Graph Products and New Solutions to Oberwolfach Problems

Graph Products and New Solutions to Oberwolfach Problems

We introduce the circle product, a method to construct simple graphs starting from known ones. The circle product can be applied in many different situations and when applied to regular graphs and to their decompositions, a new regular graph is obtained together with a new decomposition. In this paper we show how it can be used to construct infinitely many new solutions to the Oberwolfach probl...

Graph-Theoretic Solutions to Computational Geometry Problems

Many problems in computational geometry are not stated in graphtheoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm depends on the special properties of the graph constructed in this way. We survey the art gallery problem, partition into rectangles, minimum-diameter clustering, ...

Solutions to the Oberwolfach problem for orders 18 to 40

The Oberwolfach problem (OP) asks whether Kn (for n odd) or Kn minus a 1-factor (for n even) admits a 2-factorization where each 2-factor is isomorphic to a given 2-factor F . The order n and the type of the 2-factor F are the parameters of the problem. For n ≤ 17, the existence of a solution has been resolved for all possible parameters. There are also many special types of 2-factors for which...

Colouring problems related to graph products and coverings

The Oberwolfach Problem in Graph Theory

During the summer of 2007, I participated in a Research Experience for Undergraduates (REU) at Illinois State University. I was one of eight undergraduates, all of whom were mathematics education majors. During my eight week experience, we worked with four in-service teachers from the Normal-Bloomington area, as well as math and math education professors from Illinois State and surrounding univ...