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.
Displaying 1 - 5 of 5
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- Creators: Czygrinow, Andrzej
Description
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. Some of our main contributions to the areas of Gray codes and universal cycles include a new Gray code algorithm for fixed weight m-ary words, and results on the existence of universal cycles for weak orders on [n]. Overlap cycles are a relatively new structure with very few published results. We prove the existence of s-overlap cycles for k-permutations of [n], which has been an open research problem for several years, as well as constructing 1- overlap cycles for Steiner triple and quadruple systems of every order. Also included are various other results of a similar nature covering other structures such as binary strings, m-ary strings, subsets, permutations, weak orders, partitions, and designs. These listing structures lend themselves readily to some classes of combinatorial objects, such as binary n-tuples and m-ary n-tuples. Others require more work to find an appropriate structure, such as k-subsets of an n-set, weak orders, and designs. Still more require a modification in the representation of the objects to fit these structures, such as partitions. Determining when and how we can fit these sets of objects into our three listing structures is the focus of this dissertation.
ContributorsHoran, Victoria E (Author) / Hurlbert, Glenn H. (Thesis advisor) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Colbourn, Charles (Committee member) / Sen, Arunabha (Committee member) / Arizona State University (Publisher)
Created2012
Description
A storage system requiring file redundancy and on-line repairability can be represented as a Steiner system, a combinatorial design with the property that every $t$-subset of its points occurs in exactly one of its blocks. Under this representation, files are the points and storage units are the blocks of the Steiner system, or vice-versa. Often, the popularities of the files of such storage systems run the gamut, with some files receiving hardly any attention, and others receiving most of it. For such systems, minimizing the difference in the collective popularity between any two storage units is nontrivial; this is the access balancing problem. With regard to the representative Steiner system, the access balancing problem in its simplest form amounts to constructing either a point or block labelling: an assignment of a set of integer labels (popularity ranks) to the Steiner system's point set or block set, respectively, requiring of the former assignment that the sums of the labelled points of any two blocks differ as little as possible and of the latter that the sums of the labels assigned to the containing blocks of any two distinct points differ as little as possible. The central aim of this dissertation is to supply point and block labellings for Steiner systems of block size greater than three, for which up to this point no attempt has been made. Four major results are given in this connection. First, motivated by the close connection between the size of the independent sets of a Steiner system and the quality of its labellings, a Steiner triple system of any admissible order is constructed with a pair of disjoint independent sets of maximum cardinality. Second, the spectrum of resolvable Bose triple systems is determined in order to label some Steiner 2-designs with block size four. Third, several kinds of independent sets are used to point-label Steiner 2-designs with block size four. Finally, optimal and close to optimal block labellings are given for an infinite class of 1-rotational resolvable Steiner 2-designs with arbitrarily large block size by exploiting their underlying group-theoretic properties.
ContributorsLusi, Dylan (Author) / Colbourn, Charles J (Thesis advisor) / Czygrinow, Andrzej (Committee member) / Fainekos, Georgios (Committee member) / Richa, Andrea (Committee member) / Arizona State University (Publisher)
Created2021
Description
Let $G=(V,E)$ be a graph. A \emph{list assignment} $L$ for $G$ is a function from
$V$ to subsets of the natural numbers. An $L$-\emph{coloring} is a function $f$
with domain $V$ such that $f(v)\in L(v)$ for all vertices $v\in V$ and $f(x)\ne f(y)$
whenever $xy\in E$. If $|L(v)|=t$ for all $v\in V$ then $L$ is a $t$-\emph{list
assignment}. The graph $G$ is $t$-choosable if for every $t$-list assignment $L$
there is an $L$-coloring. The least $t$ such that $G$ is $t$-choosable is called
the list chromatic number of $G$, and is denoted by $\ch(G)$. The complete multipartite
graph with $k$ parts, each of size $s$ is denoted by $K_{s*k}$. Erd\H{o}s et al.
suggested the problem of determining $\ensuremath{\ch(K_{s*k})}$, and showed that
$\ch(K_{2*k})=k$. Alon gave bounds of the form $\Theta(k\log s)$. Kierstead proved
the exact bound $\ch(K_{3*k})=\lceil\frac{4k-1}{3}\rceil$. Here it is proved that
$\ch(K_{4*k})=\lceil\frac{3k-1}{2}\rceil$.
