Khovanov type homologies for graphs (Q2869283)

In [Algebr. Geom. Topol. 5, 1365--1388 (2005; Zbl 1081.05034)], the authors gave a categorification of the chromatic polynomial. The underlying algebra of this cohomology theory is \(\mathbb{Z}/(x^2)\). In the present paper, the authors extend their work by showing that any commutative, graded algebra with finite dimension in each degree can be used as the underlying algebra in their construction. The authors go on to show that various fundamental properties of the chromatic polynomial have cohomological analogues. For example, it is shown that the deletion-contraction relation for the chromatic polynomial corresponds to a long exact sequence for the graph cohomology. The paper also includes computations of the cohomology groups of several graphs.