{"id":{"repo_id":"birmingham","oai_identifier":"oai:etheses.bham.ac.uk:940"},"canonical_url":"https://search.dev.ndltd.org/etd/birmingham/oai:etheses.bham.ac.uk:940","repository":{"repo_id":"birmingham","name":"University of Birmingham","base_url":"https://etheses.bham.ac.uk/cgi/oai2"},"display":{"title":"On cycles in directed graphs","abstract":"The main results of this thesis are the following. We show that for each alpha > 0 every sufficiently large oriented graph G with minimum indegree and minimum outdegree at least 3 |G| / 8 + alpha |G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. Furthermore, answering completely a conjecture of Haggkvist and Thomason, we show that we get every possible orientation of a Hamilton cycle. We also deal extensively with short cycles, showing that for each l > 4 every sufficiently large oriented graph G with minimum indegree and minimum outdegree at least |G| / 3 + 1 contains an l-cycle. This is best possible for all those l > 3 which are not divisible by 3. Surprisingly, for some other values of l, an l-cycle is forced by a much weaker minimum degree condition. We propose and discuss a conjecture regarding the precise minimum degree which forces an l-cycle (with l > 3 divisible by 3) in an oriented graph. We also give an application of our results to pancyclicity.","abstract_html":"The main results of this thesis are the following. We show that for each alpha > 0 every sufficiently large oriented graph G with minimum indegree and minimum outdegree at least 3 |G| / 8 + alpha |G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. Furthermore, answering completely a conjecture of Haggkvist and Thomason, we show that we get every possible orientation of a Hamilton cycle. We also deal extensively with short cycles, showing that for each l > 4 every sufficiently large oriented graph G with minimum indegree and minimum outdegree at least |G| / 3 + 1 contains an l-cycle. This is best possible for all those l > 3 which are not divisible by 3. Surprisingly, for some other values of l, an l-cycle is forced by a much weaker minimum degree condition. We propose and discuss a conjecture regarding the precise minimum degree which forces an l-cycle (with l > 3 divisible by 3) in an oriented graph. We also give an application of our results to pancyclicity.","abstract_has_math":false,"creators":["Kelly, Luke Tristian"],"institution":"University of Birmingham","degree_name":"d_ph","degree_level":"d_ph","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010-07","date_published":"2010-07","updated_at":"2026-07-24T01:11:15Z","subjects":["QA Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.sponsor","label":"Sponsor","values":["na"]},{"key":"dc:creator","label":"Author","values":["Kelly, Luke Tristian"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2010-07"]},{"key":"dc:date.issued","label":"Date","values":["2010-07"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["College of Engineering & Physical Sciences","School of Mathematics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Birmingham"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["http://etheses.bham.ac.uk//id/eprint/940/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["d_ph"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["d_ph"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["QA Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://etheses.bham.ac.uk//id/eprint/940/1/kelly_10_PhD.pdf","http://etheses.bham.ac.uk//id/eprint/940/2/Decl_IS_Kelly_10_PhD.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The main results of this thesis are the following. We show that for each alpha > 0 every sufficiently large oriented graph G with minimum indegree and minimum outdegree at least 3 |G| / 8 + alpha |G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. Furthermore, answering completely a conjecture of Haggkvist and Thomason, we show that we get every possible orientation of a Hamilton cycle. We also deal extensively with short cycles, showing that for each l > 4 every sufficiently large oriented graph G with minimum indegree and minimum outdegree at least |G| / 3 + 1 contains an l-cycle. This is best possible for all those l > 3 which are not divisible by 3. Surprisingly, for some other values of l, an l-cycle is forced by a much weaker minimum degree condition. We propose and discuss a conjecture regarding the precise minimum degree which forces an l-cycle (with l > 3 divisible by 3) in an oriented graph. We also give an application of our results to pancyclicity."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["On cycles in directed graphs"]}]}],"canonical_facts":{"dc:contributor.sponsor":["na"],"dc:creator":["Kelly, Luke Tristian"],"dc:date":["2010-07"],"dc:date.issued":["2010-07"],"dc:description.abstract":["The main results of this thesis are the following. We show that for each alpha > 0 every sufficiently large oriented graph G with minimum indegree and minimum outdegree at least 3 |G| / 8 + alpha |G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. Furthermore, answering completely a conjecture of Haggkvist and Thomason, we show that we get every possible orientation of a Hamilton cycle. We also deal extensively with short cycles, showing that for each l > 4 every sufficiently large oriented graph G with minimum indegree and minimum outdegree at least |G| / 3 + 1 contains an l-cycle. This is best possible for all those l > 3 which are not divisible by 3. Surprisingly, for some other values of l, an l-cycle is forced by a much weaker minimum degree condition. We propose and discuss a conjecture regarding the precise minimum degree which forces an l-cycle (with l > 3 divisible by 3) in an oriented graph. We also give an application of our results to pancyclicity."],"dc:format":["application/pdf"],"dc:identifier.uri":["http://etheses.bham.ac.uk//id/eprint/940/1/kelly_10_PhD.pdf","http://etheses.bham.ac.uk//id/eprint/940/2/Decl_IS_Kelly_10_PhD.pdf"],"dc:publisher.department":["College of Engineering & Physical Sciences","School of Mathematics"],"dc:publisher.institution":["University of Birmingham"],"dc:relation.isreferencedby":["http://etheses.bham.ac.uk//id/eprint/940/"],"dc:subject":["QA Mathematics"],"dc:title":["On cycles in directed graphs"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["d_ph"],"dc:type.qualificationname":["d_ph"]},"updated_at":"2026-07-24T01:11:15Z"}