# Higher connectivity of graph coloring complexes

@article{Cukic2004HigherCO, title={Higher connectivity of graph coloring complexes}, author={Sonja Lj. Cukic and Dmitry N. Kozlov}, journal={International Mathematics Research Notices}, year={2004}, volume={2005}, pages={1543-1562} }

The main result of this paper is a proof of the following conjecture of Babson & Kozlov:
Theorem. Let G be a graph of maximal valency d, then the complex Hom(G,K_n) is at least (n-d-2)-connected.
Here Hom(-,-) denotes the polyhedral complex introduced by Lov\'asz to study the topological lower bounds for chromatic numbers of graphs.
We will also prove, as a corollary to the main theorem, that the complex Hom(C_{2r+1},K_n) is (n-4)-connected, for $n\geq 3$.

## 27 Citations

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In this article, we consider the bipartite graphs $K_2 \times K_n$. We prove that the connectedness of the complex $\displaystyle \text{Hom}(K_2\times K_{n}, K_m) $ is $m-n-1$ if $m \geq n$ and $m-3$…

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For any two graphs G and H Lovasz has defined a cell complex Hom (G,H) having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph…

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