Knight move in chromatic cohomology

Michael S Chmutov, Sergei Chmutov, Yongwu Rong

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Abstract

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 Γ with n vertices the only non-trivial cohomology groups Hi, n - i (Γ), Hi, n - i - 1 (Γ) come in isomorphic pairs: Hi, n - i (Γ) ≅ Hi + 1, n - i - 2 (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.

Original languageEnglish (US)
Pages (from-to)311-321
Number of pages11
JournalEuropean Journal of Combinatorics
Volume29
Issue number1
DOIs
StatePublished - Jan 1 2008

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