UVM Theses and Dissertations
Format:
Print
Author:
Stor, Kirsten
Dept./Program:
Mathematics
Year:
2010
Degree:
PhD
Abstract:
A drawing of a graph is said to be a superthrackle if every pair of edges (including adjcent edges) cross exactly once. This is a variant of the previously studied thrackle, in which only pairs of non-adjacent edges must cross. There is yet no proof of a classification of graphs which can be drawn as thrackles in the plane. Our main theorem is a characterization of the class of graphs which can be drawn as superthrackles in the plane. An interesting corollary of our characterization is that a graph which can be drawn so that every pair of edges cross an odd number of times can also be drawn so that every pair of edges cross exactly once, and in fact, at the same point in the plane. Another corollary we obtain from the proof of our main theorem is a Kuratowski-like theorem for which graphs can be drawn as superthrackles in the plane. We look at the relationships between superthrackles and other types of thrackles on the sphere. We end with some results about drawing graphs as superthrackles on surfaces of higher genus.