Knight move in chromatic cohomology

In this paper we prove the knight move theorem for the chromatic graph cohomologies with rational coefficients introduced by L. Helme-Guizon and Y. Rong. Namely, for a connected graph ΓΓ with nn vertices the only non-trivial cohomology groups Hi,n−i(Γ),Hi,n−i−1(Γ)Hi,n−i(Γ),Hi,n−i−1(Γ) come in isomorphic pairs: Hi,n−i(Γ)≅Hi+1,n−i−2(Γ)Hi,n−i(Γ)≅Hi+1,n−i−2(Γ) for i⩾0i⩾0 if ΓΓ is non-bipartite, and for i>0i>0 if ΓΓ 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.