We will discuss different categorifications of the chromatic polynomial, how they are related to each other as well as how they relate to the Hochschild homology of algebras and Khovanov link homology.

The matching polynomial of a graph is the generating function of the numbers of its matchings with respect to their cardinality. A graph polynomial is polynomial reconstructible, if its value for a graph can be determined from its values for the vertex-deleted subgraphs of the same graph. This note discusses the polynomial reconstructibility of the matching polynomial. We collect previous resul...

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

One remarkable feature of the chromatic polynomial χ(Q) is Tutte’s golden identity. This relates χ(φ+ 2) for any triangulation of the sphere to (χ(φ+ 1)) for the same graph, where φ denotes the golden ratio. We explain how this result fits in the framework of quantum topology and give a proof using the chromatic algebra, whose Markov trace is the chromatic polynomial of an associated graph. We ...