The last temptation of William T. Tutte

In 1999, at one of his last public lectures, Tutte discussed a question he had considered since the times Four Color Conjecture. He asked whether 4-coloring complex planar triangulation could have two components in which all colorings same parity. this note we answer Tutte’s contrary to speculations by showing that there are triangulations plane whose coloring complexes arbitrarily many even and odd components.