On the first group of the chromatic cohomology of graphs

The algebra of truncated polynomials A_m=Z[x]/(x^m) plays an important role in the theory of Khovanov and Khovanov-Rozansky homology of links. We have demonstrated that Hochschild homology is closely related to Khovanov homology via comultiplication free graph cohomology. It is not difficult to compute Hochschild homology of A_m and the only torsion, equal to Z_m, appears in gradings (i,m(i+1)/2) for any positive odd i. We analyze here the grading of graph cohomology which is producing torsion for a polygon. We find completely the cohomology H^{1,v-1}_{A_2}(G) and H^{1,2v-3}_{A_3}(G). The group H^{1,v-1}_{A_2}(G) is closely related to the standard graph cohomology, except that the boundary of an edge is the sum of endpoints instead of the difference. The result about H^{1,v-1}_{A_2}(G) gives as a corollary a fact about Khovanov homology of alternating and + or - adequate link diagrams. The group H^{1,2v-3}_{A_3}(G) can be computed from the homology of a cell complex, X_{\Delta,4}(G), built from the graph G. In particular, we prove that A_3 cohomology can have any torsion. We give a simple and complete characterization of those graphs which have torsion in cohomology H^{1,2v-3}_{A_3}(G) (e.g. loopless graphs which have a 3-cycle). We also construct graphs which have the same (di)chromatic polynomial but different H^{1,2v-3}_{A_3}(G). Finally, we give examples of calculations of width of H^{1,*}_{A_3}(G) and of cohomology H^{1,(m-1)(v-2)+1}_{A_m}(G) for m>3.