Real-graceful labellings: a generalisation of graceful labellings

Every graph can be associated to a characteristic exponential equation involving powers of (say) 2, whose unknowns represent vertex 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 rigidity to be non-graceful, also asymptotically.