The pebbling number of Fibonacci cubes

Abstract

The $n$-th Fibonacci cube $Γ_n$ is the subgraph of the hypercube $Q_n$ induced by binary strings with no two consecutive ones. We determine $π(Γ_n) = 2^n$ for $n \le 6$, so the pebbling number of $Γ_n$ equals that of the ambient hypercube $Q_n$ despite $Γ_n$ having far fewer vertices. The lower bound is a standard potential argument. For the upper bound, the Weight Function Lemma yields $2^n+1$ -- one too many -- so we close the gap by exhaustive MILP verification. We conjecture $π(Γ_n) = 2^n$ for all $n$.