Knight move for chromatic graph cohomology

2005/11/24 by Michael Chmutov, Chmutov, Michael, Sergei Chmutov +3
Mathematics · #05C15 #57M27 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.CO #math.GT #math.QA #msc:05C15 #msc:57M27

paper · pdf · doi:10.48550/arxiv.math/0511598

14 pages

openalex publication_date 2005/11/24 · arxiv created 2006/09/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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)≅ 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.

Related

• Reduced chromatic graph cohomology
• A splitting property of the chromatic homology of the complete graph
• Graph Cohomologies from Arbitrary Algebras
• On the homology theory for the chromatic polynomials
• A categorification for the signed chromatic polynomial