Proofs of two conjectures on generalized Fibonacci cubes

Abstract: A binary string $f$ is a factor of string $u$ if $f$ appears as a sequence of $|f|$ consecutive bits of $u$, where $|f|$ denotes the length of $f$. Generalized Fibonacci cube $Q_{d}(f)$ is the graph obtained from the $d$-cube $Q_{d}$ by removing all vertices that contain a given binary string $f$ as a factor. A binary string $f$ is called good if $Q_{d}(f)$ is an isometric subgraph of $Q_{d}$ for all $d\geq1$, it is called bad otherwise. The index of a binary string $f$, denoted by $B(f)$, is the smallest integer $d$ such that $Q_{d}(f)$ is not an isometric subgraph of $Q_{d}$. Ili\'{c}, Klav\v{z}ar and Rho conjectured that $B(f)<2|f|$ for any bad string $f$. They also conjectured that if $Q_{d}(f)$ is an isometric subgraph of $Q_{d}$, then $Q_{d}(ff)$ is an isometric subgraph of $Q_{d}$. We confirm the two conjectures by obtaining a basic result: if there exist $p$-critical words for $Q_{B(f)}(f)$, then $p$=2 or $p=3$.