Hopf-algebraic structures on mixed graphs

TL;DR

This paper develops a Hopf algebraic framework for mixed graphs with both oriented and unoriented edges, introducing coproducts and invariants that unify and extend classical graph polynomials and combinatorial results.

Contribution

It introduces two coproducts on mixed graphs forming a double bialgebra, leading to a polynomial invariant that generalizes the chromatic polynomial and relates to various graph invariants.

Findings

Defines a double bialgebra structure on mixed graphs.

Proves the polynomial invariant is the strong chromatic polynomial.

Provides an algebraic proof linking polynomial evaluations to acyclic orientations.

Abstract

We introduce two coproducts on mixed graphs (that is to say graphs with both oriented and unoriented edges), the first one by separation of the vertices into two parts, and the second one given by contraction and extractions of subgraphs. We show that, with the disjoint union product, this gives a double bialgebra, that is to say that the first coproduct makes it a Hopf algebra in the category of right comodules over the second coproduct. This structure implies the existence of a unique polynomial invariant on mixed graphs compatible with the product and both coproducts: we prove that it is the (strong) chromatic polynomial of Beck, Bogart and Pham.Using the action of the monoid of characters, we relate it to the weak chromatic polynomial, as well to Ehrhart polynomials and to a polynomial invariants related to linear extensions. As applications, we give an algebraic proof of the link between the values of the strong chromatic polynomial at negative values and acyclic orientations (a result due to Beck, Blado, Crawford, Jean-Louis and Young) and obtain a combinatorial description of the antipode of the Hopf algebra of mixed graphs.