2 00 4 Proof of the Lovász Conjecture

In this paper we prove the Lovász Conjecture: If Hom (C2r+1,H) is k-connected, then χ(H) ≥ k + 4, where H is a finite undirected graph, C2r+1 is a cycle with 2r+1 vertices, r, k ∈ Z, r ≥ 1, k ≥ −1, and Hom (G,H) is the cell complex with the vertex set being the set of all graph homomorphisms from G to H , and cells all allowed list H-colorings of G. Our method is to compute, by means of spectral sequences, the obstructions to graph colorings, which lie either directly in the cohomology groups of Hom (C2r+1,Kn), or in the vanishing of the certain powers of Stiefel-Whitney classes of Hom (C2r+1,Kn), viewed as Z2-spaces, resulting in proving even sharper statements.