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 (f↽) is the graph obtained from Qd by removing all the vertices that have a circulation containing as a factor. The Fibonacci cube Γd and the Lucas cube Λd are the graphs Qd (11) and Qd (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.