Existence of a Moore graph of degree 57 is still open
Abstract
In 2020, a paper [arXiv:2010.13443] appeared in the arXiv claiming to prove that a Moore graph of diameter 2 and degree 57 does not exist. (The paper is in Russian; we include a link to a translation of this paper kindly provided to us by Konstantin Selivanov.) The proof technique is reasonable. It employs the fact that such a graph must be distance regular and that there exists a large set of relations which such a graph must satisfy. The argument proceeds by a case analysis that shows that this set of relations cannot be satisfied. We show that this seems not to be correct. The system of equations factors into small diagonal blocks all of which have solutions. As an alternative, we show that there is a family of systems of permutations with the property that the Moore graph exists if and only if there is a member of the family with no solutions.
 Publication:

arXiv eprints
 Pub Date:
 October 2022
 DOI:
 10.48550/arXiv.2210.09577
 arXiv:
 arXiv:2210.09577
 Bibcode:
 2022arXiv221009577F
 Keywords:

 Mathematics  Combinatorics;
 05C12;
 05C30