Coloring Planar Graphs via Colored Paths in the Associahedra

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.