Edge Preserving Transformations of Graceful Labelings

Author

Nathan Risher, Southern Illinois University CarbondaleFollow

Date of Award

12-1-2023

Degree Name

Master of Science

Department

Mathematics

First Advisor

Lauderdale, Lindsey-Kay

Abstract

Let $l$ be a graceful label of a graceful graph $G$ with $n$ nodes. We outline a procedure to generate a graceful label $l$ and its graph $G$ by constructing a sequence of labeled edges $(a_k)_{k=1}^{n-1}$ where the $k$th term of the sequence corresponds to an edge labeled $k$. We use the complement of the label generated to identify a class of transformations on graceful labels that can produce additional graceful labelings on $G$. We then identify a subset of labels generated this way with properties that limit the number of graceful labels such a graph can have and study some properties of those labels. We prove that all edge-preserving transformations of these labels fix over half of all node labels, and after establishing criteria necessary for such a transformation to leave some node labels unfixed, we show that for $n\leq9$ these transformations fix all node labels.