Unlabeled Signed Graph Coloring

TL;DR

This paper generalizes the chromatic polynomial concept to unlabeled signed graphs, providing explicit formulas, analyzing signed graph quotients, and connecting to classical reciprocity theorems.

Contribution

It introduces the unlabeled chromatic polynomial for signed graphs, extending Hanlon's work, and explores signed graph quotients and arrangements with new formulas.

Findings

Explicit formulas for signed chromatic polynomials

Relation between signed graph quotients and arrangements

Formula for unlabeled acyclic orientations

Abstract

We extend the work of Hanlon on the chromatic polynomial of an unlabeled graph to define the unlabeled chromatic polynomial of an unlabeled signed graph. Explicit formulas are presented for labeled and unlabeled signed chromatic polynomials as summations over distinguished order-ideals of the signed partition lattice. We also define the quotient of a signed graph by a signed permutation, and show that its signed graphic arrangement is closely related to an induced arrangement on a distinguished subspace. Lastly, a formula for the number of unlabeled acyclic orientations of a signed graph is presented which recalls classical reciprocity theorems of Stanley and Zaslavsky.