  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.
