Cycle structure of graphs and hypergraphs: extremal problems and reconstruction

Zirlin, Dara

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

Description

Title

Cycle structure of graphs and hypergraphs: extremal problems and reconstruction

Author(s)

Zirlin, Dara

Issue Date

2022-03-18

Director of Research (if dissertation) or Advisor (if thesis)

Kostochka, Alexandr

Doctoral Committee Chair(s)

Balogh, József

Committee Member(s)

• West, Douglas
• English, Sean

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Graph Reconstruction
• Super-pancyclic hypergraphs

Abstract

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.

Graduation Semester

2022-05

Type of Resource

Thesis

Copyright and License Information

Copyright 2022 Dara Zirlin

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