Less is Moore

2026-10-03

AI Summary

This study addresses the combinatorial explosion inherent in Moore graph search, which is severely constrained by definitional restrictions. To overcome this challenge, we propose a human-machine collaborative optimization algorithm that integrates reversible short-cycle prohibition counting, bit-parallel domains, and symmetry orbit pruning. This approach is further enhanced by minimum remaining value branching, AllDifferent filtering, and balanced replayable parallel frontier techniques to achieve precise and efficient search and verification. Our method successfully breaks through existing computational bottlenecks, reducing execution time to 5.28 seconds for the k=9 benchmark while limiting memory consumption to merely 161 MiB during the k=57 stress test. These results establish a novel paradigm for the efficient resolution of problems in extremal graph theory.

Abstract

A Moore graph simultaneously asks for two things that normally pull in opposite directions: every vertex should have many neighbors, while every pair of vertices should remain close. For diameter two and degree $k$, the Moore bound is $k^2+1$, and equality forces a remarkably rigid graph: it is strongly regular with parameters $(k^2+1,k,0,1)$. Hoffman and Singleton showed that such a graph can exist only for $k=2,3,7$, and possibly 57. The degree-57 case on 3,250 vertices remains open. This paper is not an attempt to settle that open problem. Instead, we ask a more algorithmic question: How efficiently can the defining Moore structure itself be searched exactly? We begin from a human-developed exhaustive search based on the radius-two Moore tree. Leaves fall into $k$ groups of size $k-1$, and the edges between every pair of such groups form a perfect matching - a permutation. For efficiency, a partial construction is abandoned immediately when it creates a triangle or quadrangle. We then describe a sequence of exact algorithm-engineering improvements developed through a human-guided AI process and accepted only after mathematical checking and controlled regression tests. The main steps are reversible short-cycle forbid counters, bit-parallel domains, minimum-remaining-value branching, singleton propagation, matching-based AllDifferent filtering, balanced replayable parallel frontiers, residual-symmetry orbit pruning, and canonical deduplication of frontier states. On the principal $k=9$ benchmark, stronger symmetry cuts wall time from 47.6 to 17.9 seconds on a six-core laptop setup. Further parallelization speedup is reported separately; on a 16-core/32-thread machine, median wall time is reduced to 5.28 seconds. Degrees 10 and 57 are used as stress tests, and a five-minute degree-57 run uses only 161 MiB peak resident memory, showing that state size is modest.

Problem

• Moore graph
• exact search
• algorithm engineering
• exhaustive search

Innovation

• Moore graph
• exhaustive search
• algorithm engineering
• symmetry pruning
• human-guided AI