Acta Mathematica Sinica, English Series

, Volume 33, Issue 6, pp 851–860

All good (bad) words consisting of 5 blocks



Generalized Fibonacci cube Qd(f), introduced by Ilić, Klavžar and Rho, is the graph obtained from the hypercube Qd by removing all vertices that contain f as factor. A word f is good if Qd(f) is an isometric subgraph of Qd for all d ≥ 1, and bad otherwise. A non-extendable sequence of contiguous equal digits in a word μ is called a block of μ. Ilić, Klavžar and Rho shown that all the words consisting of one block are good, and all the words consisting of three blocks are bad. So a natural problem is to study the words consisting of other odd number of blocks. In the present paper, a necessary condition for a word consisting of odd number of blocks being good is given, and all the good (bad) words consisting of 5 blocks is determined.


Generalized Fibonacci cube isometric subgraph good word bad word 

MR(2010) Subject Classification

05C12 05C60 


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  1. [1]
    Azarija, J., Klavžzar, S., Lee, J., et al.: On isomorphism classes of generalized Fibonacci cubes. European J. Combin., 51, 372–379 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    Azarija, J., Klavžzar, S., Lee, J., et al.: Connectivity of Fibonacci cubes, Lucas cubes and generalized cubes. Discrete Math. Theoret. Comput. Sci., 17 (1), 79–88 (2015)MathSciNetMATHGoogle Scholar
  3. [3]
    Dedó, E., Torri, D., Salvi, N. Z.: The observability of the Fibonacci and the Lucas cubes. Discrete Math., 255, 55–63 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    Gregor P.: Recursive fault-tolerance of Fibonacci cube in hypercubes. Discrete Math., 306, 1327–1341 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Hsu, W. J.: Fibonacci cubes — a new interconnection topology. IEEE Trans. Parallel Distrib. Syst., 4 (1), 3–12 (1993)CrossRefGoogle Scholar
  6. [6]
    Hsu, W. J., Liu, J.: Distributed algorithms for shortest-path, deadlock-free routing and broadcasting in a class of interconnection topologies. Parallel Processing Symposium, 1992, 589–596Google Scholar
  7. [7]
    Ilić A., Klavžzar, S., Rho, Y.: Generalized Fibonacci cubes. Discrete Math., 312, 2–11 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    Ilić A., Klavžzar, S., Rho, Y.: Parity index of binary words and powers of prime words. Electron. J. Combin., 19(3), #P44 (2012)MathSciNetMATHGoogle Scholar
  9. [9]
    Ilić A., Klavžzar, S., Rho, Y.: The index of a binary word. Theoret. Comput. Sci., 452, 100–106 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    Imrich, W., Klavžzar, S.: Product Graphs: Structure and Recognition, Wiley, New York, 2000Google Scholar
  11. [11]
    Klavžzar, S.: On median nature and enumerative properties of Fibonacci-like cubes. Discrete Math., 299, 145–153 (2005)MathSciNetCrossRefGoogle Scholar
  12. [12]
    Klavžzar, S.: Structure of Fibonacci cubes: a survey. J. Comb. Optim., 25, 505–522 (2013)MathSciNetCrossRefGoogle Scholar
  13. [13]
    Klavžzar, S., Shpectorov, S.: Asymptotic number of isometric generalized Fibonacci cubes. European J. Combin., 33, 220–226 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Klavžzar, S., Rho, Y.: On the Wiener index of generalized Fibonacci cubes and Lucas cubes. Discrete Appl. Math., 187, 155–160 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Klavžzar, S., Žigert, P.: Fibonacci cubes are the resonance graphs of fibonaccenes. Fibonacci Quart., 43 (3), 269–276 (2005)MathSciNetMATHGoogle Scholar
  16. [16]
    Liu, J., Hsu, W. J., Chung, M. J.: Generalized Fibonacci cubes are mostly Hamiltonian. J. Graph Theory, 18 (8), 817–829 (1994)MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Munarini, E., Salvi, N. Z.: Structural and enumerative properties of the Fibonacci cubes. Discrete Math., 255, 317–324 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    Salvi, N. Z.: On the existence of cycles of every even length on generalized Fibonacci cubes. Matematiche (Catania), 51, 241–251 (1996)MathSciNetMATHGoogle Scholar
  19. [19]
    Wei, J. X.: The structures of bad words. European J. Combin., 59, 204–214 (2017)MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    Wei, J. X., Zhang, H. P.: Solution to a conjecture on words that are bad and 2-isometric. Theoret. Comput. Sci., 562, 243–251 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    Wei, J. X., Zhang, H. P.: Proofs of two conjectures on generalized Fibonacci cubes. European J. Combin., 51, 419–432 (2016)MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    Wei, J. X., Zhang, H. P.: A negative answer to a problem on generalized Fibonacci cubes. Discrete Math., 340, 81–86 (2017)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of Mathematics and Statistics ScienceLudong UniversityYantaiP. R. China

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