#### Withdraw

Loading…

# Coloring and constructing (hyper)graphs with restrictions

#### Reiniger, Benjamin M

Loading…

## Permalink

https://hdl.handle.net/2142/87975

## Description

- Title
- Coloring and constructing (hyper)graphs with restrictions
- Author(s)
- Reiniger, Benjamin M
- Issue Date
- 2015-07-06
- Director of Research (if dissertation) or Advisor (if thesis)
- Kostochka, Alexandr V.
- Doctoral Committee Chair(s)
- West, Douglas B.
- Committee Member(s)
- Reznick, Bruce A.
- Molla, Theodore

- 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 coloring
- hypergraph coloring
- critical graphs
- list coloring
- hypergraph degrees
- poset dimension

- Abstract
- We consider questions regarding the existence of graphs and hypergraphs with certain coloring properties and other structural properties. In Chapter 2 we consider color-critical graphs that are nearly bipartite and have few edges. We prove a conjecture of Chen, Erdős, Gyárfás, and Schelp concerning the minimum number of edges in a “nearly bipartite” 4-critical graph. In Chapter 3 we consider coloring and list-coloring graphs and hypergraphs with few edges and no small cycles. We prove two main results. If a bipartite graph has maximum average degree at most 2(k−1), then it is colorable from lists of size k; we prove that this is sharp, even with an additional girth requirement. Using the same approach, we also provide a simple construction of graphs with arbitrarily large girth and chromatic number (first proved to exist by Erdős). In Chapter 4 we consider list-coloring the family of kth power graphs. Kostochka and Woodall conjectured that graph squares are chromatic-choosable, as a strengthening of the Total List Coloring Conjecture. Kim and Park disproved this stronger conjecture, and Zhu asked whether graph kth powers are chromatic-choosable for any k. We show that this is not true: we construct families of graphs based on affine planes whose choice number exceeds their chromatic number by a logarithmic factor. In Chapter 5 we consider the existence of uniform hypergraphs with prescribed degrees and codegrees. In Section 5.2, we show that a generalization of the graphic 2-switch is insufficient to connect realizations of a given degree sequence. In Section 5.3, we consider an operation on 3-graphs related to the octahedron that preserves codegrees; this leads to an inductive definition for 2-colorable triangulations of the sphere. In Section 5.4, we discuss the notion of fractional realizations of degree sequences, in particular noting the equivalence of the existence of a realization and the existence of a fractional realization in the graph and multihypergraph cases. In Chapter 6 we consider a question concerning poset dimension. Dorais asked for the maximum guaranteed size of a subposet with dimension at most d of an n-element poset. A lower bound of sqrt(dn) was observed by Goodwillie. We provide a sublinear upper bound.
- Graduation Semester
- 2015-8
- Type of Resource
- text
- Permalink
- http://hdl.handle.net/2142/87975
- Copyright and License Information
- Copyright 2015 Benjamin M. Reiniger

## Owning Collections

##### Graduate Dissertations and Theses at Illinois PRIMARY

Graduate Theses and Dissertations at Illinois#### Manage Files

Loading…

#### Edit Collection Membership

Loading…

#### Edit Metadata

Loading…

#### Edit Properties

Loading…

#### Embargoes

Loading…