  There are several famous unsolved conjectures about the chromatic number that
were relaxed and already proven to hold for the fractional chromatic number. We
discuss similar relaxations for the topological lower bound(s) of the chromatic
number. In particular, we prove that such a relaxed version is true for the
Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of
Hadwiger from this point of view. For the latter, a similar statement was
already proven in an earlier paper of the first author with G. Tardos, our main
concern here is that the so-called odd Hadwiger conjecture looks much more
difficult in this respect. We prove that the statement of the odd Hadwiger
conjecture holds for large enough Kneser graphs and Schrijver graphs of any
fixed chromatic number.
