Nowhere-Harmonic Colorings of Graphs

TL;DR

This paper introduces nowhere-harmonic colorings of graphs, extending classical concepts like proper colorings and chromatic polynomials, and explores their properties and enumeration complexities.

Contribution

It defines nowhere-harmonic colorings using the graph Laplacian and develops analogues of chromatic polynomials and Stanley's theorem for these colorings.

Findings

Nowhere-harmonic colorings generalize proper colorings.

Enumeration of nowhere-harmonic colorings is more complex.

Analogues of classical theorems are established for these colorings.

Abstract

Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley's theorem relating negative evaluations of the chromatic polynomial to acyclic orientations. Further, we discuss some examples demonstrating that nowhere-harmonic colorings are more complicated from an enumerative perspective than proper colorings.