Connectivity of Fibonacci cubes, Lucas cubes and generalized cubes

Abstract : 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.