On some Hopf monoids in graphical species

TL;DR

This paper explores Hopf monoids in graphical species, providing a graph-theoretic framework that extends previous work on combinatorial Hopf algebras and relates to graph coloring and hyperplane arrangements.

Contribution

It introduces a novel graph-theoretic approach to Hopf monoids in species, expanding the theoretical landscape and providing concrete examples linked to graph coloring and hyperplane arrangements.

Findings

Several new Hopf monoids in graphical species are constructed.

Connections to graph coloring and hyperplane arrangements are established.

The work extends the algebraic framework to a graph-theoretic context.

Abstract

Combinatorial Hopf algebras arise in a variety of applications. Recently, Aguiar and Mahajan showed how many well-studied Hopf algebras are closely related to Hopf monoids in species. In this paper, we study Hopf monoids in graphical species, giving a `graph-theoretic' analogue to the work of Aguiar and Mahajan. In particular, several examples of Hopf monoids in graphical species are detailed, most of which are related to graph coloring, or hyperplane arrangements associated to graphs.