  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.
