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Total mutual visibility in Fibonacci and Lucas cubes

Primarily AI-generated textHuman understanding: no partsmath.CO — Combinatorics

Contributed by Daniel Adam Pyle ↗

Version 1 / Oct 21, 2026 / CC BY 4.0

Abstract

A set of vertices is total mutual-visible if every pair of graph vertices has a shortest path whose internal vertices avoid the set. We prove that the total mutual-visibility numbers of the n-dimensional Fibonacci and Lucas cubes both have order ρⁿ/n, where ρ > 1 satisfies ρ³ = ρ² + 1. The eligible Fibonacci words are exactly those avoiding 11 for which zero padding avoids 0000. Their induced graph decomposes into fibres of recognizing subsets of maximal independent sets of a path, and each fibre is a Cartesian product of smaller Fibonacci cubes. Total visibility becomes open packing in these fibres. An elementary coloring and counting argument proves the order of growth. Typical degree and free-dimension estimates, together with a classical binary-code asymptotic, sharpen the normalized bounds to constants 4.032058… and 6.346255…. We give the cyclic analogue, an exact component inventory, and reproducible exact values through dimension 20 for Fibonacci cubes and 16 for Lucas cubes. Neither a leading asymptotic constant nor a closed formula in every dimension is established.

Provenance statement

The literature research, proof development, computational verification, and manuscript drafting were performed by an OpenAI-powered AI assistant called Lyra. Daniel A. Pyle initiated and directed the project and is submitting the work as an independent researcher. Verification included separate AI-assisted reviews, exact integer computations, and independent numerical optimization checks. Numerical solver results are distinguished from exact proofs. These checks do not constitute human peer review or proof-assistant formalization. Established results and techniques are credited to their sources. Bibliographic novelty remains provisional, and neither a leading asymptotic constant nor a closed formula in every dimension is claimed. The manuscript and reproducibility materials are available in the project repository: https://github.com/dapyle11/Total-mutual-visibility-in-Fibonacci-and-Lucas-cubes-proofs-code-and-results.

Tools used

OpenAI
OpenAI-powered assistant (Lyra) Python, SymPy, SciPy / HiGHS

References

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Version history

  1. v1Submitted by Daniel Adam PyleInitial depositCurrentOct 21, 2026