  In 1999, at one of his last public lectures, Tutte discussed a question he
had considered since the times of the Four Color Conjecture. He asked whether
the 4-coloring complex of a planar triangulation could have two components in
which all colorings had the same parity. In this note we answer Tutte's
question to the contrary of his speculations by showing that there are
triangulations of the plane whose coloring complexes have arbitrarily many even
and odd components. We end up with a closely related conjecture, which is based
on an extensive computation, and which claims that for every planar
triangulation whose 4-coloring complex is disconnected has a component of even
parity and one of odd parity.
