Abstract:
An unproven claim is that all trees may be gracefully labeled. However there are some special classes of trees that are proven to have graceful labelings. A path is the simplest form of a tree, and it has been proven that all paths can be gracefully labeled. The focus of this study is on the characteristics of gracefully labeled paths and a method for producing graceful labelings of P sub n with given properties. We report on progress towards a proof that labelings of paths of any size may assign the label 1 to any node and be completed as graceful labelings. Representations such as the Edge Tree Diagram and the Matrix-Entry Choosing methods are developed. We also prove that certain assignments of labels to the first two vertices of a path guarantee that a labeling may not be completed to form a graceful labeling. Finally we develop a computer program to generate gracefully labeled paths to assist in examining the results and identifying conjectures worthy of further study.