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Kostochka, Alexandr V. West, Douglas B. Reznick, Bruce A. Molla, Theodore Reiniger, Benjamin M 2015-09-29T20:37:48Z 2015-09-29T20:37:48Z 2015-08 2015-07-06 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. Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2015-09-29 without embargo terms The student, Benjamin Reiniger, accepted the attached license on 2015-06-24 at 13:19. The student, Benjamin Reiniger, submitted this Dissertation for approval on 2015-06-24 at 17:34. This Dissertation was approved for publication on 2015-07-06 at 13:15. DSpace SAF Submission Ingestion Package generated from Vireo submission #8308 on 2015-09-29 at 13:21:38 Made available in DSpace on 2015-09-29T20:37:48Z (GMT). No. of bitstreams: 2 REINIGER-DISSERTATION-2015.pdf: 538625 bytes, checksum: d7338ace4423aa92753c22c1f3d06e37 (MD5) LICENSE.txt: 4214 bytes, checksum: 26bc2776a71e605c05818c216b129d0c (MD5) Previous issue date: 2015-07-06 application/pdf http://hdl.handle.net/2142/87975 en Copyright 2015 Benjamin M. Reiniger graph coloring hypergraph coloring critical graphs list coloring hypergraph degrees poset dimension Coloring and constructing (hyper)graphs with restrictions text text 2015-8 Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.