An online version of the list coloring problem was introduced independently by Schauz
and Zhu. It can be formulated as a game between two players, Alice and Bob. Alice
designs lists of colors for all vertices, but does not tell Bob, and is allowed to
change her mind about unrevealed colors as the game progresses. On her $i$-th turn
Alice reveals all vertices with $i$ in their list. On his $i$-th turn Bob decides,
irrevocably, which (independent set) of these vertices to color with $i$. For a
function $l$ from $V$ to the natural numbers, Bob wins the $l$-\emph{game} if
eventually he colors every vertex $v$ before $v$ has had $l(v)+1$ colors of its
list revealed by Alice; otherwise Alice wins. The graph $G$ is $l$-\emph{online
choosable} or \emph{$l$-paintable} if Bob has a strategy to win the $l$-game. If
$l(v)=t$ for all $v\in V$ and $G$ is $l$-paintable, then $G$ is t-paintable.
The \emph{online list chromatic number }of $G$ is the least $t$ such that $G$
is $t$-paintable, and is denoted by $\ensuremath{\ch^{\mathrm{OL}}(G)}$. Evidently,
$\ch^{\mathrm{OL}}(G)\geq\ch(G)$. Zhu conjectured that the gap $\ch^{\mathrm{OL}}(G)-\ch(G)$
can be arbitrarily large. However there are only a few known examples with this gap
equal to one, and none with larger gap. This conjecture is explored in this thesis.
One of the obstacles is that there are not many graphs whose exact list coloring
number is known. This is one of the motivations for establishing new cases of Erd\H{o}s'
problem. Here new examples of graphs with gap one are found, and related technical
results are developed as tools for attacking Zhu's conjecture.
The square $G^{2}$ of a graph $G$ is formed by adding edges between all vertices
at distance $2$. It was conjectured that every graph $G$ satisfies $\chi(G^{2})=\ch(G^{2})$.
This was recently disproved for specially constructed graphs. Here it is shown that
a graph arising naturally in the theory of cellular networks is also a counterexample.
$V$ to subsets of the natural numbers. An $L$-\emph{coloring} is a function $f$
with domain $V$ such that $f(v)\in L(v)$ for all vertices $v\in V$ and $f(x)\ne f(y)$
whenever $xy\in E$. If $|L(v)|=t$ for all $v\in V$ then $L$ is a $t$-\emph{list
assignment}. The graph $G$ is $t$-choosable if for every $t$-list assignment $L$
there is an $L$-coloring. The least $t$ such that $G$ is $t$-choosable is called
the list chromatic number of $G$, and is denoted by $\ch(G)$. The complete multipartite
graph with $k$ parts, each of size $s$ is denoted by $K_{s*k}$. Erd\H{o}s et al.
suggested the problem of determining $\ensuremath{\ch(K_{s*k})}$, and showed that
$\ch(K_{2*k})=k$. Alon gave bounds of the form $\Theta(k\log s)$. Kierstead proved
the exact bound $\ch(K_{3*k})=\lceil\frac{4k-1}{3}\rceil$. Here it is proved that
$\ch(K_{4*k})=\lceil\frac{3k-1}{2}\rceil$.
An online version of the list coloring problem was introduced independently by Schauz
and Zhu. It can be formulated as a game between two players, Alice and Bob. Alice
designs lists of colors for all vertices, but does not tell Bob, and is allowed to
change her mind about unrevealed colors as the game progresses. On her $i$-th turn
Alice reveals all vertices with $i$ in their list. On his $i$-th turn Bob decides,
irrevocably, which (independent set) of these vertices to color with $i$. For a
function $l$ from $V$ to the natural numbers, Bob wins the $l$-\emph{game} if
eventually he colors every vertex $v$ before $v$ has had $l(v)+1$ colors of its
list revealed by Alice; otherwise Alice wins. The graph $G$ is $l$-\emph{online
choosable} or \emph{$l$-paintable} if Bob has a strategy to win the $l$-game. If
$l(v)=t$ for all $v\in V$ and $G$ is $l$-paintable, then $G$ is t-paintable.
