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Generalized Line Graphs, Mohra Abdullah Z. Alqahtani

Dissertations

With every nonempty graph, there are associated many graphs. One of the best known and most studied of these is the line graph L (G) of a graph G, whose vertices are the edges of G and where two vertices of L (G) are adjacent if the corresponding edges of G are adjacent. This concept was implicitly introduced by Whitney in 1932. Over the years, characterizations of graphs that are line graphs have been given, as well as graphs whose line graphs have some specified property. For example, Beineke characterized graphs that are line graphs by forbidding certain graphs …

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, …

The Bellringer Sequence: Investigating What And How Preservice Mathematics Teachers Learn Through Pedagogies Of Enactment, Mary A. Ochieng

Dissertations

This study examines preservice teacher learning through pedagogies of enactment—approaches to teacher education that allow preservice teachers to learn by doing what teachers do. Preservice teacher (PST) learning is examined through the implementation of the Bellringer Sequence (BRS), a pedagogy of enactment conceptualized in the study. The BRS is centered around bellringers—brief mathematical tasks implemented as students arrive for class. The BRS is a sequence of four activities centered on a bellringer: preparation (for teaching a bellringer) implementation (of the bellringer with peers), debriefing (discussing the implementation as colleagues), and written reflection (about the effectiveness of the bellringer).

Practice-based approaches …

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 …

Graceful Colorings And Connection In Graphs, Alexis D. Byers

Dissertations

For a graph G of size m, a graceful labeling of G is an injective function f : V (G) → {0, 1, . . . , m} that gives rise to a bijective function f ¹ : E(G) → {1, 2, . . . , m} defined by f ¹(uv) = |f (u) − f (v)|. A graph is graceful if it has a graceful labeling. Over the years, a number of variations of graceful …

Edge Induced Weightings Of Uniform Hypergraphs And Related Problems, Laars C. Helenius

Dissertations

The starting point of the research is the so called 1-2-3 Conjecture formulated in 2004 by Karoński, Luczak, and Thomason. Roughly speaking it says that the edges of any graph can be weighted from {1, 2, 3} so that the induced vertex coloring (as the sum of weights adjacent to a given vertex) is proper. The conjecture has attracted a lot of interest from researchers over the last decade but is still unanswered. More recently, the conjecture has been studied for hypergraphs.

The main result of this dissertation shows in particular that an analogous conjecture holds for almost all uniform …