{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/124304"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/124304","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On optimal structures in hypergraphs","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2024-09-16 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled &#x27;Open Access&#x27;. The submission was exported from vireo on 2024-09-16 without embargo terms","abstract_has_math":false,"creators":["McCourt, Grace"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Kostochka, Alexandr","Balogh, Jozsef","Reznick, Bruce","Wigal, Michael"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-05","date_published":"2024-05","updated_at":"2026-07-22T22:25:00Z","subjects":["Extremal Hypergraph Theory","Berge Cycles"],"languages":["en","eng"],"rights":["Copyright 2024 Grace McCourt"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/124304","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Kostochka, Alexandr","Balogh, Jozsef","Reznick, Bruce","Wigal, Michael"]},{"key":"dc:creator","label":"Author","values":["McCourt, Grace"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2024-05","2024-04-18"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"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":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Extremal Hypergraph Theory","Berge Cycles"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2024 Grace McCourt"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/124304"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2024-09-16 without embargo terms","The student, Grace McCourt, accepted the attached license on 2024-04-16 at 15:38.","The student, Grace McCourt, submitted this Dissertation for approval on 2024-04-16 at 15:48.","This Dissertation was approved for publication on 2024-04-18 at 13:31.","DSpace SAF Submission Ingestion Package generated from Vireo submission #20444 on 2024-09-16 at 00:34:41","The famous Dirac's Theorem gives an exact bound on the minimum degree of an n-vertex graph guaranteeing the existence of a hamiltonian cycle, namely minimum degree at least n/2. In the same paper, Dirac also observed that a graph with minimum degree at least k \\geq 2 contains a cycle of length at least k+1, and that for 2-connected graphs, we obtain a cycle of length at least \\min{2k,n}. In this thesis, we prove exact bounds of similar type for hamiltonian Berge cycles as well as for Berge cycles of length at least k in r-uniform, n-vertex hypergraphs for all combinations of k, r and n with 3 \\leq r, k \\leq n. We also provide bounds that generalize Dirac's result on 2-connected graphs to r-uniform, n-vertex hypergraphs. The bounds for each result differ for different ranges of r compared to n and k. Dirac's Theorem is part of a larger family of results in graph theory that give tight degree bounds which guarantee the presence of long paths or cycles in graphs. One such result involves graphs being hamiltonian-connected. A hypergraph H is hamiltonian-connected if for any distinct vertices x and y, H contains a hamiltonian Berge path from x to y. We find for all 3 \\leq r < n, exact lower bounds on minimum degree \\delta(n,r) of an n-vertex r-uniform hypergraph H guaranteeing that H is hamiltonian-connected and prove a related result on the codiameter of an n-vertex r-uniform hypergraph. Additionally, we consider a variation of Ryser's Conjecture by studying the diameter cover number. Let the diameter cover number, D^t_r(G), denote the least integer $d$ such that under any r-coloring of the edges of the graph G, there exists a collection of t monochromatic subgraphs of diameter at most d such that every vertex of G is contained in at least one of the subgraphs. We explore the diameter cover number D_2^2(G) when G is a complete multipartite graph. Specifically, we determine exactly the value of D_2^2(G) for all complete tripartite graphs G, and almost all complete multipartite graphs with more than three parts."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["On optimal structures in hypergraphs"]}]}],"canonical_facts":{"dc:contributor":["Kostochka, Alexandr","Balogh, Jozsef","Reznick, Bruce","Wigal, Michael"],"dc:creator":["McCourt, Grace"],"dc:date":["2024-05","2024-04-18"],"dc:description":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2024-09-16 without embargo terms","The student, Grace McCourt, accepted the attached license on 2024-04-16 at 15:38.","The student, Grace McCourt, submitted this Dissertation for approval on 2024-04-16 at 15:48.","This Dissertation was approved for publication on 2024-04-18 at 13:31.","DSpace SAF Submission Ingestion Package generated from Vireo submission #20444 on 2024-09-16 at 00:34:41","The famous Dirac's Theorem gives an exact bound on the minimum degree of an n-vertex graph guaranteeing the existence of a hamiltonian cycle, namely minimum degree at least n/2. In the same paper, Dirac also observed that a graph with minimum degree at least k \\geq 2 contains a cycle of length at least k+1, and that for 2-connected graphs, we obtain a cycle of length at least \\min{2k,n}. In this thesis, we prove exact bounds of similar type for hamiltonian Berge cycles as well as for Berge cycles of length at least k in r-uniform, n-vertex hypergraphs for all combinations of k, r and n with 3 \\leq r, k \\leq n. We also provide bounds that generalize Dirac's result on 2-connected graphs to r-uniform, n-vertex hypergraphs. The bounds for each result differ for different ranges of r compared to n and k. Dirac's Theorem is part of a larger family of results in graph theory that give tight degree bounds which guarantee the presence of long paths or cycles in graphs. One such result involves graphs being hamiltonian-connected. A hypergraph H is hamiltonian-connected if for any distinct vertices x and y, H contains a hamiltonian Berge path from x to y. We find for all 3 \\leq r < n, exact lower bounds on minimum degree \\delta(n,r) of an n-vertex r-uniform hypergraph H guaranteeing that H is hamiltonian-connected and prove a related result on the codiameter of an n-vertex r-uniform hypergraph. Additionally, we consider a variation of Ryser's Conjecture by studying the diameter cover number. Let the diameter cover number, D^t_r(G), denote the least integer $d$ such that under any r-coloring of the edges of the graph G, there exists a collection of t monochromatic subgraphs of diameter at most d such that every vertex of G is contained in at least one of the subgraphs. We explore the diameter cover number D_2^2(G) when G is a complete multipartite graph. Specifically, we determine exactly the value of D_2^2(G) for all complete tripartite graphs G, and almost all complete multipartite graphs with more than three parts."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/124304"],"dc:language":["en","eng"],"dc:rights":["Copyright 2024 Grace McCourt"],"dc:subject":["Extremal Hypergraph Theory","Berge Cycles"],"dc:title":["On optimal structures in hypergraphs"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:00Z"}