The \emph{online list chromatic number }of $G$ is the least $t$ such that $G$
is $t$-paintable, and is denoted by $\ensuremath{\ch^{\mathrm{OL}}(G)}$. Evidently,
$\ch^{\mathrm{OL}}(G)\geq\ch(G)$. Zhu conjectured that the gap $\ch^{\mathrm{OL}}(G)-\ch(G)$
can be arbitrarily large. However there are only a few known examples with this gap
equal to one, and none with larger gap. This conjecture is explored in this thesis.
One of the obstacles is that there are not many graphs whose exact list coloring
number is known. This is one of the motivations for establishing new cases of Erd\H{o}s'
problem. Here new examples of graphs with gap one are found, and related technical
results are developed as tools for attacking Zhu's conjecture.
The square $G^{2}$ of a graph $G$ is formed by adding edges between all vertices
at distance $2$. It was conjectured that every graph $G$ satisfies $\chi(G^{2})=\ch(G^{2})$.
This was recently disproved for specially constructed graphs. Here it is shown that
a graph arising naturally in the theory of cellular networks is also a counterexample.
ContributorsWang, Ran (Author) / Kierstead, H.A. (Thesis advisor) / Colbourn, Charles (Committee member) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Sen, Arunabha (Committee member) / Arizona State University (Publisher)
Created2015
Description
The chromatic number $\chi(G)$ of a graph $G=(V,E)$ is the minimum
number of colors needed to color $V(G)$ such that no adjacent vertices
receive the same color. The coloring number $\col(G)$ of a graph
$G$ is the minimum number $k$ such that there exists a linear ordering
of $V(G)$ for which each vertex has at most $k-1$ backward neighbors.
It is well known that the coloring number is an upper bound for the
chromatic number. The weak $r$-coloring number $\wcol_{r}(G)$ is
a generalization of the coloring number, and it was first introduced
by Kierstead and Yang \cite{77}. The weak $r$-coloring number $\wcol_{r}(G)$
is the minimum integer $k$ such that for some linear ordering $L$
of $V(G)$ each vertex $v$ can reach at most $k-1$ other smaller
vertices $u$ (with respect to $L$) with a path of length at most
$r$ and $u$ is the smallest vertex in the path. This dissertation proves that $\wcol_{2}(G)\le23$ for every planar graph $G$.
The exact distance-$3$ graph $G^{[\natural3]}$ of a graph $G=(V,E)$
is a graph with $V$ as its set of vertices, and $xy\in E(G^{[\natural3]})$
if and only if the distance between $x$ and $y$ in $G$ is $3$.
This dissertation improves the best known upper bound of the
chromatic number of the exact distance-$3$ graphs $G^{[\natural3]}$
of planar graphs $G$, which is $105$, to $95$. It also improves
the best known lower bound, which is $7$, to $9$.
A class of graphs is nowhere dense if for every $r\ge 1$ there exists $t\ge 1$ such that no graph in the class contains a topological minor of the complete graph $K_t$ where every edge is subdivided at most $r$ times. This dissertation gives a new characterization of nowhere dense classes using generalized notions of the domination number.
number of colors needed to color $V(G)$ such that no adjacent vertices
receive the same color. The coloring number $\col(G)$ of a graph
$G$ is the minimum number $k$ such that there exists a linear ordering
of $V(G)$ for which each vertex has at most $k-1$ backward neighbors.
It is well known that the coloring number is an upper bound for the
chromatic number. The weak $r$-coloring number $\wcol_{r}(G)$ is
a generalization of the coloring number, and it was first introduced
by Kierstead and Yang \cite{77}. The weak $r$-coloring number $\wcol_{r}(G)$
is the minimum integer $k$ such that for some linear ordering $L$
of $V(G)$ each vertex $v$ can reach at most $k-1$ other smaller
vertices $u$ (with respect to $L$) with a path of length at most
$r$ and $u$ is the smallest vertex in the path. This dissertation proves that $\wcol_{2}(G)\le23$ for every planar graph $G$.
