Results on the K-Equitable Labeling of Complete Binary Trees

Abstract

When it comes to graphs, there have always been questions about different ways to label the vertices and edges, and the properties that a given method will have. These questions have led to many different results regarding a myriad of different types of graphs. Some of the more popular derivations are in regards to radio labeling, network flows, minimum and maximum matching algorithms and other optimization problems that can have real-world applications to things like traffic control, infrastructure, logistics, radio frequency assignment, computer networks, micro-chip architecture and many, many more. As is the case with many areas of mathematics, many of these properties and methods were explored well before there were any known applications. Such is sometimes the nature of mathematics research. This thesis will cover some topics that have been developed around the graceful and k-equitable labeling of graphs and trees (focusing more on trees, specifically binary trees) and their relationship to other well-known problems including the n-queens problems. First, we must start with some definitions and examples. Please note that this thesis will only deal with standard graphs (graphs that do not contain multiple edges between any two given vertices).