A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph

Abstract: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 $\lambda=1$ and $\mu=2$ 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.

Artificial Intelligence (cs.AI); Symbolic Computation (cs.SC); Combinatorics (math.CO)

arXiv:2608.11211 [cs.AI]

(or arXiv:2608.11211v2 [cs.AI] for this version)

https://doi.org/10.48550/arXiv.2608.11211 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.