### Abstract

We determine that there is no partial geometry G with parameters (s,t,α)=(4,27,2). The existence of such a geometry has been a challenging open problem of interest to researchers for almost 40 years. The particular interest in G is due to the fact that it would have the exceptional McLaughlin graph as its point graph. Our proof makes extensive use of symmetry and high-performance distributed computing, and details of our techniques and checks are provided. One outcome of our work is to show that a pseudogeometric strongly regular graph achieving equality in the Krein bound need not be the point graph of any partial geometry.

Original language | English |
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Pages (from-to) | 27-41 |

Number of pages | 15 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 155 |

DOIs | |

Publication status | Published - 1 Apr 2018 |

MoE publication type | A1 Journal article-refereed |

### Keywords

- Backtrack search
- Krein bound
- McLaughlin geometry
- McLaughlin graph
- Partial geometry
- Pseudogeometric graph

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## Cite this

*Journal of Combinatorial Theory. Series A*,

*155*, 27-41. https://doi.org/10.1016/j.jcta.2017.10.004