Finding the number of regions in the graph

In a connected planar graph, the plane is divided into distinct areas called regions (or faces), including the outer unbounded region. The number of regions can be found using Euler's formula for planar graphs, which relates vertices, edges, and regions.

Key Formulas

Sum of Degrees Theorem − The sum of the degrees of all vertices equals twice the number of edges −

∑ deg(V_i) = 2|E|

Euler's Formula − For any connected planar graph −

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

Where |V| is the number of vertices, |E| is the number of edges, and |R| is the number of regions.

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

Step 1: Find the Number of Edges

Using the sum of degrees theorem −

? deg(V?) = 2|E|     (for i = 1 to 20)

20 × 3 = 2|E|

60 = 2|E|

|E| = 30

Since all 20 vertices have degree 3, the total degree sum is 60. Each edge contributes 2 to the total degree, so there are 30 edges.

Step 2: Find the Number of Regions

Using Euler's formula −

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

20 + |R| = 30 + 2

20 + |R| = 32

|R| = 32 - 20

|R| = 12

Hence, the number of regions is 12.

Conclusion

To find the number of regions in a connected planar graph, first use the sum of degrees theorem to determine the number of edges, then apply Euler's formula |V| + |R| = |E| + 2 to solve for |R|.