- graceful labelings of partial cubes

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. © 2006 Elsevier B.V. All rights reserved.

[1]  W.-J. Hsu,et al.  FIBONACCI CUBES—A CLASS OF SELF-SIMILAR GRAPHS , 1993 .

[2]  W. Imrich,et al.  Product Graphs: Structure and Recognition , 2000 .

[3]  Vo D. Chepoi,et al.  Isometric subgraphs of Hamming graphs and d-convexity , 1988 .

[4]  Wilfried Imrich,et al.  Tree-like isometric subgraphs of hypercubes , 2003, Discuss. Math. Graph Theory.

[5]  Norma Zagaglia Salvi,et al.  Structural and enumerative properties of the Fibonacci cubes , 2002, Discret. Math..

[6]  Peter Winkler,et al.  Isometric embedding in products of complete graphs , 1984, Discret. Appl. Math..

[7]  Charles Delorme,et al.  Two sets of graceful graphs , 1980, J. Graph Theory.

[8]  Bojan Mohar,et al.  Cubic inflation, mirror graphs, regular maps, and partial cubes , 2004, Eur. J. Comb..

[9]  Sandi Klavzar,et al.  On cubic and edge-critical isometric subgraphs of hypercubes , 2003, Australas. J Comb..

[10]  D. Djoković.  Distance-preserving subgraphs of hypercubes , 1973 .

[11]  Wen-Jing Hsu,et al.  Fibonacci Cubes-A New Interconnection Technology , 1993, IEEE Trans. Parallel Distributed Syst..

[12]  Maryvonne Mahéo.  Strongly graceful graphs , 1980, Discret. Math..

[13]  Jie Wu,et al.  Extended Fibonacci cubes , 1995, Proceedings.Seventh IEEE Symposium on Parallel and Distributed Processing.

[14]  Norma Zagaglia Salvi,et al.  On a Problem on Generalised Fibonacci Cubes , 2003, Ars Comb..

[15]  Christian Barrientos.  Graceful labelings of cyclic snakes , 2001, Ars Comb..

[16]  Sandi Klavzar.  On median nature and enumerative properties of Fibonacci-like cubes , 2005, Discret. Math..

[17]  Sandi Klavzar,et al.  Partial Cubes and Crossing Graphs , 2002, SIAM J. Discret. Math..

[18]  Cemi,et al.  A Zoo of L 1 -embeddable Polytopal Graphs , 1996 .

[19]  Ronald L. Graham,et al.  Concrete mathematics - a foundation for computer science (2. ed.) , 1994 .

[20]  Sandi Klavar.  On median nature and enumerative properties of Fibonacci-like cubes , 2005 .

[21]  Michel Deza,et al.  Geometry of cuts and metrics , 2009, Algorithms and combinatorics.