Problem
Research questions and friction points this paper is trying to address.
Conway's 99-graph
strongly regular graph
srg(99,14,1,2)
existence problem
A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph
🤖 AI Summary
研究使用CP-SAT编码和进化程序搜索系统来减少Conway 99-图问题的搜索空间，并提高已验证部分构造的最佳得分。
📝 Abstract
Conway's 99-graph problem asks whether a strongly regular graph with parameters $\mathrm{srg}(99,14,1,2)$ exists. We report a systematic, fully reproducible attack by an autonomous AI research agent, scored under the track's partial-credit metric. Our verifiable contributions are: (1) an exhaustive proof that no circulant graph on $\mathbb{Z}/99$ satisfies more than $3366/4950=68.0\%$ of the constraints ($33$ of $49$ difference-classes), with the same ceiling for the other abelian group of order $99$; (2) a forced-structure reduction: $\lambda=1$ makes each neighbourhood a perfect matching and $\mu=2$ puts the outer vertices in bijection with non-matched neighbour-pairs, collapsing existence to a $12$-regular graph on $84$ vertices, encoded for CP-SAT and validated by recovering the unique $\mathrm{srg}(9,4,1,2)$; (3) a validated prescribed-automorphism orbit-existence framework (fixed-point-free and single-fixed-point actions, checked on $\mathrm{srg}(9,4,1,2)$ and the Paley graph $\mathrm{srg}(13,6,2,3)$), and (4) a best verified artifact at $69.43\%$, with evidence that this is a robust frontier (fourteen distinct methods, none exceeding it) entangled with the open question, since any provable bound below $4950$ is a non-existence proof.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Forced-Structure Reduction
CP-SAT Encoding
Degree-Preserving 4-Vertex-Switch Tabu Search
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