Cycle structure of graphs and hypergraphs: extremal problems and reconstruction

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Zirlin, Dara

Contributors dc:contributor

• Kostochka, Alexandr
• Balogh, József
• West, Douglas
• English, Sean

• Graph Reconstruction
• Super-pancyclic hypergraphs

Statement dc:rights

• Copyright 2022 Dara Zirlin

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/115348

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F115348