康威99-图的强制结构约简与可验证边界
A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph
- Ashoka University（阿育王大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究通过自主AI智能体对康威99-图问题展开系统性可复现研究，完成强制结构约简、可验证边界等贡献，获最佳验证成果69.43%。
AI中文摘要：
康威99-图问题询问是否存在参数为$\text{srg}(99,14,1,2)$的强正则图。我们报告了由自主AI研究智能体开展的系统性、完全可复现的研究，按该赛道的部分信用指标评分。我们的可验证贡献包括：(1) 详尽证明$\boldsymbol{Z}/99$上的循环图最多满足$3366/4950=68.0\\%$的约束（49个差类中的33个），且99阶的另一阿贝尔群也有相同上限；(2) 强制结构约简：$\boldsymbol{\lambda}=1$使每个邻域成为完美匹配，$\boldsymbol{\mu}=2$使外部顶点与未匹配邻接对一一对应，将存在性问题归约为84个顶点上的12正则图，编码为CP-SAT并通过唯一$\text{srg}(9,4,1,2)$验证；(3) 已验证的指定自同构轨道存在性框架（无不动点和单不动点作用，在$\text{srg}(9,4,1,2)$和Paley图$\text{srg}(13,6,2,3)$上验证）；(4) 最佳验证成果为69.43%，有证据表明这是与该开放问题纠缠的稳健边界（14种不同方法均未超过该值），因为任何低于4950的可证边界都是非存在性证明。
英文摘要：
Conway's 99-graph problem asks whether a strongly regular graph with parameters $\mathrm{srg}(99,14,1,2)$ exists. We develop two complementary lines of attack. Fixing one vertex, the conditions $λ=1$ and $μ=2$ force its neighbourhood to be a perfect matching and determine every edge between that neighbourhood and the remaining vertices. For $(99,14,1,2)$, the unresolved part is therefore a constrained $12$-regular graph on $84$ vertices. We encode this reduction in CP-SAT and validate it by recovering the unique $\mathrm{srg}(9,4,1,2)$. We also prove by exhaustive enumeration that no circulant graph on $\mathbb{Z}/99$ satisfies more than $68.0\%$ of the CAISc constraints, and we give a validated orbit formulation for prescribed automorphisms. We then study the partial-score search problem. Fourteen human-designed search configurations reached at most $69.43\%$. Separately, we supplied the scoring function to an evolutionary program-search system. It produced a degree-preserving $4$-vertex-switch tabu search whose best verified artifact scores $70.73\%$. The generated move differs from those used in our own searches and crosses a plateau that was stable under them. These results do not resolve the existence problem, but they reduce the exact search space and improve the best verified partial construction found in our experiments.
