  The local chromatic number of a graph was introduced by Erdos et al. in 1986.
It is in between the chromatic and fractional chromatic numbers. This motivates
the study of the local chromatic number of graphs for which these quantities
are far apart. Such graphs include Kneser graphs, their vertex color-critical
subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their
generalizations; and Borsuk graphs. We give more or less tight bounds for the
local chromatic number of many of these graphs.
  We use an old topological result of Ky Fan which generalizes the Borsuk-Ulam
theorem. It implies the existence of a multicolored copy of the balanced
complete bipartite graph on t points in every proper coloring of many graphs
whose chromatic number t is determined via a topological argument. (This was in
particular noted for Kneser graphs by Ky Fan.) This yields a lower bound of
t/2+1 for the local chromatic number of these graphs. We show this bound to be
tight or almost tight in many cases.
  As another consequence of the above we prove that the graphs considered here
have equal circular and ordinary chromatic numbers if the latter is even. This
partially proves a conjecture of Johnson, Holroyd, and Stahl and was
independently attained by F. Meunier. We also show that odd chromatic Schrijver
graphs behave differently, their circular chromatic number can be arbitrarily
close to the other extreme.
