Growing Graceful and Harmonious Trees

TL;DR

This paper introduces symbolic methods for listing graphs with identical label sequences, confirms a conjecture on graceful labeled trees, and presents a new algorithm for graceful labeling.

Contribution

It provides a constructive approach to enumerate gracefully labeled trees and confirms Whitty's conjecture on determinantal constructions.

Findings

Confirmed Whitty's conjecture on graceful trees

Developed symbolic constructions for graph enumeration

Proposed a new graceful labeling algorithm

Abstract

We describe symbolic constructions for listing and enumerating graphs having the same induced edge label sequence. We settle in the affirmative R. Whitty's [W08] conjectured existence of determinantal constructions for listing and enumerating gracefully labeled trees. We conclude the paper with a description of a new graceful labeling algorithm.