Generalized Fibonacci cubes

Generalized Fibonacci cube Q d ( f ) is introduced as the graph obtained from the d -cube Q d by removing all vertices that contain a given binary string f as a substring. In this notation, the Fibonacci cube ? d is Q d ( 11 ) . The question whether Q d ( f ) is an isometric subgraph of Q d is studied. Embeddable and non-embeddable infinite series are given. The question is completely solved for strings f of length at most five and for strings consisting of at most three blocks. Several properties of the generalized Fibonacci cubes are deduced. Fibonacci cubes are, besides the trivial cases Q d ( 10 ) and Q d ( 01 ) , the only generalized Fibonacci cubes that are median closed subgraphs of the corresponding hypercubes. For admissible strings f , the f -dimension of a graph is introduced. Several problems and conjectures are also listed.

[1]  David Eppstein.  The lattice dimension of a graph , 2005, Eur. J. Comb..

[2]  Christian Lavault,et al.  Distributed Algorithms for AllPairs Shortest Paths and Minimum Diameter Spanning Tree Problems , 1998, OPODIS.

[3]  David Offner.  Some Turán type results on the hypercube , 2009, Discret. Math..

[4]  Petra Zigert,et al.  FIBONACCI CUBES ARE THE RESONANCE GRAPHS OF FIBONACCENES , 2003 .

[5]  Petr Gregor,et al.  Embedding Fibonacci Cubes into hypercubes with Ω(2cn) faulty nodes , 2000 .

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

[7]  Wen-Jing Hsu,et al.  Generalized fibonacci cubes are mostly hamiltonian , 1994, J. Graph Theory.

[8]  Ernesto Dedó,et al.  The observability of the Fibonacci and the Lucas cubes , 2002, Discret. Math..

[9]  David Eppstein,et al.  The Fibonacci Dimension of a Graph , 2011, Electron. J. Comb..

[10]  Norma Zagaglia Salvi.  On the existence of cycles of every even length on generalized Fibonacci cubes , 1997 .

[11]  Petr Gregor,et al.  Embedding Fibonacci Cubes into Hypercubes with Omega(2cn) Faulty Nodes , 2000, MFCS.

[12]  Wen-Jing Hsu,et al.  Distributed algorithms for shortest-path, deadlock-free routing and broadcasting in a class of interconnection topologies , 1992, Proceedings Sixth International Parallel Processing Symposium.

[13]  David A. Pike,et al.  Decycling of Fibonacci cubes , 2006, The Australasian Journal of Combinatorics.

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

[15]  David Eppstein,et al.  Journal of Graph Algorithms and Applications Recognizing Partial Cubes in Quadratic Time 270 Eppstein Recognizing Partial Cubes in Quadratic Time , 2022 .

[16]  Aleksander Vesel,et al.  Fast Recognition of Fibonacci Cubes , 2007, Algorithmica.

[17]  Martyn Mulder,et al.  The structure of median graphs , 1978, Discret. Math..

[18]  Emden R. Gansner,et al.  On the lattice of order ideals of an up-down poset , 1982, Discret. Math..

[19]  Petr Gregor.  Recursive fault-tolerance of Fibonacci cube in hypercubes , 2006, Discret. Math..

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

[21]  Iztok Peterin.  Edge-counting vectors, Fibonacci cubes, and Fibonacci triangle , 2007 .

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

[23]  Sandi Klavzar,et al.  Theta-graceful labelings of partial cubes , 2006, Discret. Math..