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 polynomials. We also consider chromatic symmetric functions and their noncommutative versions.
Highlights
[5] in order to treat the four color theorem, is a polynomial invariant attached to a graph; its values at X “ k gives the number of valid colorings of the graph with k colors, for any integer k ě 1
Our main tools, presented in the first section, will be a Hopf algebra pHG, m, ∆q and a bialgebra pHG, m, δq, both based on graphs
The morphism φ1 is described in the second section: for any graph G, φ1pGq is the chromatic polynomial PchrpGq (Theorem 3.5)
Summary
The chromatic polynomial, introduced by Birkhoff and Lewis [5] in order to treat the four color theorem, is a polynomial invariant attached to a graph; its values at X “ k gives the number of valid colorings of the graph with k colors, for any integer k ě 1. Our main tools, presented, will be a Hopf algebra pHG, m, ∆q and a bialgebra pHG, m, δq, both based on graphs They share the same product, given by disjoint union; the first (cocommutative) coproduct, denoted by ∆, is given by partitions of vertices into two parts; the second (not cocommutative) one, denoted by δ, is. These two bialgebras are in cointeraction, a notion described in [7,10,19,20]: we obtain that pHG, m, ∆q is a bialgebra-comodule over pHG, m, δq, see Theorem 2.13 Another example of interacting bialgebras is the pair pQrXs, m, ∆q and pQrXs, m, δq, where m is the usual product of QrXs and the two coproducts ∆ and δ are defined by:. The morphism φ1 is described in the second section: for any graph G, φ1pGq is the chromatic polynomial PchrpGq (Theorem 3.5) This characterizes the chromatic polynomial as the unique polynomial invariant on graphs compatible with both bialgebraic structures. PQrXs, m, ∆q is a Hopf algebra, of antipode S sending any P pXq P QrXs to P pXq
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