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}
}
  • S. Cukic, D. Kozlov
  • Published 14 October 2004
  • Mathematics
  • International Mathematics Research Notices
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 $n\geq 3$. 

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References

SHOWING 1-10 OF 19 REFERENCES

Topological obstructions to graph colorings

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

Kneser's Conjecture, Chromatic Number, and Homotopy

Kneser's Conjecture

A combinatorial conjecture formulated by Kneser (1955). It states that whenever the n-subsets of a (2n+k)-set are divided into k+1 classes, then two disjoint subsets end up in the same class. Lovasz

Algebraic Topology

The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.

Complexes of graph homomorphisms, preprint

  • 2003

KOZLOV This fact is a direct corollary of the previous theorem since the maximal valency of C 2r+1 is equal to 2. Finally, we are able to put the pieces together and prove the Conjecture 1

  • KOZLOV This fact is a direct corollary of the previous theorem since the maximal valency of C 2r+1 is equal to 2. Finally, we are able to put the pieces together and prove the Conjecture 1

We know that the result of Conjecture 1.1 is sharp for several classes of graphs, for example for odd cycles and complete graphs

  • We know that the result of Conjecture 1.1 is sharp for several classes of graphs, for example for odd cycles and complete graphs

Proof of the Lovász Conjecture, preprint, 40 pages