New Categorifications of the Chromatic and the Dichromatic Polynomials for Graphs

In this paper, for each graph $G$, we def\mbox{}ine 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 def\mbox{}ine 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 categorif\mbox{}ications of HOMFLYPT polynomial. We also give simplif\mbox{}ied def\mbox{}inition of this triply-graded link homology theory.