A short proof of w₁ⁿ(Hom(C₂ᵣ₊₁, Kₙ₊₂))=0 for all n and a graph colouring theorem by Babson and Kozlov

Original · EN

We show that the n-th power of the first Stiefel-Whitney class of the Z₂-operation on the graph complex Hom(C₂ᵣ₊₁,Kₙ₊₂) is zero, confirming a conjecture by Babson and Kozlov. This proves the strong form of their graph colouring theorem, which they had only proven for odd n. Our proof is also considerably simpler than their proof of the weak form of the theorem, which is also known as the Lovász conjecture.