  A graph with chromatic number $k$ is called $k$-chromatic. Using
computational methods, we show that the smallest triangle-free 6-chromatic
graphs have at least 32 and at most 40 vertices.
  We also determine the complete set of all triangle-free 5-chromatic graphs up
to 24 vertices. This implies that Reed's conjecture holds for triangle-free
graphs up to at least this order. We also establish that the smallest regular
triangle-free 5-chromatic graphs have 24 vertices. Finally, we show that the
smallest 5-chromatic graphs of girth at least 5 have at least 29 vertices and
that the smallest 4-chromatic graphs of girth at least 6 have at least 25
vertices.
