Chromatic polynomials and bialgebras of graphs

TL;DR

This paper characterizes the chromatic polynomial as a unique graph invariant compatible with two bialgebra structures, providing algebraic proofs of classical results and extending to non-commutative versions with indexed graphs.

Contribution

It introduces a bialgebraic framework for chromatic polynomials, offering new algebraic proofs and a non-commutative generalization involving indexed graphs.

Findings

Provides Hopf-algebraic proofs of Rota's sign conjecture.

Interprets Stanley's results on chromatic polynomial values at negative integers.

Extends the framework to non-commutative algebra with indexed graphs.

Abstract

The chromatic polynomial is characterized as the unique polynomial invariant of graphs, compatible with two interacting bialgebras structures: the first coproduct is given by partitions of vertices into two parts, the second one by a contraction-extraction process. This gives Hopf-algebraic proofs of Rota's result on the signs of coefficients of chromatic polynomials and of Stanley's interpretation of the values at negative integers of chromatic polynomi-als. We also give non-commutative version of this construction, replacing graphs by indexed graphs and Q[X] by the Hopf algebra WSym of set partitions.