On the Lucas Cubes

A Lucas cube 3^ can be defined as the graph whose vertices are the binary strings of length n without either two consecutive l's or a 1 in the first and in the last position, and in which the vertices are adjacent when their Hamming distance is exactly 1. A Lucas cube 5E„ is very similar to the Fibonacci cube Tn which is the graph defined as 2J, except for the fact that the vertices are binary strings of length n without two consecutive ones. The Fibonacci cube has been introduced as a new topology for the interconnection of parallel multicomputers alternative to the classical one given by the Boolean cube [4]. An attractive property of the Lucas cube of order n is the decomposition, which can be carried out recursively into two disjoint subgraphs isomorphic to Fibonacci cubes of order n-\ and n-3; on the other hand, the Lucas cube of order n can be embedded in the Boolean cube of order n. This implies that certain topologies commonly used, as the linear array, particular types of meshes and trees and the Boolean cubes, directly embedded in the Fibonacci cube, can also be embedded in the Lucas cube. Thus, the Lucas cube can also be used as a topology for multiprocessor systems. Among many different interpretations, Fn+2 can be regarded as the cardinality of the set formed by the subsets of {1,...,«} which do not contain a pair of consecutive integers; i.e., the set of the binary strings of length n without two consecutive ones, the Fibonacci strings. If C„ is the set of the Fibonacci strings of order n, then Cn+2 = 0Cn+1 + \0Cn and \C„\ = Fn+2. A Lucas string is a Fibonacci string with the further condition that there is no 1 in the first and in the last position simultaneously. If C„ is the set of Lucas strings of order n, then \Cn\ = Ln, where Ln are the Lucas numbers for every n> 0. For n>\Ln can be regarded as the cardinality of the family of the subsets of {1, ...,n} without two consecutive integers and without the couple l,n. We have