On a Problem on Generalised Fibonacci Cubes

Abstract

A Fibonacci string of order \(n\) is a binary string of length \(n\) with no two consecutive ones. The Fibonacci cube \(\Gamma_n\) is the subgraph of the hypercube \(Q_n\) induced by the set of Fibonacci strings of order \(n\). For positive integers \(i, n\), with \(n \geq i\), the \(i\)th extended Fibonacci cube is the vertex-induced subgraph of \(Q_n\) for which \(V(\Gamma_{i}^{n}) = V_i\) is defined recursively by

\[V_{n+2}^{i} = 0 V_{n+1}^{i} + 10V_n^{i},\]

with initial conditions \(V_i^i = B_i, V_{i+1}^{i} = B_{i+1}\), where \(B_k\) denotes the set of binary strings of length \(k\). In this study, we answer in the affirmative a conjecture of Wu [10] that the sequences \(\{|V_n^i|\}_{i={1+2}}^\infty\) are pairwise disjoint for all \(i \geq 0\), where \(V_n^0 = V(\Gamma_n)\).