On the homology theory for the chromatic polynomials

TL;DR

This paper explores a homology theory for chromatic polynomials, revealing its structure over rationals and introducing a new Lee-type homology with a spectral sequence connecting them.

Contribution

It demonstrates the support of rational homology in two lines and develops a Lee-type homology for graphs, establishing a spectral sequence linking the two theories.

Findings

Rational homology $H^*(G;Q)$ is supported in two lines.

Introduces a new homology $H_{Lee}(G)$ analogous to Lee's theory.

Establishes a spectral sequence from $H^*(G)$ to $H_{Lee}(G)$.

Abstract

In \cite{10.2140/agt.2005.5.1365}, Rong and Helme-Guizon defined a categorification for the chromatic polynomial $P_G(x)$ of graphs $G$, i.e. a homology theory $H^*(G)$ whose Euler characteristic equals $P_G(x)$. In this paper, we showed that the rational homoology $H^*(G;\mathbb{Q})$ is supported in two lines, and develop an analogy of Lee's theory for Khovanov homology. In particular, we develop a new homology theory $H_{Lee}(G)$, and showed that there is a spectral sequence whose $E_2$ -term is isomorphic to $H^*(G)$ converges to $H_{Lee}(G)$.