Knight move for chromatic graph cohomology

Details Knight move for chromatic graph cohomology Knight move for chromatic graph cohomology

2005-11-24

arxiv.org/abs/math/0511598v2

Mathematics

Combinatorics

14 pages

Scientific paper

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 G with n vertices the only non-trivial cohomology groups $H^{i,n-i}(G)$, $H^{i,n-i-1}(G)$ come in isomorphic pairs: $H^{i,n-i}(G)\cong H^{i+1,n-i-2}(G)$ for i >= 0 if G is non-bipartite, and for i > 0 if G 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 Poincare polynomial in terms of the chromatic polynomial and a deletion-contraction formula for the Poincare polynomial.

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