A categorification for the signed chromatic polynomial

TL;DR

This paper introduces a cohomology theory for signed graphs that categorifies the signed chromatic polynomial, extending the concept of graph polynomial categorification to signed graphs.

Contribution

It constructs graded cohomology groups for signed graphs whose Euler characteristic equals the signed chromatic polynomial, and establishes a long exact sequence related to deletion-contraction.

Findings

Cohomology groups match the signed chromatic polynomial

Long exact sequence corresponds to deletion-contraction rule

Extends categorification from unsigned to signed graphs

Abstract

By coloring a signed graph by signed colors, one obtains the signed chromatic polynomial of the signed graph. For each signed graph we construct graded cohomology groups whose graded Euler characteristic yields the signed chromatic polynomial of the signed graph. We show that the cohomology groups satisfy a long exact sequence which corresponds to signed deletion-contraction rule. This work is motivated by Helme-Guizon and Rong's construction of the categorification for the chromatic polynomial of unsigned graphs.