## ASU Electronic Theses and Dissertations

This collection includes most of the ASU Theses and Dissertations from 2011 to present. ASU Theses and Dissertations are available in downloadable PDF format; however, a small percentage of items are under embargo. Information about the dissertations/theses includes degree information, committee members, an abstract, supporting data or media.

In addition to the electronic theses found in the ASU Digital Repository, ASU Theses and Dissertations can be found in the ASU Library Catalog.

Dissertations and Theses granted by Arizona State University are archived and made available through a joint effort of the ASU Graduate College and the ASU Libraries. For more information or questions about this collection contact or visit the Digital Repository ETD Library Guide or contact the ASU Graduate College at gradformat@asu.edu.

Every graph can be colored with one more color than its maximum degree. A well-known theorem of Brooks gives the precise conditions under which a graph can be colored with maximum degree colors. It is natural to ask for the required conditions on a graph to color with one less color than the maximum degree; in 1977 Borodin and Kostochka conjectured a solution for graphs with maximum degree at least 9: as long as the graph doesn't contain a maximum-degree-sized clique, it can be colored with one fewer than the maximum degree colors. This study attacks the conjecture on multiple ...

Contributors
Rabern, Landon, Kierstead, Henry, Colbourn, Charles, et al.
Created Date
2013

The primary focus of this dissertation lies in extremal combinatorics, in particular intersection theorems in finite set theory. A seminal result in the area is the theorem of Erdos, Ko and Rado which finds the upper bound on the size of an intersecting family of subsets of an n-element set and characterizes the structure of families which attain this upper bound. A major portion of this dissertation focuses on a recent generalization of the Erdos--Ko--Rado theorem which considers intersecting families of independent sets in graphs. An intersection theorem is proved for a large class of graphs, namely chordal graphs which ...

Contributors
Kamat, Vikram Mahendra, Hurlbert, Glenn, Colbourn, Charles, et al.
Created Date
2011

Gray codes are perhaps the best known structures for listing sequences of combinatorial objects, such as binary strings. Simply defined as a minimal change listing, Gray codes vary greatly both in structure and in the types of objects that they list. More specific types of Gray codes are universal cycles and overlap sequences. Universal cycles are Gray codes on a set of strings of length n in which the first n-1 letters of one object are the same as the last n-1 letters of its predecessor in the listing. Overlap sequences allow this overlap to vary between 1 and n-1. ...

Contributors
Horan, Victoria E., Hurlbert, Glenn H, Czygrinow, Andrzej, et al.
Created Date
2012

The Tamari lattice T(n) was originally defined on bracketings of a set of n+1 objects, with a cover relation based on the associativity rule in one direction. Since then it has been studied in various areas of mathematics including cluster algebras, discrete geometry, algebraic combinatorics, and Catalan theory. Although in several related lattices the number of maximal chains is known, the enumeration of these chains in Tamari lattices is still an open problem. This dissertation defines a partially-ordered set on equivalence classes of certain saturated chains of T(n) called the Tamari Block poset, TB(lambda). It further proves TB(lambda) is a ...

Contributors
Treat, Kevin, Fishel, Susanna, Czygrinow, Andrzej, et al.
Created Date
2016

Extremal graph theory results often provide minimum degree conditions which guarantee a copy of one graph exists within another. A perfect $F$-tiling of a graph $G$ is a collection $\mathcal{F}$ of subgraphs of $G$ such that every element of $\mathcal{F}$ is isomorphic to $F$ and such that every vertex in $G$ is in exactly one element of $\mathcal{F}$. Let $C^{3}_{t}$ denote the loose cycle on $t = 2s$ vertices, the $3$-uniform hypergraph obtained by replacing the edges $e = \{u, v\}$ of a graph cycle $C$ on $s$ vertices with edge triples $\{u, x_e, v\}$, where $x_e$ is uniquely assigned ...

Contributors
Oursler, Roy, Czygrinow, Andrzej, Kierstead, Hal A, et al.
Created Date
2019

A central concept of combinatorics is partitioning structures with given constraints. Partitions of on-line posets and on-line graphs, which are dynamic versions of the more familiar static structures posets and graphs, are examined. In the on-line setting, vertices are continually added to a poset or graph while a chain partition or coloring (respectively) is maintained. %The optima of the static cases cannot be achieved in the on-line setting. Both upper and lower bounds for the optimum of the number of chains needed to partition a width $w$ on-line poset exist. Kierstead's upper bound of $\frac{5^w-1}{4}$ was improved to \$w^{14 \lg ...

Contributors
Smith, Matthew Earl, Kierstead, Henry A, Colbourn, Charles, et al.
Created Date
2012

Since the seminal work of Tur ́an, the forbidden subgraph problem has been among the central questions in extremal graph theory. Let ex(n;F) be the smallest number m such that any graph on n vertices with m edges contains F as a subgraph. Then the forbidden subgraph problem asks to find ex(n; F ) for various graphs F . The question can be further generalized by asking for the extreme values of other graph parameters like minimum degree, maximum degree, or connectivity. We call this type of question a Tura ́n-type problem. In this thesis, we will study Tura ́n-type ...

Contributors
Yie, Jangwon, Czygrinow, Andrzej, Kierstead, Henry, et al.
Created Date
2018