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Journal Articles International Electronic Journal of Algebra Year : 2021

Chromatic polynomials and bialgebras of 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.
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Dates and versions

hal-01396199 , version 1 (14-11-2016)
hal-01396199 , version 2 (29-11-2016)
hal-01396199 , version 3 (30-04-2021)

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Cite

Loïc Foissy. Chromatic polynomials and bialgebras of graphs. International Electronic Journal of Algebra, 2021, 30, pp.116-167. ⟨10.24330/ieja.969651⟩. ⟨hal-01396199v3⟩
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