Suppose that a connected planar graph has six vertices, each of degree four. Into how many regions is the plane divided by a planar representation of this graph?
Method 1 (brute force) :-
Method 2 :-
We know that,
sum of degree of vertices = 2 * number of edges (handshaking theorem)
$\implies 4+4+4+4+4+4 = 2 * e$
$\implies e = 12$
Also according to Euler's formula,
$\implies r= 12 -6 +2 $
$\implies r= 8$
$\therefore$ Option $2.$ is correct.