I'm answering this hoping you know the basics of graph theory.
Going with these notations, $r =$degree of region, $v= $ degree of vertex and $e = $ degree of an edge.
Now,
- Exactly 5 edges in boundary of each region : degree of a region is $5$. So $5r = 2e$
- Cubic planar graph : degree of every vertex is $3$. So $3v = 2e$.
- Cubic planar graph : Using Euler's Formula, we have $r = e - v + 2$
Now we just need to solve for $e$, using the equations above.
$\frac{2}{5} \times e = e - \frac{2}{3} \times e + 2 \Rightarrow 6e = 15e - 10e + 30 \Rightarrow e = 30 $
And here's is your answer = 30 edges are present in your graph.
:) sudoankit