{"id":{"repo_id":"wvu","oai_identifier":"oai:researchrepository.wvu.edu:etd-2453"},"canonical_url":"https://search.dev.ndltd.org/etd/wvu/oai:researchrepository.wvu.edu:etd-2453","repository":{"repo_id":"wvu","name":"West Virginia University","base_url":"https://researchrepository.wvu.edu/do/oai/"},"display":{"title":"Chords of longest circuits of graphs","abstract":"This thesis is on a long standing open conjecture proposed by one of the most prominent mathematicians, Dr. C. Thomassen: Every longest circuit of 3-connected graph has a chord. In 1987, C. Q. Zhang proved that every longest circuit of a 3-connected planar graph G has a chord if G is cubic or if the minimum degree is at least 4. In 1997, Carsten Thomassen proved that every longest circuit of 3-connected cubic graph has a chord.;In this dissertation, we prove the following three independent partial results: (1) Every longest circuit of a 3-connected graph embedded in a projective plane with minimum degree at least has a chord (Theorem 2.3.1). (2) Every longest circuit of a 3-connected cubic graph has at least two chords. Furthermore if the graph is also a planar, then every longest circuit has at least three chords (Theorem 3.2.6, 3.2.7). (3) Every longest circuit of a 4-connected graph embedded in a torus or Klein bottle has a chord.;We get these three independent results with three totally different approaches: Connectivity (Tutte circuit), second Hamilton circuit, and charge and discharge methods.","abstract_html":"This thesis is on a long standing open conjecture proposed by one of the most prominent mathematicians, Dr. C. Thomassen: Every longest circuit of 3-connected graph has a chord. In 1987, C. Q. Zhang proved that every longest circuit of a 3-connected planar graph G has a chord if G is cubic or if the minimum degree is at least 4. In 1997, Carsten Thomassen proved that every longest circuit of 3-connected cubic graph has a chord.;In this dissertation, we prove the following three independent partial results: (1) Every longest circuit of a 3-connected graph embedded in a projective plane with minimum degree at least has a chord (Theorem 2.3.1). (2) Every longest circuit of a 3-connected cubic graph has at least two chords. Furthermore if the graph is also a planar, then every longest circuit has at least three chords (Theorem 3.2.6, 3.2.7). (3) Every longest circuit of a 4-connected graph embedded in a torus or Klein bottle has a chord.;We get these three independent results with three totally different approaches: Connectivity (Tutte circuit), second Hamilton circuit, and charge and discharge methods.","abstract_has_math":false,"creators":["Li, Xuechao"],"institution":null,"degree_name":"PhD","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Cun-Quan Zhang."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2001,"date_issued":"2001-12-01T08:00:00Z","date_published":"2001-12-01T08:00:00Z","updated_at":"2026-07-24T06:15:47Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://researchrepository.wvu.edu/etd/1450"],"render_values":[{"text":"https://researchrepository.wvu.edu/etd/1450","href":"https://researchrepository.wvu.edu/etd/1450","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.33915/etd.1450","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Cun-Quan Zhang."]},{"key":"dc:creator","label":"Author","values":["Li, Xuechao"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2019-01-17T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["PhD"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://doi.org/10.33915/etd.1450","https://researchrepository.wvu.edu/etd/1450"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis is on a long standing open conjecture proposed by one of the most prominent mathematicians, Dr. C. Thomassen: Every longest circuit of 3-connected graph has a chord. In 1987, C. Q. Zhang proved that every longest circuit of a 3-connected planar graph G has a chord if G is cubic or if the minimum degree is at least 4. In 1997, Carsten Thomassen proved that every longest circuit of 3-connected cubic graph has a chord.;In this dissertation, we prove the following three independent partial results: (1) Every longest circuit of a 3-connected graph embedded in a projective plane with minimum degree at least has a chord (Theorem 2.3.1). (2) Every longest circuit of a 3-connected cubic graph has at least two chords. Furthermore if the graph is also a planar, then every longest circuit has at least three chords (Theorem 3.2.6, 3.2.7). (3) Every longest circuit of a 4-connected graph embedded in a torus or Klein bottle has a chord.;We get these three independent results with three totally different approaches: Connectivity (Tutte circuit), second Hamilton circuit, and charge and discharge methods."]},{"key":"dc:title","label":"Title","values":["Chords of longest circuits of graphs"]}]}],"canonical_facts":{"dc:contributor":["Cun-Quan Zhang."],"dc:creator":["Li, Xuechao"],"dc:date.available":["2019-01-17T08:00:00Z"],"dc:description.abstract":["This thesis is on a long standing open conjecture proposed by one of the most prominent mathematicians, Dr. C. Thomassen: Every longest circuit of 3-connected graph has a chord. In 1987, C. Q. Zhang proved that every longest circuit of a 3-connected planar graph G has a chord if G is cubic or if the minimum degree is at least 4. In 1997, Carsten Thomassen proved that every longest circuit of 3-connected cubic graph has a chord.;In this dissertation, we prove the following three independent partial results: (1) Every longest circuit of a 3-connected graph embedded in a projective plane with minimum degree at least has a chord (Theorem 2.3.1). (2) Every longest circuit of a 3-connected cubic graph has at least two chords. Furthermore if the graph is also a planar, then every longest circuit has at least three chords (Theorem 3.2.6, 3.2.7). (3) Every longest circuit of a 4-connected graph embedded in a torus or Klein bottle has a chord.;We get these three independent results with three totally different approaches: Connectivity (Tutte circuit), second Hamilton circuit, and charge and discharge methods."],"dc:identifier":["https://doi.org/10.33915/etd.1450","https://researchrepository.wvu.edu/etd/1450"],"dc:subject":["Mathematics"],"dc:title":["Chords of longest circuits of graphs"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["PhD"]},"updated_at":"2026-07-24T06:15:47Z"}