The Last Temptation of William T. Tutte

TL;DR

This paper investigates the structure of 4-coloring complexes in planar triangulations, disproving Tutte's speculation by showing they can have arbitrarily many components of each parity, and proposes a related conjecture based on computational evidence.

Contribution

It provides the first counterexamples to Tutte's question and introduces a new conjecture about the parity components in 4-coloring complexes of planar triangulations.

Findings

Existence of triangulations with arbitrarily many even and odd components.

Disproof of Tutte's speculation regarding 4-coloring complex connectivity.

Proposal of a new conjecture supported by extensive computation.

Abstract

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.