# Who proved that a triangulation is 3-colourable implies its dual is bipartite

Let $$G$$ be a maximal planar graph (also called a triangulation); i.e, $$G$$ is a planar graph every face of which is a triangle. It is well known that the following three statements are equivalent:
(i) $$G$$ is 3-colourable
(ii) dual graph $$G^*$$ of $$G$$ is bipartite
and (iii) $$G$$ is Eulerian (i.e., every vertex has even degree).

I am interested in direct constructive proof of (i) $$\iff$$ (iii).
I would like to know who came up with this proof; esp. the proof of (i)$$\implies$$ (ii) given below

(i) $$\implies$$ (ii):
Suppose $$G^*$$ has a 3-face colouring $$f$$ with colours 1,2,3. Then, one can obtain a 2-colouring $$f^*$$ of $$G^*$$ by assigning colour +1 at a vertex $$v$$ if colours 1,2,3 appear clockwise on faces around $$v$$, and colour -1 at $$v$$ if 1,2,3 appear counterclockwise around $$v$$. So, $$G^*$$ is bipartite.

David Gale wrote to Shen giving a proof for (i) $$\iff$$ (iii) using homology theory (for triangulations of the sphere). In that letter, he proves that the mapping $$f\to f^*$$ given in the proof above is in fact a bijection (answer to this question explains why). Shen explains these in Mathematical Entertainments 20(3). I think that Gale's proof is different for the direction (i)$$\implies (ii)$$ (I am not sure, homology theory is Greek to me). I guess this proof should be known already because it is easier than the other direction. So, like I said, I would like to know who came up with this proof of (i)$$\implies (ii)$$ first.

Tsai and West's paper A new proof of 3-colorability of Eulerian triangulations refer to Lovász (Combinatorial Problems and Exercises, Problem number is 9.52 I suppose) and to Shen (reference given above).

Thank you.

I saw a paper that states this as a folklore result and cites Ore. Interestingly, the book gives a different proof for (ii)$$\implies$$(i). It seems that at that time, it wasn't known that the mapping $$f\longmapsto f^*$$ is a bijection.