The Genius Conjectures (via Bell Polynomials)

TL;DR

This paper introduces two conjectures related to i-matchings in random r-regular bipartite graphs, involving Bell polynomials and properties of convergent power series, suggesting deep combinatorial insights.

Contribution

The paper formulates two novel conjectures connecting graph matchings, Bell polynomials, and partition properties, proposing a challenging open problem in combinatorics.

Findings

Conjectures involve basic properties of convergent power series.

Connections between i-matchings and Bell polynomials are established.

Proposed conjectures are believed to require deep combinatorial proofs.

Abstract

We present two related conjectures, arising in work on i-matchings in random r-regular bipartite graphs. The conjectures themselves are easily stated and involve only basic properties of convergent power series. One formulation involves Bell's polynomials. The conjectures name was chosen since we earnestly believe only a truly genius mathematician will prove them. We advise others not to try. A further belief is that the proof will arise from some deep properties of partitions.