Paper 5. Lower Bounds for g(r) at Every r: 3r - 3, and 3r - 2 from r = 16

Osman, Mohamed A.

Paper 5. Lower Bounds for g(r) at Every r: 3r - 3, and 3r - 2 from r = 16

Proposed proofs that g(r) is at least 3r - 3 for every r and at least 3r - 2 from r = 16

Mohamed A. Osman · ORCID 0009-0004-5912-999X · independent researcher · 2026-10-19

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Abstract. In the framework of Sivashankar (arXiv:2606.24878), the cover number of the remainder of maximum degree three is bounded by 4 tau <= q + r + 3, answering his question whether +4 can be +3; hence g(r) >= 3r - 3 for every r, attained at r = 3 and 4. Excluding the equality case gives g(r) >= 3r - 2 for every r >= 16, with one case left open at r = 15. Every finite step is checked exhaustively by a named script; the proofs are unreviewed.

No priority is claimed for any result, and this is not presented as a settled answer. The lower bound is computer-assisted and has not been independently reviewed by a human or by a proof assistant. AI tools were used throughout the work and its review; every computational claim is regenerated by a script or certified by a proof file in code/ and results/ of the repository. Licence: CC BY 4.0 (text), MIT (code).