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Discrete Mathematics & Theoretical Computer Science |
Jernej Azarija ; Sandi Klavžar ; Jaehun Lee ; Yoomi Rho - Connectivity of Fibonacci cubes, Lucas cubes and generalized cubes
dmtcs:2115 - Discrete Mathematics & Theoretical Computer Science, February 12, 2015, Vol. 17 no. 1 - https://doi.org/10.46298/dmtcs.2115- 1 Institute of Mathematics, Physics and Mechanics [Ljubljana]
- 2 Faculty of Mathematics and Physics [Ljubljana]
- 3 Faculty of Natural Sciences and Mathematics [Maribor]
- 4 Department of Mathematics [Incheon]
If f is a binary word and d a positive integer, then the generalized Fibonacci cube Qd(f) is the graph obtained from the d-cube Qd by removing all the vertices that contain f as a factor, while the generalized Lucas cube Qd(lucas(f)) is the graph obtained from Qd by removing all the vertices that have a circulation containing f as a factor. The Fibonacci cube Γd and the Lucas cube Λd are the graphs Qd(11) and Qd(lucas(11)), respectively. It is proved that the connectivity and the edge-connectivity of Γd as well as of Λd are equal to ⌊ d+2 / 3⌋. Connected generalized Lucas cubes are characterized and generalized Fibonacci cubes are proved to be 2-connected. It is asked whether the connectivity equals minimum degree also for all generalized Fibonacci/Lucas cubes. It was checked by computer that the answer is positive for all f and all d≤9.