Proofs of Conjectures 2.4 and 2.5 on Generalized Fibonacci Cubes

This preprint proves Conjectures 2.4 and 2.5 of Azarija et al. on generalized Fibonacci cubes. If two forbidden binary words define isomorphic cubes in dimension n ≥ 1, they also define isomorphic cubes in dimension n − 1. For two distinct forbidden words unrelated by reversal, bitwise complementation, or their composition, isomorphism in dimension n requires each word to have length at least 2(n + 1)/3. The conjectures were posed in J. Azarija, S. Klavžar, J. Lee, J. Pantone, and Y. Rho, “On isomorphism classes of generalized Fibonacci cubes,” European Journal of Combinatorics 51 (2016), 372–379. DOI 10.1016/j.ejc.2015.05.011