Enumerating Colorings, Tensions and Flows in Cell Complexes

TL;DR

This paper generalizes graph coloring, tension, and flow polynomials to cell complexes, providing formulas, recurrence relations, and reciprocity theorems using geometric methods like Ehrhart theory.

Contribution

It introduces a unified framework for enumerating colorings, tensions, and flows in cell complexes, extending classical graph invariants with new formulas and reciprocity results.

Findings

Derived deletion-contraction recurrences for quasipolynomials.

Established closed-form formulas under unimodularity conditions.

Proved reciprocity theorems with combinatorial interpretations.

Abstract

We study quasipolynomials enumerating proper colorings, nowhere-zero tensions, and nowhere-zero flows in an arbitrary CW-complex $X$, generalizing the chromatic, tension and flow polynomials of a graph. Our colorings, tensions and flows may be either modular (with values in $\mathbb{Z}/k\mathbb{Z}$ for some $k$) or integral (with values in $\{-k+1,\dots,k-1\}$). We obtain deletion-contraction recurrences and closed formulas for the chromatic, tension and flow quasipolynomials, assuming certain unimodularity conditions. We use geometric methods, specifically Ehrhart theory and inside-out polytopes, to obtain reciprocity theorems for all of the aforementioned quasipolynomials, giving combinatorial interpretations of their values at negative integers as well as formulas for the numbers of acyclic and totally cyclic orientations of $X$.