PROOF OF THE LOVÁSZ CONJECTURE 967 If φ ∈

To any two graphs G and H one can associate a cell complex Hom (G,H) by taking all graph multihomomorphisms 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.