Matching Items (3)
Filtering by
- All Subjects: Theoretical mathematics
- All Subjects: covering perfect hash families
- Creators: Czygrinow, Andrzej
Description
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 problems and their variants for graphs and hypergraphs.
Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs.
In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures.
Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved.
Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs.
In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures.
Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved.
ContributorsYie, Jangwon (Author) / Czygrinow, Andrzej (Thesis advisor) / Kierstead, Henry (Committee member) / Colbourn, Charles (Committee member) / Fishel, Susanna (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2018
Description
Modern software and hardware systems are composed of a large number of components. Often different components of a system interact with each other in unforeseen and undesired ways to cause failures. Covering arrays are a useful mathematical tool for testing all possible t-way interactions among the components of a system.
The two major issues concerning covering arrays are explicit construction of a covering array, and exact or approximate determination of the covering array number---the minimum size of a covering array. Although these problems have been investigated extensively for the last couple of decades, in this thesis we present significant improvements on both of these questions using tools from the probabilistic method and randomized algorithms.
First, a series of improvements is developed on the previously known upper bounds on covering array numbers. An estimate for the discrete Stein-Lovász-Johnson bound is derived and the Stein- Lovász -Johnson bound is improved upon using an alteration strategy. Then group actions on the set of symbols are explored to establish two asymptotic upper bounds on covering array numbers that are tighter than any of the presently known bounds.
Second, an algorithmic paradigm, called the two-stage framework, is introduced for covering array construction. A number of concrete algorithms from this framework are analyzed, and it is shown that they outperform current methods in the range of parameter values that are of practical relevance. In some cases, a reduction in the number of tests by more than 50% is achieved.
Third, the Lovász local lemma is applied on covering perfect hash families to obtain an upper bound on covering array numbers that is tightest of all known bounds. This bound leads to a Moser-Tardos type algorithm that employs linear algebraic computation over finite fields to construct covering arrays. In some cases, this algorithm outperforms currently used methods by more than an 80% margin.
Finally, partial covering arrays are introduced to investigate a few practically relevant relaxations of the covering requirement. Using probabilistic methods, bounds are obtained on partial covering arrays that are significantly smaller than for covering arrays. Also, randomized algorithms are provided that construct such arrays in expected polynomial time.
The two major issues concerning covering arrays are explicit construction of a covering array, and exact or approximate determination of the covering array number---the minimum size of a covering array. Although these problems have been investigated extensively for the last couple of decades, in this thesis we present significant improvements on both of these questions using tools from the probabilistic method and randomized algorithms.
First, a series of improvements is developed on the previously known upper bounds on covering array numbers. An estimate for the discrete Stein-Lovász-Johnson bound is derived and the Stein- Lovász -Johnson bound is improved upon using an alteration strategy. Then group actions on the set of symbols are explored to establish two asymptotic upper bounds on covering array numbers that are tighter than any of the presently known bounds.
Second, an algorithmic paradigm, called the two-stage framework, is introduced for covering array construction. A number of concrete algorithms from this framework are analyzed, and it is shown that they outperform current methods in the range of parameter values that are of practical relevance. In some cases, a reduction in the number of tests by more than 50% is achieved.
Third, the Lovász local lemma is applied on covering perfect hash families to obtain an upper bound on covering array numbers that is tightest of all known bounds. This bound leads to a Moser-Tardos type algorithm that employs linear algebraic computation over finite fields to construct covering arrays. In some cases, this algorithm outperforms currently used methods by more than an 80% margin.
Finally, partial covering arrays are introduced to investigate a few practically relevant relaxations of the covering requirement. Using probabilistic methods, bounds are obtained on partial covering arrays that are significantly smaller than for covering arrays. Also, randomized algorithms are provided that construct such arrays in expected polynomial time.
ContributorsSarakāra, Kauśika (Author) / Colbourn, Charles J. (Thesis advisor) / Czygrinow, Andrzej (Committee member) / Richa, Andréa W. (Committee member) / Syrotiuk, Violet R. (Committee member) / Arizona State University (Publisher)
Created2016
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