New categorifications of the chromatic and dichromatic polynomials for graphs
For each graph $G$, we define a chain complex of graded modules over the ring of polynomials whose graded Euler characteristic is equal to the chromatic polynomial of $G$. Furthermore, we define a chain complex of doubly-graded modules whose (doubly) graded Euler characteristic is equal to the dichromatic polynomial of $G$. Both constructions use Koszul complexes, and are similar to the new Khovanov–Rozansky categorifications of the HOMFLYPT polynomial. We also give a simplified definition of this triply-graded link homology theory.