Graph colourings, spaces of edges and spaces of circuits

TL;DR

This paper provides a new proof of the strong form of the graph coloring theorem using cohomological indices of Hom complexes, and describes the homotopy type of these complexes for certain graphs, showing bounds on chromatic numbers.

Contribution

It introduces a new, simpler proof of Babson and Kozlov's graph coloring theorem and generalizes the results to other graphs, expanding understanding of Hom complexes and chromatic bounds.

Findings

Cohomological index of Hom(K_2,G) bounds chromatic number from below.

Hom(C_{2r+1}, G) complexes have related cohomological indices, providing bounds.

The homotopy type of the direct limit of Hom(C_{2r+1}, G) relates closely to Hom(K_2, G).

Abstract

By Lovasz' proof of the Kneser conjecture, the chromatic number of a graph G is bounded from below by the index of the Z_2-space Hom(K_2,G) plus two. We show that the cohomological index of Hom(K_2,G) is also greater than the cohomological index of the Z_2-space Hom(C_{2r+1}, G) for r>0. This gives a new and simple proof of the strong form of the graph colouring theorem by Babson and Kozlov, which had been conjectured by Lovasz, and at the same time shows that it never gives a stronger bound than can be obtained by Hom(K_2, G). The proof extends ideas introduced by Zivaljevic in a previous elegant proof of a special case. We then generalise the arguments and obtain conditions under which corresponding results hold for other graphs in place of C_{2r+1}. This enables us to find an infinite family of test graphs of chromatic number 4 among the Kneser graphs. Our main new result is a description of the Z_2-homotopy type of the direct limit of the system of all the spaces Hom(C_{2r+1}, G) in terms of the Z_2-homotopy type of Hom(K_2, G). A corollary is that the coindex of Hom(K_2, G) does not exceed the coindex of Hom(C_{2r+1}, G) by more then one if r is chosen sufficiently large. Thus the graph colouring bound in the theorem by Babson & Kozlov is also never weaker than that from Lovasz' proof of the Kneser conjecture.