Knight move for chromatic graph cohomology

In this paper we prove the knight move theorem for the chromatic graph cohomologies with rational coefficients introduced by L. HelmeGuizon and Y. Rong. Namely, for a connected graph Γ with n vertices the only non-trivial cohomology groups H(Γ), H(Γ) come in isomorphic pairs: H(Γ) = H(Γ) for i > 0 if Γ is non-bipartite, and for i > 0 if Γ is bipartite. As a corollary, the ranks of the cohomology groups are determined by the chromatic polynomial. At the end, we give an explicit formula for the Poincaré polynomial in terms of the chromatic polynomial and a deletion-contraction formula for the Poincaré polynomial. Introduction Recently, motivated by the Khovanov cohomology in knot theory [Kho], Laure Helme-Guizon and Yongwu Rong [HGR1] developed a bigraded cohomology theory for graphs. Its main property is that the Euler characteristic with respect to one grading and the Poincaré polynomial with respect to the other grading give the chromatic polynomial of the graph. There is a long exact sequence relating the cohomology of a graph with the cohomologies of graphs obtained from it by contraction and deletion of an edge. It generalizes the classical contraction-deletion rule for the chromatic polynomial. This sequence is an important tool in proving various properties of the cohomology (see [HGR1]). In particular, for a connected graph Γ with n vertices, it allows to prove that the cohomologies are concentrated on two diagonals H(Γ) and H(Γ). We prove that H(Γ) is isomorphic to H(Γ) (“knight move”) for all i with the exception of i = 0 for a bipartite graph Γ. An analogous theorem for Khovanov cohomology in knot theory was proved in [Lee]. Our proof follows the same idea of considering an additional differential Φ on the chromatic cochain complex of the graph Γ. The associated spectral sequence collapses at the term E2, and so this term is given by the cohomologies of Γ with respect to the differential Φ+d. They turn out to be trivial with a small exception. On the other hand, the term E1 of the spectral sequence is represented by the cohomology of our graph with the differential Φ. So Φ gives the desired isomorphism. Its existence implies that the chromatic