A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph
TL;DR
These results do not resolve the existence problem, but they reduce the exact search space and improve the best verified partial construction found in the authors' experiments.
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.
