Proofs of two conjectures on generalized Fibonacci cubes

TL;DR

This paper proves two conjectures regarding the properties of generalized Fibonacci cubes, specifically about the bounds on the index of bad strings and the isometric nature of doubled strings, confirming longstanding hypotheses in graph theory.

Contribution

The paper confirms two conjectures on generalized Fibonacci cubes, establishing bounds on the index of bad strings and the isometric property of doubled strings, advancing understanding in graph embedding theory.

Findings

Confirmed that for bad strings, the index is less than twice the string length.

Proved that if a string's cube is isometric, then doubling the string preserves this property.

Established that p-critical words in the minimal non-isometric case are only of length 2 or 3.

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 $p=3$.