Higher connectivity of graph coloring complexes
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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 Lovász 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 $n\geq 3$.
16 pages, 6 figures
16 pages, 6 figures