Graceful and Prime Labelings -- Algorithms, Embeddings and Conjectures

TL;DR

This paper introduces algorithms and conditions for graceful graph labelings, explores embeddings, and proposes conjectures, including generalizations of the Ringel-RosaKotzig conjecture, advancing understanding of graceful and prime graphs.

Contribution

It presents new algorithms for generating and embedding graceful graphs, generalizes existing conjectures, and offers novel properties and conditions for highly graceful and prime graphs.

Findings

Four algorithms for graceful graphs are described.

Conditions for highly graceful and critical graphs are provided.

Conjectures on graceful, critical, and highly graceful graphs are proposed.

Abstract

Four algorithms giving rise to graceful graphs from a known (non)graceful graph are described. Some necessary conditions for a graph to be highly graceful and critical are given. Finally some conjectures are made on graceful, critical and highly graceful graphs. The RingelRosaKotzig Conjecture is generalized to highly graceful graphs. MayedaSeshu Tree Generation Algorithm is modified to generate all possible graceful labelings of trees of order p. An alternative algorithm in terms of integers modulo p is described which includes all possible graceful labelings of trees of order p and some interesting properties are observed. Optimal and graceful graph embeddings (not necessarily connected) are given. Alternative proofs for embedding a graph into a graceful graph as a subgraph and as an induced subgraph are included. An algorithm to obtain an optimal graceful embedding is described. A necessary condition for a graph to be supergraceful is given. As a consequence some classes of nonsupergraceful graphs are obtained. Embedding problems of a graph into a supergraceful graph are studied. A catalogue of super graceful graphs with at most five nodes is appended. Optimal graceful and supergraceful embeddings of a graph are given. Graph theoretical properties of prime and superprime graphs are listed. Good upper bound for minimum number of edges in a nonprime graph is given and some conjectures are proposed which in particular includes Entringers prime tree conjecture. A conjecture for regular prime graphs is also proposed.