Types of Graphs
Before you go through this article, make sure that you have gone through the previous article on various Types of Graphs in Graph Theory.
We have discussed
 A graph is a collection of vertices connected to each other through a set of edges.
 The study of graphs is known as Graph Theory.
In this article, we will discuss about Planar Graphs.
Planar Graph
A planar graph may be defined as
In graph theory,
Planar graph is a graph that can be drawn in a plane such that none of its edges cross each other. 
Planar Graph Example
The following graph is an example of a planar graph
Here,
 In this graph, no two edges cross each other.
 Therefore, it is a planar graph.
Regions of Plane
The planar representation of the graph splits the plane into connected areas called as Regions of the plane.
Each region has some degree associated with it given as
 Degree of Interior region = Number of edges enclosing that region
 Degree of Exterior region = Number of edges exposed to that region
Example
Consider the following planar graph
Here, this planar graph splits the plane into 4 regions R1, R2, R3 and R4 where
 Degree (R1) = 3
 Degree (R2) = 3
 Degree (R3) = 3
 Degree (R4) = 5
Planar Graph Chromatic Number
 Chromatic Number of any planar graph is always less than or equal to 4.
 Thus, any planar graph always requires maximum 4 colors for coloring its vertices.
Planar Graph Properties
Property01:
In any planar graph, Sum of degrees of all the vertices = 2 x Total number of edges in the graph
Property02:
In any planar graph, Sum of degrees of all the regions = 2 x Total number of edges in the graph
Special Cases
Case01:
In any planar graph, if degree of each region is K, then
Case02:
In any planar graph, if degree of each region is at least K (>=K), then
Case03:
In any planar graph, if degree of each region is at most K (<=K), then

Property03:
If G is a connected planar simple graph with ‘e’ edges, ‘v’ vertices and ‘r’ number of regions in the planar representation of G, then
r = e – v + 2 
This is known as Euler’s Formula.
It remains same in all the planar representations of the graph.
Property04:
If G is a planar graph with k components, then
r = e – v + (k + 1) 
Also Read Bipartite Graph
PRACTICE PROBLEMS BASED ON PLANAR GRAPH IN GRAPH THEORY
Problem01:
Let G be a connected planar simple graph with 25 vertices and 60 edges. Find the number of regions in G.
Solution
Given
 Number of vertices (v) = 25
 Number of edges (e) = 60
By Euler’s formula, we know r = e – v + 2.
Substituting the values, we get
Number of regions (r)
= 60 – 25 + 2
= 37
Thus, Total number of regions in G = 37.
Problem02:
Let G be a planar graph with 10 vertices, 3 components and 9 edges. Find the number of regions in G.
Solution
Given
 Number of vertices (v) = 10
 Number of edges (e) = 9
 Number of components (k) = 3
By Euler’s formula, we know r = e – v + (k+1).
Substituting the values, we get
Number of regions (r)
= 9 – 10 + (3+1)
= 1 + 4
= 3
Thus, Total number of regions in G = 3.
Problem03:
Let G be a connected planar simple graph with 20 vertices and degree of each vertex is 3. Find the number of regions in G.
Solution
Given
 Number of vertices (v) = 20
 Degree of each vertex (d) = 3
Calculating Total Number Of Edges (e)
By sum of degrees of vertices theorem, we have
Sum of degrees of all the vertices = 2 x Total number of edges
Number of vertices x Degree of each vertex = 2 x Total number of edges
20 x 3 = 2 x e
∴ e = 30
Thus, Total number of edges in G = 30.
Calculating Total Number Of Regions (r)
By Euler’s formula, we know r = e – v + 2.
Substituting the values, we get
Number of regions (r)
= 30 – 20 + 2
= 12
Thus, Total number of regions in G = 12.
Problem04:
Let G be a connected planar simple graph with 35 regions, degree of each region is 6. Find the number of vertices in G.
Solution
Given
 Number of regions (n) = 35
 Degree of each region (d) = 6
Calculating Total Number Of Edges (e)
By sum of degrees of regions theorem, we have
Sum of degrees of all the regions = 2 x Total number of edges
Number of regions x Degree of each region = 2 x Total number of edges
35 x 6 = 2 x e
∴ e = 105
Thus, Total number of edges in G = 105.
Calculating Total Number Of Vertices (v)
By Euler’s formula, we know r = e – v + 2.
Substituting the values, we get
35 = 105 – v + 2
∴ v = 72
Thus, Total number of vertices in G = 72.
Problem05:
Let G be a connected planar graph with 12 vertices, 30 edges and degree of each region is k. Find the value of k.
Solution
Given
 Number of vertices (v) = 12
 Number of edges (e) = 30
 Degree of each region (d) = k
Calculating Total Number Of Regions (r)
By Euler’s formula, we know r = e – v + 2.
Substituting the values, we get
Number of regions (r)
= 30 – 12 + 2
= 20
Thus, Total number of regions in G = 20.
Calculating Value Of k
By sum of degrees of regions theorem, we have
Sum of degrees of all the regions = 2 x Total number of edges
Number of regions x Degree of each region = 2 x Total number of edges
20 x k = 2 x 30
∴ k = 3
Thus, Degree of each region in G = 3.
Problem06:
What is the maximum number of regions possible in a simple planar graph with 10 edges?
Solution
In a simple planar graph, degree of each region is >= 3.
So, we have 3 x R <= 2 x E.
Substituting the value E = 10, we get
3 x R <= 2 x 10
R <= 6.67
R <= 6
Thus, Maximum number of regions in G = 6.
Problem07:
What is the minimum number of edges necessary in a simple planar graph with 15 regions?
Solution
In a simple planar graph, degree of each region is >= 3.
So, we have 3 x R <= 2 x E.
Substituting the value R = 15, we get
3 x 15 <= 2 x E
E >= 22.5
E >= 23
Thus, Minimum number of edges required in G = 23.
To gain better understanding about Planar Graphs in Graph Theory,
Next Article Euler Graph
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