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Highly Hamiltonian Graphs And Digraphs, Zhenming Bi
Highly Hamiltonian Graphs And Digraphs, Zhenming Bi
Dissertations
A cycle that contains every vertex of a graph or digraph is a Hamiltonian cycle. A graph or digraph containing such a cycle is itself called Hamiltonian. This concept is named for the famous Irish physicist and mathematician Sir William Rowan Hamilton. These graphs and digraphs have been the subject of study for over six decades. In this dissertation, we study graphs and digraphs with even stronger Hamiltonian properties, namely highly Hamiltonian graphs and digraphs.
The Regularity Lemma And Its Applications, Elizabeth Sprangel
The Regularity Lemma And Its Applications, Elizabeth Sprangel
Honors Theses
The regularity lemma (also known as Szemerédi's Regularity Lemma) is one of the most powerful tools used in extremal graph theory. In general, the lemma states that every graph has some structure. That is, every graph can be partitioned into a finite number of classes in a way such that the number of edges between any two parts is “regular." This thesis is an introduction to the regularity lemma through its proof and applications. We demonstrate its applications to extremal graph theory, Ramsey theory, and number theory.
Sum-Defined Colorings In Graphs, James Hallas
Sum-Defined Colorings In Graphs, James Hallas
Honors Theses
There have been numerous studies using a variety of methods for the purpose of uniquely distinguishing every two adjacent vertices of a graph. Many of these methods have involved graph colorings. The most studied colorings are proper colorings. A proper coloring of a graph G is an assignment of colors to the vertices of G such that adjacent vertices are assigned distinct colors. The minimum number of colors required in a proper coloring of G is the chromatic number of G. In our work, we introduce a new coloring that induces a (nearly) proper coloring. Two vertices u and …