Abstract
The Fibonacci cube of dimension n, denoted as Γn, is the subgraph of the n-cube Qn induced by vertices with no consecutive 1’s. Ashrafi and his co-authors proved the non-existence of perfect codes in Γn for n ≥ 4. As an open problem the authors suggest to consider the existence of perfect codes in generalizations of Fibonacci cubes. The most direct generalization is the family Γn(1s) of subgraphs induced by strings without 1s as a substring where s ≥ 2 is a given integer. In a precedent work we proved the existence of a perfect code in Γn(1s) for n = 2p − 1 and s ≥ 3.2p − 2 for any integer p ≥ 2. The Lucas cube Λn is obtained from Γn by removing vertices that start and end with 1. Very often the same problems are studied on Fibonacci cubes and Lucas cube. In this note we prove the non-existence of perfect codes in Λn for n ≥ 4 and prove the existence of perfect codes in some generalized Lucas cube Λn(1s).
Highlights
Introduction and notationsAn interconnection topology can be represented by a graph G = (V, E), where V denotes the processors and E the communication links
The Fibonacci cube was introduced in [8] as a new interconnection network. This graph is an isometric subgraph of the hypercube which is inspired in the Fibonacci numbers
As an open problem the authors suggest to consider the existence of perfect codes in generalizations of Fibonacci cubes
Summary
The Fibonacci cube was introduced in [8] as a new interconnection network This graph is an isometric subgraph of the hypercube which is inspired in the Fibonacci numbers. A perfect code of a graph G is both a dominating set and a code It is a set of vertices C such that every vertex of G belongs to the closed neighbourhood of exactly one vertex of C. The vertex set of the n-cube Qn is the set Bn of binary strings of length n, two vertices being adjacent if they differ in precisely one position. Let Fn and Ln be the set of strings of Fibonacci strings and Lucas strings of length n. As an open problem the authors suggest to consider the existence of perfect codes in generalizations of Fibonacci cubes. The Lucas cube Λn, n ≥ 0, admits a perfect code if and only if n ≤ 3
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