On optimal structures in hypergraphs

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.

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

2024

Author dc:creator

• McCourt, Grace

Contributors dc:contributor

• Kostochka, Alexandr
• Balogh, Jozsef
• Reznick, Bruce
• Wigal, Michael

• Extremal Hypergraph Theory
• Berge Cycles

Statement dc:rights

• Copyright 2024 Grace McCourt

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/124304

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F124304