Finding the number of regions in the graph

Problem Statement

Let 'G' be a connected planar graph with 20 vertices and the degree of each vertex is 3. Find the number of regions in the graph.

Solution

By the sum of degrees theorem,

 20 ∑ i=1  deg(V_i) = 2|E|

20(3) = 2|E|

|E| = 30

By Euler’s formula,

|V| + |R| = |E| + 2

20+ |R| = 30 + 2

|R| = 12

Hence, the number of regions is 12.