  Hassler Whitney's theorem of 1931 reduces the task of finding proper, vertex
4-colorings of triangulations of the 2-sphere to finding such colorings for the
class \(\mathfrak H\) of triangulations of the 2-sphere that have a Hamiltonian
circuit. This has been used by Whitney and others from 1936 to the present to
find equivalent reformulations of the 4 Color Theorem (4CT). Recently there has
been activity to try to use some of these reformuations to find a shorter proof
of the 4CT. Every triangulation in \(\mathfrak H\) has a dual graph that is a
union of two binary trees with the same number of leaves. Elements of a group
known as Thompson's group \(F\) are equivalence classes of pairs of binary
trees with the same number of leaves. This paper explores this resemblance and
finds that some recent reformulations of the 4CT are essentially attempting to
color elements of \(\mathfrak H\) using expressions of elements of \(F\) as
words in a certain generating set for \(F\). From this, we derive information
about not just the colorability of certain elements of \(\mathfrak H\), but
also about all possible ways to color these elements. Because of this we raise
(and answer some) questions about enumeration. We also bring in an extension
\(E\) of the group \(F\) and ask whether certain elements ``parametrize'' the
set of all colorings of the elements of \(\mathfrak H\) that use all four
colors.
