Proof of a conjecture of Fomichev and Karev

TL;DR

This paper proves a conjecture by demonstrating the equality of two graph invariants, one based on colorings and the other from an algebraic weight system, linking combinatorics and algebra.

Contribution

It establishes the equality of the graph invariants , defined through colorings, and , derived from the  weight system, confirming a conjecture in graph theory.

Findings

Proved the conjecture relating  and  invariants.

Connected graph colorings with  weight systems.

Validated the conjecture in the context of algebraic graph invariants.

Abstract

We prove a conjecture of Fomichev and Karev [{European J. Combin.} 127 (2025) 104160] by showing the equality of two graph invariants: $\varphi$, defined via graph colorings, and $\psi$, derived from the $\mathfrak{sl}(2)$-weight system of its 2-dimensional irreducible representation.