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Irregular Domination In Graphs, Caryn Mays
Irregular Domination In Graphs, Caryn Mays
Dissertations
Domination in graphs has been a popular area of study due in large degree to its applications to modern society as well as the mathematical beauty of the topic. While this area evidently began with the work of Claude Berge in 1958 and Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of a survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then, a large number of variations of domination have surfaced and provided numerous applications to different areas of science and real-life problems. Among these variations are domination parameters …
Zonality In Graphs, Andrew Bowling
Zonality In Graphs, Andrew Bowling
Dissertations
Graph labeling and coloring are among the most popular areas of graph theory due to both the mathematical beauty of these subjects as well as their fascinating applications. While the topic of labeling vertices and edges of graphs has existed for over a century, it was not until 1966 when Alexander Rosa introduced a labeling, later called a graceful labeling, that brought the area of graph labeling to the forefront in graph theory. The subject of graph colorings, on the other hand, goes back to 1852 when the young British mathematician Francis Guthrie observed that the countries in a map …
On Problems In Random Structures, Ryan Cushman
On Problems In Random Structures, Ryan Cushman
Dissertations
This work addresses two problems in optimizing substructures within larger random structures. In the first, we study the triangle-packing number v(G), which is the maximum size of a set of edge-disjoint triangles in a graph. In particular we study this parameter for the random graph G(n,m). We analyze a random process called the online triangle packing process in order to bound v(G). The lower bound on v(G(n,m)) that this produces allows for the verification of a conjecture of Tuza for G( …
Extremal Problems On Induced Graph Colorings, James Hallas
Extremal Problems On Induced Graph Colorings, James Hallas
Dissertations
Graph coloring is one of the most popular areas of graph theory, no doubt due to its many fascinating problems and applications to modern society, as well as the sheer mathematical beauty of the subject. As far back as 1880, in an attempt to solve the famous Four Color Problem, there have been numerous examples of certain types of graph colorings that have generated other graph colorings of interest. These types of colorings only gained momentum a century later, however, when in the 1980s, edge colorings were studied that led to vertex colorings of various types, led by the introduction …
Variations In Ramsey Theory, Drake Olejniczak
Variations In Ramsey Theory, Drake Olejniczak
Dissertations
The Ramsey number R(F,H) of two graphs F and H is the smallest positive integer n for which every red-blue coloring of the (edges of a) complete graph of order n results in a graph isomorphic to F all of whose edges are colored red (a red F) or a blue H. Beineke and Schwenk extended this concept to a bipartite version of Ramsey numbers, namely the bipartite Ramsey number BR(F,H) of two bipartite graphs F and H is the smallest positive integer rsuch that every red-blue coloring of the r-regular complete bipartite graph results in either …
Probabilistic And Extremal Problems In Combinatorics, Sean English
Probabilistic And Extremal Problems In Combinatorics, Sean English
Dissertations
Graph theory as a mathematical branch has been studied rigorously for almost three centuries. In the past century, many new branches of graph theory have been proposed. One important branch of graph theory involves the study of extremal graph theory. In 1941, Turán studied one of the first extremal problems, namely trying to maximize the number of edges over all graphs which avoid having certain structures. Since then, a large body of work has been created in the study of similar problems. In this dissertation, a few different extremal problems are studied, but for hypergraphs rather than graphs. In particular, …
Induced Graph Colorings, Ian Hart
Induced Graph Colorings, Ian Hart
Dissertations
An edge coloring of a nonempty graph G is an assignment of colors to the edges of G. In an unrestricted edge coloring, adjacent edges of G may be colored the same. If every two adjacent edges of G are colored differently, then this edge coloring is proper and the minimum number of colors in a proper edge coloring of G is the chromatic index χ/(G) of G. A proper vertex coloring of a nontrivial graph G is an assignment of colors to the vertices of G such that every two adjacent vertices of …
Color-Connected Graphs And Information-Transfer Paths, Stephen Devereaux
Color-Connected Graphs And Information-Transfer Paths, Stephen Devereaux
Dissertations
The Department of Homeland Security in the United States was created in 2003 in response to weaknesses discovered in the transfer of classied information after the September 11, 2001 terrorist attacks. While information related to national security needs to be protected, there must be procedures in place that permit access between appropriate parties. This two-fold issue can be addressed by assigning information-transfer paths between agencies which may have other agencies as intermediaries while requiring a large enough number of passwords and rewalls that is prohibitive to intruders, yet small enough to manage. Situations such as this can be represented by …
Chromatic Connectivity Of Graphs, Elliot Laforge
Third Order Degree Regular Graphs, Leslie D. Hayes
Third Order Degree Regular Graphs, Leslie D. Hayes
Honors Theses
A graph G is regular of degree d if for every vertex v in G there exist exactly d vertices at distance 1 from v. A graph G is kth order regular of degree d if for every vertex v in G, there exist exactly d vertices at distance k from v. In this paper, third order regular graphs of degree 1 with small order are characterized.