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 Lov´sz conjecture which states that a 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.
