Conjectures implying Four Color Theorem Four Color Theorem (4CT) states that every planar graph is four colorable. There are two proofs given by [Appel,Haken 1976] and [Robertson,Sanders,Seymour,Thomas 1997]. Both these proofs are computer-assisted and quite intimidating.

There are several conjectures in graph theory that imply 4CT. Resolution of these conjectures probably requires a better understanding of the proofs of 4CT. Here is one such conjecture :

Conjecture : Let $G$ be a planar graph, let $C$ be a set of colors and $f : C \rightarrow C$ a fixed-point free involution. Let $L = (L_v : v \in V(G))$ be such that 


$|L_v| \geq 4$ for all $v \in V$ and
if $\alpha \in L_v$ then $f(\alpha) \in L_v$ for all $v \in V$, for all $\alpha \in C$.


Then there exists an $L$-coloring of the graph $G$.

If you know such conjectures implying 4CT, please list them one in each answer. I could not find a comprehensive list of such conjectures.
4CT is equivalent to:


Coloring a hypergraph induced by a finite family of discs (not necessarily interior disjoint but also discs that might have arbitrary overlaps) with at most four colors $[1]$.
The proposition that every planar triangulation with more than three vertices is the union of two connected bipartite graphs, each with no isthmus $[2]$.
A combinatorial problem about the three-dimensional vector cross product algebra $[3]$.
The proposition that the grammar G is totally ambiguous $[4]$.
For any positive integer $n$, there exists at least a signable path joining two permutations of $S_n$.
An algebraic equivalent of the four-color theorem is presented. The equivalent is the assertion of non-membership of a family of polynomials in a family of polynomial ideals over a particular finite field $[6]$.


Smorodinsky S. On the chromatic number of geometric hypergraphs.
Mabry R. Bipartite Graphs and the Four-Color Theorem.
Kauffman L.H. Map coloring and the vector cross product.
Cooper B. J. et al. Toward a language theoretic proof of the four color theorem.
Eliahou S. et al. Signed permutations and the four color theorem.
Howard L. An Algebraic Reformulation of the Four Color Theorem.