An elementary chromatic reduction for gain graphs and special hyperplane arrangements

TL;DR

This paper introduces a new elementary reduction method for gain graphs and hyperplane arrangements, providing formulas and relations that facilitate the calculation of various chromatic functions and polynomials.

Contribution

It develops a universal framework for weak chromatic functions of gain graphs, generalizing chromatic polynomial expressions and applying them to important hyperplane arrangements.

Findings

Derived formulas for gain graphs in terms of minors without neutral edges

New evaluations of chromatic polynomials for Shi, Linial, and Catalan arrangements

Calculated total chromatic polynomials for key gain graphs

Abstract

A gain graph is a graph whose edges are labelled invertibly by "gains" from a group. "Switching" is a transformation of gain graphs that generalizes conjugation in a group. A "weak chromatic function" of gain graphs with gains in a fixed group satisfies three laws: deletion-contraction for links with neutral gain, invariance under switching, and nullity on graphs with a neutral loop. The laws lead to the "weak chromatic group" of gain graphs, which is the universal domain for weak chromatic functions. We find expressions, valid in that group, for a gain graph in terms of minors without neutral-gain edges, or with added complete neutral-gain subgraphs, that generalize the expression of an ordinary chromatic polynomial in terms of monomials or falling factorials. These expressions imply relations for chromatic functions of gain graphs. We apply our relations to some special integral gain graphs including those that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining new evaluations of and new ways to calculate the zero-free chromatic polynomial and the integral and modular chromatic functions of these gain graphs, hence the characteristic polynomials and hypercubical lattice-point counting functions of the arrangements. We also calculate the total chromatic polynomial of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.