## Planer Graph in Discrete MathematicsIf we want to learn the planer graph, we have to know about the graph. The graph can be described as a collection of vertices, which are connected to each other with the help of a set of edges. We can also call the study of a graph as the Graph theory. In this section, we are able to learn about the definition of a planer graph, its properties, and examples of a Planer graph. ## Planer GraphA planer graph can be described as a graph that will be drawn in a plane in such a way that none of its edges cross each other. The simple example of a planer graph is described as follows: The above graph does not contain two edges that cross each other. Hence this graph is a planer graph. ## Regions of plane:In the process of planer representation of a graph, the plane is split into the connected areas, which are known as the regions of the plane. There is some amount of degree which is associated with each region. These degrees are described as follows: - Degree of Interior region is equal to the Number of edges enclosing that region.
- Same as, Degree of Exterior region is equal to the Number of edges exposed to that region.
In this example, we will consider a planer graph, which is described as follows: In the above planer graph, we can see that the graph splits the plane into 4 regions, i.e., R1, R2, R3, and R4, and the degree of these regions are described as follows: Degree (R1) = 3 Degree (R2) = 3 Degree (R3) = 3 Degree (R4) = 5 ## Chromatic Number of Planer graph- In any planer graph, the chromatic number must always be less than or equal to 4.
- Thus, to color the vertices of any planer graph, we always need a maximum of 4 colors.
## Properties of Planer graphThere are various properties of a planer graph, and some of them are described as follows:
There are some special cases, which are described as follows:
K * |R| = 2 * |E|
K * |R| <= 2 * |E|
K * |R| >= 2 * |E|
r = e - v + 2 We can call the above formula as Euler's formula. In all the forms of planer representation of a graph, this formula remains the same.
r = e-v+ (k+1) ## Examples of Planer graph:There are various examples of planer graphs, and some of them are described as follows:
Number of vertices (v) = 25 Number of edges (e) = 60 With the help of Euler's formula, we have r = e - v + 2. Now we will put the values of v and e in this formula and get the following details: r = 60 - 25 + 2 r = 37 Thus, the total number of region in graph G = 37.
Number of vertices (v) = 10 Number of components (k) = 3 Number of edges (e) = 9 With the help of Euler's formula, we have r = e - v + (k+1). Now we will put the values of v, e and k in this formula, and get the following details: r = 9-10+ (3+1) r = -1+4 r = 3 Thus, the total number of region in graph G = 3.
Degree of each vertex (d) = 3 Number of vertices (v) = 20
When we do the addition of a degree of vertices theorem, we have the following relation: The Sum of degrees of all the vertices is equal to the Total number of edges * 2 Degree of each vertex * Number of vertices is equal to the Total number of edges * 2 So we will use the formula of degree of each vertex and get the following: 20*3 = 2*e e = 30 Thus, in graph G, the total number of edges = 30.
With the help of Euler's formula, we have r = e - v + 2. When we put the values of v and e in this formula, we get the following details: Number of regions (r) = 30-20+2 r = 12 Thus, in graph G, the total number of regions = 12.
Degree of each regions (d) = 6 Number of regions (n) = 35
When we do the addition of a degree of regions theorem, we have the following relation: The Sum of degrees of all the regions is equal to the Total number of edges * 2 Degree of each region * Number of regions is equal to the Total number of edges * 2 35*6 = 2*e e = 105 Thus, in graph G, the total number of edges = 105.
With the help of Euler's formula, we have r = e - v + 2. When we will put the values of r and e in this formula, we get the following details: 35 = 105 - v + 2 v = 72 Thus, in graph G, the total number of vertices = 72.
Number of edges (e) = 30 Degree of each regions (d) = k Number of vertices (v) = 12
With the help of Euler's formula, we have r = e - v + 2. When we put the values of v, d, and e in this formula, we get the following details: Number of regions (r) = 30-12+2 r = 20 Thus, in graph G, the total number of regions = 20.
When we do the addition of a degree of regions theorem, we have the following relation: The Sum of degrees of all the regions is equal to the Total number of edges * 2 Degree of each region * Number of regions is equal to the Total number of edges * 2 20*k = 2*30 k = 30 Thus, in graph G, the Degree of each region = 3.
Degree of each region is >= 3. So we have the following details: 3 * |R| <= 2 * |E| Now we will put the value of |E| = 10, and get the following details: 3 * |R| <= 2* 10 |R| <= 6.67 |R| <= 6 Thus, in graph G, the maximum number of regions = 6.
Degree of each region is >= 3 So we have the following details: 3 * |R| <= 2 * |E| Now we will put the value of |R| = 15, and get the following details: 3 * 15 <= 2* |E| |E| >= 22.5 |E| >= 23 Thus, in graph G, the maximum number of edges = 23. Next TopicBipartite Graph in Discrete mathematics |