Connectivity of Fibonacci cubes, Lucas cubes and generalized cubes

Jernej Azarija (1) , Sandi Klavžar (2, 3, 1) , Jaehun Lee (4) , Yoomi Rho (4)

1 Institute of Mathematics, Physics and Mechanics [Ljubljana]
2 FMF - 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.

Mots clés: Fibonacci cube; Lucas cube; generalized Fibonacci cube; generalized Lucas cube; connectivity; combinatorics on words

Domaines: Mathématique discrète [cs.DM]; Interface homme-machine [cs.HC]

Jernej Azarija, Sandi Klavžar, Jaehun Lee, Yoomi Rho. Connectivity of Fibonacci cubes, Lucas cubes and generalized cubes. Discrete Mathematics and Theoretical Computer Science, 2015, Vol. 17 no. 1 (1), pp.79--88. ⟨10.46298/dmtcs.2115⟩. ⟨hal-01196859⟩