A short proof of w_1^n(Hom(C_{2r+1}, K_{n+2}))=0 for all n and a graph colouring theorem by Babson and Kozlov

TL;DR

This paper proves that the n-th power of the first Stiefel-Whitney class of a certain graph complex is zero, confirming a conjecture and strengthening a graph coloring theorem with a simpler proof.

Contribution

It provides a short, simplified proof of a conjecture by Babson and Kozlov, establishing the vanishing of a cohomology class for all n and strengthening their graph coloring theorem.

Findings

Confirmed the conjecture that the n-th power of the first Stiefel-Whitney class is zero for all n.

Proved the strong form of Babson and Kozlov's graph coloring theorem.

Simplified the proof of the weak form of the theorem, also known as the Lovász conjecture.

Abstract

We show that the n-th power of the first Stiefel-Whitney class of the Z_2-operation on the graph complex Hom(C_{2r+1},K_{n+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 Lov\'asz conjecture.