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arXiv · 1611.04303

Chromatic polynomials and bialgebras of graphs

Loïc Foissy
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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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Loïc Foissy. 2016-11-14. Chromatic polynomials and bialgebras of graphs. https://arxiv.org/abs/1611.04303

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