# Producing colouring of maximal planar graphs G from colouring of dual of G

A (simple) graph is said to be maximal planar if it is planar and adding one more edge loses planarity. In other words, the graph is planar and every face is a triangle. Tait gave a one-one correspondence between 4-colourings of maximal planar graphs and 3-edge colourings of their dual graphs. That is, given a 4-vertex colouring of a maximal planar graph $$G$$, one can produce a 3-edge colouring of the dual graph $$G^*$$ of $$G$$, and vice versa. The schemes to produce one from the other are elegant.

Similarly, there are schemes to produce (vertex) colouring of $$G^*$$ from colouring of $$G$$.

Is there a known scheme to produce a colouring of a maximal planar graph from a colouring of its dual graph?
(In other words, given a face colouring of a maximal planar graph, can we produce a vertex colouring?)

Clarification: I am not asking for a 4-colouring of the maximal planar graph. From a 4-colouring (or 5-coluoring) of its dual graph, is it possible to produce a colouring of the maximal planar graph (may be with 5 or even 6 colours; but it should be a colouring scheme that actually use the colouring of the dual graph).

The simple but useless answer is that I don't know of such a scheme. However, more to the point: proving that such a scheme worked would be tantamount to proving the 4-color theorem. It is very easy to prove that duals of maximal planar graphs have 4-colorings (in fact 3-colorings except for $$K_4$$ by Brooks' theorem), so if you could prove that a 3- or 4-coloring of the dual graph could always be converted into a 4-coloring of the primal graph then you would have also proved that a 4-coloring of the primal graph exists.
• The dual coloring may have interesting properties w.r.t. colouring varaints. For example, Tait's edge colouring scheme gives a one-one correspondence between 4-acyclic colourings of maximal planar graphs $G$ and Hamiltonian 3-covers of dual of $G$. (A graph has a Hamilonian 3-cover if it has 3 Hamiltonian cycles such that each edge of the graph occur in exactly two of those cycles) Jun 9 '20 at 2:40