Need help to come up with definitive proofs with regard to Planar Graphs
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 answer as $2(v+2)$.


The final answer should be $2(v-2)$.


  Since I derived the value $r=2(v-2)$ which would always remaining even while incrementing $v$.


You are right that $r$, the number of regions will always remain even since $r=2(v-2)$. However, question 2 is asking you to prove "the number of tricolored regions is always even no matter how the vertices are colored.".


  I don't really understand the point of the Hint or maybe what exactly we have to prove here.


The hint is meant to give you a clue how to reduce the number of vertices.  The proof (or my proof) is quite easy and direct once you know it. I will let you figure out your proof.