Knight move in chromatic cohomology
| Autor: | Michael Chmutov, Sergei Chmutov, Yongwu Rong |
|---|---|
| Rok vydání: | 2008 |
| Předmět: |
Discrete mathematics
Mathematics::Combinatorics 010102 general mathematics Chromatic polynomial 01 natural sciences Cohomology Graph Theoretical Computer Science Combinatorics symbols.namesake Corollary Computational Theory and Mathematics Computer Science::Discrete Mathematics 0103 physical sciences Poincaré conjecture symbols Bipartite graph Discrete Mathematics and Combinatorics Geometry and Topology 010307 mathematical physics Chromatic scale 0101 mathematics Connectivity Mathematics |
| Zdroj: | European Journal of Combinatorics. 29:311-321 |
| ISSN: | 0195-6698 |
| DOI: | 10.1016/j.ejc.2006.07.009 |
| Popis: | 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 @C with n vertices the only non-trivial cohomology groups H^i^,^n^-^i(@C),H^i^,^n^-^i^-^1(@C) come in isomorphic pairs: H^i^,^n^-^i(@C)@?H^i^+^1^,^n^-^i^-^2(@C) for i>=0 if @C is non-bipartite, and for i>0 if @C 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|
Full Text from ScienceDirect