I was working through a few problem sets and came across this question
Suppose there is a connected planar simple graph $G$ with $v$ vertices such that all its regions are triangles (a cycle consisting of three edges).
Question 1: Into how many regions does a representation of the Planar Graph $G$ Split the plane?
Incase of this question, since all the degrees of regions are said to be 3, we can say $2e=3r$. I substituted this value into the Euler equation, $v-e+r=2$ and got my final answeras $2(v+2)$.
Question 2: Suppose the vertices of the planar graph G are colored in three colors. A region is called to be tricolored (or bicolored) if its vertices are colored in exactly three (or two) different colors. Similarly, a monocolored region is the one with all its vertices colored in exactly one color. Prove that the number of tricolored regions is always even no matter how the vertices are colored.
(Hint: If you place a new vertex inside a region (triangle) of G and connect it with all vertices of that region, then all regions are still triangles and the parity of the total number of regions stays the same.)
This question seemed very straight-forward to me on account of my previous answer. Since I derived the value $r = 2(v+2)$ which would always remaining even while incrementing $v$. I don't really understand the point of the Hint or maybe what exactly we have to prove here.
Any help with regard to the same would be appreciated. Thanks!