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

Abstract

The Fibonacci cube of dimension n, denoted as $\Gamma$ n , is the subgraph of the n-cube 5 Q n induced by vertices with no consecutive 1's. Ashrafi and his co-authors proved the non-existence of perfect codes in $\Gamma$ n for n $\ge$ 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 $\Gamma$ n (1 s) of subgraphs induced by strings without 1 s as a substring where s $\ge$ 2 is a given integer. In a precedent work 10 we proved the existence of a perfect code in $\Gamma$ n (1 s) for n = 2 p -- 1 and s $\ge$ 3.2 p--2 for any integer p $\ge$ 2. The Lucas cube $\Lambda$ n is obtained from $\Gamma$ 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 $\Lambda$ n for n $\ge$ 4 and 15 prove the existence of perfect codes in some generalized Lucas cube $\Lambda$ n (1 s).