{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/115348"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/115348","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Cycle structure of graphs and hypergraphs: extremal problems and reconstruction","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-11 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-11 without embargo terms","abstract_has_math":false,"creators":["Zirlin, Dara"],"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, József","West, Douglas","English, Sean"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-05","date_published":"2022-05","updated_at":"2026-07-22T22:24:54Z","subjects":["Graph Reconstruction","Super-pancyclic hypergraphs"],"languages":["en","eng"],"rights":["Copyright 2022 Dara Zirlin"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/115348","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Kostochka, Alexandr","Balogh, József","West, Douglas","English, Sean"]},{"key":"dc:creator","label":"Author","values":["Zirlin, Dara"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-05","2022-03-18"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"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":["Graph Reconstruction","Super-pancyclic hypergraphs"]}]},{"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 2022 Dara Zirlin"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/115348"]}]},{"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 2022-11-11 without embargo terms","The student, Dara Zirlin, accepted the attached license on 2022-03-10 at 15:30.","The student, Dara Zirlin, submitted this Dissertation for approval on 2022-03-10 at 15:53.","This Dissertation was approved for publication on 2022-03-18 at 10:26.","DSpace SAF Submission Ingestion Package generated from Vireo submission #17531 on 2022-11-11 at 13:04:33","The main focus of this thesis is to study the structure of graphs and hypergraphs with cycles, with a focus on extremal problems and reconstruction. We also study 2k-factors in (2r+1)-regular graphs. In Chapter 2, we consider super-pancyclic hypergraphs and super-cyclic bipartite graphs. A hypergraph H is super-pancyclic if for each A⊆V(H) with |A| at most 3, H contains a Berge cycle with base vertex set A. A super-cyclic bipartite graph is a (X,Y)-bigraph G such that for each A ⊆X with |A| at most 3, G has a cycle C_A such that V(C_A) ∩ X=A. Super-cyclic bipartite graphs are incidence graphs of super-pancyclic hypergraphs, and our proofs use the language of such graphs. A hypergraph H is hamiltonian if it contains a Berge cycle whose base set of vertices is all of V(H). We find Dirac-type sufficient conditions for a hypergraph H with few edges to be hamiltonian. We also show that these conditions guarantee that H is super-pancyclic. We extend some results of Jackson on the existence of long cycles in bipartite graphs where the vertices in one part have high minimum degree. Moreover, we prove a conjecture of Jackson from 1981 on long cycles in 2-connected bipartite graphs. In addition, we present two natural necessary conditions for a hypergraph to be super-pancyclic, and show that in several classes of hypergraphs these necessary conditions are also sufficient. In particular, they are sufficient for every hypergraph H with delta(H) at least max{|V(H)|, (|E(H)|+10)/4}. In Chapter 3, we consider reconstruction and recognition problems. The (n-l)-deck of an n-vertex graph is the multiset of subgraphs obtained from it by deleting l vertices. A graph is l-reconstructible if it is determined by its (n-l)-deck. A family of n-vertex graphs is l-recognizable if every graph having the same (n-l)-deck as a graph in the family is also in the family. We prove that 3-regular graphs are 2-reconstructible. We also prove that the family of n-vertex graphs having no cycles is l-recognizable when n is at least 2l +1 (except for (n,l)=(5,2)). It is known that this fails when n=2l. In Chapter 4, we study 2k-factors in (2r+1)-regular graphs. Hanson, Loten, and Toft proved that every (2r+1)-regular graph with at most 2r cut-edges has a 2-factor. We generalize their result by proving for k at most (2r+1)/3 that every (2r+1)-regular graph with at most 2r-3(k-1) cut-edges has a 2k-factor. Both the restriction on k and the restriction on the number of cut-edges are sharp. We characterize the graphs that have exactly 2r-3(k-1)+1 cut-edges but no 2k-factor. For k>(2r+1)/3, there are graphs without cut-edges that have no 2k-factor, as studied by Bollobas, Saito, and Wormald."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Cycle structure of graphs and hypergraphs: extremal problems and reconstruction"]}]}],"canonical_facts":{"dc:contributor":["Kostochka, Alexandr","Balogh, József","West, Douglas","English, Sean"],"dc:creator":["Zirlin, Dara"],"dc:date":["2022-05","2022-03-18"],"dc:description":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-11 without embargo terms","The student, Dara Zirlin, accepted the attached license on 2022-03-10 at 15:30.","The student, Dara Zirlin, submitted this Dissertation for approval on 2022-03-10 at 15:53.","This Dissertation was approved for publication on 2022-03-18 at 10:26.","DSpace SAF Submission Ingestion Package generated from Vireo submission #17531 on 2022-11-11 at 13:04:33","The main focus of this thesis is to study the structure of graphs and hypergraphs with cycles, with a focus on extremal problems and reconstruction. We also study 2k-factors in (2r+1)-regular graphs. In Chapter 2, we consider super-pancyclic hypergraphs and super-cyclic bipartite graphs. A hypergraph H is super-pancyclic if for each A⊆V(H) with |A| at most 3, H contains a Berge cycle with base vertex set A. A super-cyclic bipartite graph is a (X,Y)-bigraph G such that for each A ⊆X with |A| at most 3, G has a cycle C_A such that V(C_A) ∩ X=A. Super-cyclic bipartite graphs are incidence graphs of super-pancyclic hypergraphs, and our proofs use the language of such graphs. A hypergraph H is hamiltonian if it contains a Berge cycle whose base set of vertices is all of V(H). We find Dirac-type sufficient conditions for a hypergraph H with few edges to be hamiltonian. We also show that these conditions guarantee that H is super-pancyclic. We extend some results of Jackson on the existence of long cycles in bipartite graphs where the vertices in one part have high minimum degree. Moreover, we prove a conjecture of Jackson from 1981 on long cycles in 2-connected bipartite graphs. In addition, we present two natural necessary conditions for a hypergraph to be super-pancyclic, and show that in several classes of hypergraphs these necessary conditions are also sufficient. In particular, they are sufficient for every hypergraph H with delta(H) at least max{|V(H)|, (|E(H)|+10)/4}. In Chapter 3, we consider reconstruction and recognition problems. The (n-l)-deck of an n-vertex graph is the multiset of subgraphs obtained from it by deleting l vertices. A graph is l-reconstructible if it is determined by its (n-l)-deck. A family of n-vertex graphs is l-recognizable if every graph having the same (n-l)-deck as a graph in the family is also in the family. We prove that 3-regular graphs are 2-reconstructible. We also prove that the family of n-vertex graphs having no cycles is l-recognizable when n is at least 2l +1 (except for (n,l)=(5,2)). It is known that this fails when n=2l. In Chapter 4, we study 2k-factors in (2r+1)-regular graphs. Hanson, Loten, and Toft proved that every (2r+1)-regular graph with at most 2r cut-edges has a 2-factor. We generalize their result by proving for k at most (2r+1)/3 that every (2r+1)-regular graph with at most 2r-3(k-1) cut-edges has a 2k-factor. Both the restriction on k and the restriction on the number of cut-edges are sharp. We characterize the graphs that have exactly 2r-3(k-1)+1 cut-edges but no 2k-factor. For k>(2r+1)/3, there are graphs without cut-edges that have no 2k-factor, as studied by Bollobas, Saito, and Wormald."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/115348"],"dc:language":["en","eng"],"dc:rights":["Copyright 2022 Dara Zirlin"],"dc:subject":["Graph Reconstruction","Super-pancyclic hypergraphs"],"dc:title":["Cycle structure of graphs and hypergraphs: extremal problems and reconstruction"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:54Z"}