For each graph we construct graded cohomology groups whose graded Euler characteristic is the chromatic polynomial of the graph. We show the cohomology groups satisfy a long exact sequence which corresponds to the well-known deletion-contraction rule. This work is motivated by Khovanov’s work on categorification of the Jones polynomial of knots.
"A categorification for the chromatic polynomial." Algebr. Geom. Topol. 5 (4) 1365 - 1388, 2005. https://doi.org/10.2140/agt.2005.5.1365