed curves intersecting at finite number of points. If we label the curves with 1,2,...,n, then each region is labeled with the set of labels of the curves containing that region. If every subset of [1,2,...,n] is a label for at least one region, then the collection of curves is an independent family of curves. If we rotate any curve around a point by an angle of (2π/n) radians (n-1) times and each time it coincides with one of the other curves, then the collection of curves is a symmetric independent family of curves. We solve this geometric problem by first solving a combinatorial problem of looking for symmetric chain decompositions (SCD's) of necklace-representative posets with the chain cover property (CCP). We then show the way of constructing a symmetric independent family of curves with the minimum possible number of regions from an SCD of necklace-representative poset with the CCP.