The (non-)existence of perfect codes in Lucas cubes

Abstract

The Fibonacci cube of dimension n, denoted as Γ n , is the subgraph of the n-cube Q n induced by vertices with no consecutive 1's. Ashrafi and his co-authors proved the nonexistence of perfect codes in Γ n 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 Γ n (1 s ) of subgraphs induced by strings without 1 s as a substring where s ≥ 2 is a given integer. In a precedent work we proved the existence of a perfect code in Γ n (1 s ) for n = 2 p − 1 and s ≥ 3.2 p−2 for any integer p ≥ 2. The Lucas cube Λ n is obtained from Γ n 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 Λ n for n ≥ 4 and prove the existence of perfect codes in some generalized Lucas cube Λ n (1 s ).