Θ-graceful labelings of partial cubes

Abstract

Partial cubes are graphs that allow isometric embeddings into hypercubes. -graceful labelings of partial cubes are introduced as a natural extension of graceful labelings of trees. It is shown that several classes of partial cubes are -graceful, for instance even cycles, Fibonacci cubes, and (newly introduced) lexicographic subcubes. The Cartesian product of -graceful partial cubes is again such and we wonder whether in fact any partial cube is -graceful. A connection between -graceful labelings and representations of integers in certain number systems is established. Some directions for further investigation are also proposed.