Real-Graceful Labelings: A Generalization of Graceful Labelings

Abstract

Every graph can be associated to a cheracteristic exponential equation involving powers of (say) \(2\), whose unknowns represent ver-
tex labels and whose general solution is equivalent to a graceful labelling of the graph. If we do not require that the solutions be
integers, we obtain a generalisation of a graceful labelling that uses real numbers as labels. Some graphs that are well known to be non-graceful become graceful in this more general context. Among other things, “real-graceful” labellings provide some information on the Tigidity to be non-graceful, also asymptotically.