The exact distance-$3$ graph $G^{[\natural3]}$ of a graph $G=(V,E)$
is a graph with $V$ as its set of vertices, and $xy\in E(G^{[\natural3]})$
if and only if the distance between $x$ and $y$ in $G$ is $3$.
This dissertation improves the best known upper bound of the
chromatic number of the exact distance-$3$ graphs $G^{[\natural3]}$
of planar graphs $G$, which is $105$, to $95$. It also improves
the best known lower bound, which is $7$, to $9$.
A class of graphs is nowhere dense if for every $r\ge 1$ there exists $t\ge 1$ such that no graph in the class contains a topological minor of the complete graph $K_t$ where every edge is subdivided at most $r$ times. This dissertation gives a new characterization of nowhere dense classes using generalized notions of the domination number.
ContributorsAlmulhim, Ahlam (Author) / Kierstead, Henry (Thesis advisor) / Sen, Arunabha (Committee member) / Richa, Andrea (Committee member) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Arizona State University (Publisher)
Created2020
Description
A $k$-list assignment for a graph $G=(V, E)$ is a function $L$ that assigns a $k$-set $L(v)$ of "available colors" to each vertex $v \in V$. A $d$-defective, $m$-fold, $L$-coloring is a function $\phi$ that assigns an $m$-subset $\phi(v) \subseteq L(v)$ to each vertex $v$ so that each color class $V_{i}=\{v \in V:$ $i \in \phi(v)\}$ induces a subgraph of $G$ with maximum degree at most $d$. An edge $xy$ is an $i$-flaw of $\phi$ if $i\in \phi(x) \cap \phi(y)$. An online list-coloring algorithm $\mathcal{A}$ works on a known graph $G$ and an unknown $k$-list assignment $L$ to produce a coloring $\phi$ as follows. At step $r$ the set of vertices $v$ with $r \in L(v)$ is revealed to $\mathcal{A}$. For each vertex $v$, $\mathcal{A}$ must decide irrevocably whether to add $r$ to $\phi(v)$. The online choice number $\pt_{m}^{d}(G)$ of $G$ is the least $k$ for which some such algorithm produces a $d$-defective, $m$-fold, $L$-coloring $\phi$ of $G$ for all $k$-list assignments $L$. Online list coloring was introduced independently by Uwe Schauz and Xuding Zhu. It was known that if $G$ is planar then $\pt_{1}^{0}(G) \leq 5$ and $\pt_{1}^{1}(G) \leq 4$ are sharp bounds; here it is proved that $\pt_{1}^{3}(G) \leq 3$ is sharp, but there is a planar graph $H$ with $\pt_{1}^{2}(H)\ge 4$. Zhu conjectured that for some integer $m$, every planar graph $G$ satisfies $\pt_{m}^{0}(G) \leq 5 m-1$, and even that this is true for $m=2$. This dissertation proves that $\pt_{2}^{1}(G) \leq 9$, so the conjecture is "nearly" true, and the proof extends to $\pt_{m}^{1}(G) \leq\left\lceil\frac{9}{2} m\right\rceil$. Using Alon's Combinatorial Nullstellensatz, this is strengthened by showing that $G$ contains a linear forest $(V, F)$ such that there is an online algorithm that witnesses $\mathrm{pt}_{2}^{1}(G) \leq 9$ while producing a coloring whose flaws are in $F$, and such that no edge is an $i$-flaw and a $j$-flaw for distinct colors $i$ and $j$.
Contributorshan, ming (Author) / Kierstead, Henry A. (Thesis advisor) / Czygrinow, Andrzej (Committee member) / Sen, Arunabha (Committee member) / Spielberg, John (Committee member) / Fishel, Susanna (Committee member) / Arizona State University (Publisher)
Created2021