Date of Award
Master of Science (MS)
College of Science and Mathematics
Thesis Sponsor/Dissertation Chair/Project Chair
The study of proper edge-colorings of graphs has been a popular topic in graph theory since the work of Vizing. While the proper edge-colorings of entire graphs was the topic of interest when the subject began decades ago, more recent works have focused on the study of properly colored subgraphs, as opposed to proper colorings of graphs as a whole. The types of properly colored subgraphs that we will be most concerned with are paths. The topic of finding certain types of properly colored paths within larger edge-colored graphs, though seemingly specific in nature, has been a topic of much interest lately. The study began with the work of Chartrand et ah, with what are called rainbow paths, i.e., paths in which no two edges are of the same color. Prom the study of finding properly colored subgraphs within a graph G, Chartrand et al. created a new edge-coloring problem, namely the rainbow connection problem, by adding a connectivity requirement which involves finding rainbow paths between any two vertices of G. Rainbow connection is very well-studied, and we will survey some of the more well-known results. Most of these are concerned with a particular parameter of a graph G, called the rainbow connection number, which is defined as the smallest number of edge colors needed so that between any two vertices of a graph there exists a rainbow path. The main original result of this thesis is concerned with a very natural extension of the rainbow connection number, called the proper connection number. In particular, we will look at the proper connection number of a type of bipartite graph, called a circulant graph. As well as being a result in itself, this yields progress on a conjecture of Borozan et al. We will also provide ideas for future work.
Fuentes, Melissa Marie, "Proper Connection of Bipartite Circulant Graphs" (2013). Theses, Dissertations and Culminating Projects. 848.