A (p, q)-graph is edge-graceful if the edges can be labeled 1 through q in such a way that the labels induced on the vertices by summing over incident edges modulo p are distinct. Lo showed that a graph G is edge-graceful only if p|q^2 + q - p(p - 1)/2. Since then, many families of graphs have been shown to be edge-graceful. These are exhaustively enumerated in Gallian's dynamic survey, which also contains a complete bibliography of the subject.

graceful graph