ar X iv : m at h / 04 02 39 5 v 3 [ m at h . C O ] 1 8 Ju l 2 00 5 PROOF OF THE LOVÁSZ CONJECTURE

To any two graphs G and H one can associate a cell complex Hom (G,H) by taking all graph multihomorphisms from G to H as cells. In this paper we prove the Lovász Conjecture which states that if Hom (C2r+1, G) is k-connected, then χ(G) ≥ k + 4, where r, k ∈ Z, r ≥ 1, k ≥ −1, and C2r+1 denotes the cycle with 2r + 1 vertices. The proof requires analysis of the complexes Hom (C2r+1,Kn). For even n, the obstructions to graph colorings are provided by the presence of torsion in H∗(Hom (C2r+1,Kn);Z). For odd n, the obstructions are expressed as vanishing of certain powers of Stiefel-Whitney characteristic classes of Hom (C2r+1,Kn), where the latter are viewed as Z2-spaces with the involution induced by the reflection of C2r+1.