Proof of the Lovász conjecture

Abstract

To any two graphs G and H one can associate a cell complex Horn (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+i, G) is k-connected, then X(G) ≥ k + 4, where r, k ∈ ℤ r ≥ 1, k ≥ -1, and C2r+1 denotes the cycle with 2r+l 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 Horn (C2r+1) Kn), where the latter are viewed as ℤ2-spaces with the involution induced by the reflection of C2r+1.