Hamiltonian Cycles in Sparse Graphs

Hamiltonian Cycles in Sparse Graphs Alexander Hertel Master of Science Graduate Department of Computer Science University of Toronto 2004 The subject of this thesis is the Hamiltonian Cycle problem, which is of interest in many areas including graph theory, algorithm design, and computational complexity. Named after the famous Irish mathematician Sir William Rowan Hamilton, a Hamiltonian Cycle within a graph is a simple cycle that passes through each vertex exactly once. This thesis provides a history of the problem, a survey of major results, as well as a detailed account of the author’s original contributions with respect to sparse graphs. The first of these is the “Stonecarver’s Algorithm”, which is successful in finding Hamiltonian Cycles in random regular graphs. The second gives upper and lower bounds on the creation of a specific obstruction to Hamiltonicity under the context of the Stonecarver Algorithm. Finally, the third is a theorem which strengthens Barnette’s Conjecture.