Existence of a Moore graph of degree 57 is still open

TL;DR

This paper critically examines a recent proof claiming the non-existence of a Moore graph with degree 57, suggesting that the proof's case analysis may be flawed and exploring alternative systems of permutations related to the graph's existence.

Contribution

It challenges the validity of a recent proof about the non-existence of the Moore graph of degree 57 and proposes an alternative approach involving permutation systems.

Findings

The proof's case analysis appears to be incorrect.

The system of equations factors into solvable blocks.

Existence of the Moore graph correlates with permutation systems with no solutions.

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.