{"id":{"repo_id":"wku-diss","oai_identifier":"oai:digitalcommons.wku.edu:theses-1280"},"canonical_url":"https://search.dev.ndltd.org/etd/wku-diss/oai:digitalcommons.wku.edu:theses-1280","repository":{"repo_id":"wku-diss","name":"Western Kentucky University","base_url":"https://digitalcommons.wku.edu/do/oai/"},"display":{"title":"On 4-Regular Planar Hamiltonian Graphs","abstract":"In order to research knots with large crossing numbers, one would like to be able to select a random knot from the set of all knots with n crossings with as close to uniform probability as possible. The underlying graph of a knot diagram can be viewed as a 4-regular planar graph. The existence of a Hamiltonian cycle in such a graph is necessary in order to use the graph to compute an upper bound on rope length for a given knot. The algorithm to generate such graphs is discussed and an exact count of the number of graphs is obtained. In order to allow for the existence of such a count, a somewhat technical definition of graph equivalence is used. The main result of the thesis is the asymptotic results of how fast the number of graphs with n vertices (crossings) grows with n.","abstract_html":"In order to research knots with large crossing numbers, one would like to be able to select a random knot from the set of all knots with n crossings with as close to uniform probability as possible. The underlying graph of a knot diagram can be viewed as a 4-regular planar graph. The existence of a Hamiltonian cycle in such a graph is necessary in order to use the graph to compute an upper bound on rope length for a given knot. The algorithm to generate such graphs is discussed and an exact count of the number of graphs is obtained. In order to allow for the existence of such a count, a somewhat technical definition of graph equivalence is used. The main result of the thesis is the asymptotic results of how fast the number of graphs with n vertices (crossings) grows with n.","abstract_has_math":false,"creators":["High, David"],"institution":null,"degree_name":"Master of Science","degree_level":null,"degree_discipline":"Department of Mathematics and Computer Science","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2006,"date_issued":"2006-05-01T07:00:00Z","date_published":"2006-05-01T07:00:00Z","updated_at":"2026-07-24T06:07:16Z","subjects":["Applied Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wku.edu/theses/277","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["High, David"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematics and Computer Science"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Applied Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wku.edu/theses/277"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In order to research knots with large crossing numbers, one would like to be able to select a random knot from the set of all knots with n crossings with as close to uniform probability as possible. The underlying graph of a knot diagram can be viewed as a 4-regular planar graph. The existence of a Hamiltonian cycle in such a graph is necessary in order to use the graph to compute an upper bound on rope length for a given knot. The algorithm to generate such graphs is discussed and an exact count of the number of graphs is obtained. In order to allow for the existence of such a count, a somewhat technical definition of graph equivalence is used. The main result of the thesis is the asymptotic results of how fast the number of graphs with n vertices (crossings) grows with n."]},{"key":"dc:title","label":"Title","values":["On 4-Regular Planar Hamiltonian Graphs"]}]}],"canonical_facts":{"dc:creator":["High, David"],"dc:description.abstract":["In order to research knots with large crossing numbers, one would like to be able to select a random knot from the set of all knots with n crossings with as close to uniform probability as possible. The underlying graph of a knot diagram can be viewed as a 4-regular planar graph. The existence of a Hamiltonian cycle in such a graph is necessary in order to use the graph to compute an upper bound on rope length for a given knot. The algorithm to generate such graphs is discussed and an exact count of the number of graphs is obtained. In order to allow for the existence of such a count, a somewhat technical definition of graph equivalence is used. The main result of the thesis is the asymptotic results of how fast the number of graphs with n vertices (crossings) grows with n."],"dc:identifier":["https://digitalcommons.wku.edu/theses/277"],"dc:subject":["Applied Mathematics"],"dc:title":["On 4-Regular Planar Hamiltonian Graphs"],"dc:type":["Thesis"],"thesis:degree_discipline":["Department of Mathematics and Computer Science"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T06:07:16Z"}