Abstract. The Stanley chromatic symmetric function X G of a graph G is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology of graded S n -modules, whose graded Frobenius series Frob G (q, t) reduces to the chromatic symmetric function at q = t = 1. This homology can be thought of as a categorification of the chromatic symmetric function, and provides a homological analogue of several familiar properties of X G . In particular, the decomposition formula for X G discovered recently by Orellana and Scott, and Guay-Paquet is lifted to a long exact sequence in homology.
If R is a commutative ring, M a compact R-oriented manifold and G a finite graph without loops or multiple edges, we consider the graph configuration space M G and a Bendersky-Gitler type spectral sequence converging to the homology H * (M G , R). We show that its E 1 term is given by the graph cohomology complex C A (G) of the graded commutative algebra A = H * (M, R) and its higher differentials are obtained from the Massey products of A, as conjectured by Bendersky and Gitler for the case of a complete graph G. Similar results apply to the spectral sequence constructed from an arbitrary finite graph G and a graded commutative DG algebra A.
The Hochschild homology of the algebra of truncated polynomials A m = Z[x]/(x m ) is closely related to the Khovanov-type homology as shown by the second author. In the present paper we utilize this in the study of the first graph cohomology group of an arbitrary graph G with v vertices. The complete description of this group is given for m = 2, 3. For the algebra A 2 we relate the chromatic graph cohomology with the Khovanov homology of adequate links. We describe the chromatic cohomology over the algebra A 3 using the homology of a cell complex built on the graph G. In particular we prove that tor H 1,2v−3can be isomorphic to any finite abelian group. Moreover, we give a characterization of graphs which have torsion in cohomology H 1,2v−3 A 3 (G) and construct graphs which have the same (di)chromatic polynomial but different H 1,2v−3 A 3 (G).
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