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.

Source : oai:HAL:hal-01196859v1

Volume: Vol. 17 no. 1

Section: Graph Theory

Published on: February 12, 2015

Submitted on: October 1, 2014

Keywords: Fibonacci cube,Lucas cube,generalized Fibonacci cube,generalized Lucas cube,connectivity,combinatorics on words,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[INFO.INFO-HC] Computer Science [cs]/Human-Computer Interaction [cs.HC]

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