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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm190-0-9,
author = {Marko Sto\v si\'c},
title = {New categorifications of the chromatic and dichromatic polynomials for graphs},
journal = {Fundamenta Mathematicae},
volume = {189},
year = {2006},
pages = {231-243},
zbl = {1102.57009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm190-0-9}
}
Marko Stošić. New categorifications of the chromatic and dichromatic polynomials for graphs. Fundamenta Mathematicae, Tome 189 (2006) pp. 231-243. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm190-0-9/