Abstract
Graph Theory 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.
Highlights
Fibonacci cubes [4] and Lucas cubes [16] form hypercube-like classes of graphs that have found several applications and were extensively studied so far, see the recent survey [9]
In the subsequent section we prove that the generalized Fibonacci cubes are always 2-connected while in Section 4 we characterize connected generalized Lucas cubes
Fibonacci cubes and Lucas cubes were recently extended to generalized Fibonacci cubes [6] and to generalized Lucas cubes [7] as follows
Summary
Fibonacci cubes [4] and Lucas cubes [16] form hypercube-like classes of graphs that have found several applications and were extensively studied so far, see the recent survey [9]. The topics studied include different metric aspects [1, 10, 11, 13], a number of computer science issues [3, 18, 20, 21], applications in chemistry [23, 24, 25], and a variety of additional topics [2, 15, 17] It is quite surprising, that, to the best of our knowledge, the connectivity of these cubes has not yet been established. In this paper we fill this gap by determining the vertex- and the edge-connectivity of Fibonacci cubes and of Lucas cubes, see Section 2.
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