Hyperoctahedral Eulerian Idempotents, Hodge Decompositions, and Signed Graph Coloring Complexes

TL;DR

This paper extends a known combinatorial-topological relationship from unsigned graphs to signed graphs, using hyperoctahedral Eulerian idempotents to analyze chromatic polynomials and homology decompositions.

Contribution

It introduces a type B analogue of Hanlon's result, connecting chromatic polynomials of signed graphs with Hodge-type decompositions via hyperoctahedral Eulerian idempotents.

Findings

Established a type B analogue of Hanlon's theorem.

Connected chromatic polynomial coefficients to homology summands.

Utilized hyperoctahedral Eulerian idempotents in the analysis.

Abstract

Phil Hanlon proved that the coefficients of the chromatic polynomial of a graph G are equal (up to sign) to the dimensions of the summands in a Hodge-type decomposition of the top homology of the coloring complex for G. We prove a type B analogue of this result for chromatic polynomials of signed graphs using hyperoctahedral Eulerian idempotents.