Kronheimer and Mrowka recently defined an instanton invariant of embedded trivalent graphs (webs) in $\mathbb{R}^3$. This can be regarded as (roughly) counting the number of representations of the fundamental group of the complement of the graph to $SO(3)$ such that the meridian of each edge is sent to an involution. They conjecture that for planar webs their invariant gives the number of Tait colorings (it's easy to see that a Tait coloring gives a representation of this sort to the Klein four group). They show this is true for Eulerian planar trivalent graphs among others. They show that their invariant is always non-zero, and hence this conjecture implies the four color theorem. They also give various refinements of this conjecture in this paper and a sequel.

Kronheimer and Mrowka recently defined an instanton invariant of embedded trivalent graphs (webs) in $\mathbb{R}^3$. This can be regarded as (roughly) counting the number of representations of the fundamental group of the complement of the graph to $SO(3)$ such that the meridian of each edge is sent to an involution. They conjecture that for planar webs their invariant gives the number of Tait colorings (it's easy to see that a Tait coloring gives a representation of this sort to the Klein four group). They show this is true for Eulerian planar trivalent graphs among others. They show that their invariant is always non-zero, and hence this conjecture implies the four color theorem. They also give various refinements of this conjecture in this paper and a sequel. See also this blog-post.