three graceful operations

a graph of size n is said to be graceful when is possible toassign distinct integers from {0, 1, . . . , n} to its verticesand {|f(u)−f(v)| : uv ∈ e(g)} consists of n integers. inthis paper we present broader families of graceful graphs; these families are obtained via three different operations: the third power of a caterpillar, the symmetric product of g and k2 , and the disjoint :union: of g and pm, where g is a special type of graceful graph named - graph. moreover, the majority of the graceful labelings obtained here correspond to the most restrictive kind, they are -labelings. these labelings are in the core of this research area due to the fact that they can be used to create other types of graph labelings, almost independently of the nature of these labelings.