Connectivity of Fibonacci cubes, Lucas cubes and generalized cubes

Graph Theory International audience 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.

The Larger Bound on the Domination Number of Fibonacci Cubes and Lucas Cubes

LetΓnandΛnbe then-dimensional Fibonacci cube and Lucas cube, respectively. Denote byΓ[un,k,z]the subgraph ofΓninduced by the end-vertexun,k,zthat has no up-neighbor. In this paper, the number of end-vertices and domination numberγofΓnandΛnare studied. The formula of calculating the number of end-vertices is given and it is proved thatγ(Γ[un,k,z])≤2k-1+1. Using these results, the larger bound on the domination numberγofΓnandΛnis determined.

Generalized Lucas cubes

Let f be is a binary string and d?1. Then the generalized Lucas cube Qd(f?)is introduced as the graph obtained from the Qd by removing all vertices that have a circulation containing f as a substring. The question for which f and d, the generalized Lucas cube Qd(f?) is an isometric subgraph of the d-cube Qd is solved for all binary strings of length at most five. Several isometrically embeddable and non-embeddable infinite series where f is of arbitrary length are given. Some structural properties of generalized Lucas cubes are also presented.