| A function f is a graceful valuation of a graph G with n edges if f is an injection from the vertices of G to the set {dollar}rm{lcub}0,1,...,n{rcub}{dollar} such that, when each edge {dollar}rm{lcub}u, v{rcub}{dollar} is assigned the label {dollar}vert f{dollar} (u) {dollar}-f{dollar} (v){dollar}vert,{dollar} the resulting edge labels are distinct. This labeling provides a sequential labeling of the edges from 1 to n. Any graph that can be gracefully labeled is a graceful graph. If, in addition, there is a number {dollar}gamma{dollar} so that for each edge {dollar}{lcub}{dollar}u, v{dollar}{rcub}{dollar} either f (u) {dollar}legamma |
