Conjectures implying Four Color Theorem
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.