  The main result of this paper is a proof of the following conjecture of
Babson & Kozlov:
  Theorem. Let G be a graph of maximal valency d, then the complex Hom(G,K_n)
is at least (n-d-2)-connected.
  Here Hom(-,-) denotes the polyhedral complex introduced by Lov\'asz to study
the topological lower bounds for chromatic numbers of graphs.
  We will also prove, as a corollary to the main theorem, that the complex
Hom(C_{2r+1},K_n) is (n-4)-connected, for $n\geq 3$.
