# Bipartite Graph in Discrete mathematics

If we want to learn the Euler 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. In this section, we are able to learn about the definition of a bipartite graph, complete bipartite graph, chromatic number of a bipartite graph, its properties, and examples.

### Definition of Bipartite graph

The bipartite graph can be described as a special type of graph which has the following properties:

• This graph always has two sets, X and Y, with the vertices.
• In this graph, the vertices of set X can only have a connection with the set Y.
• We cannot join the vertices within the same set.

The example of a Bipartite graph is described as follows: In the above graph, we have the following things:

• We can decompose the vertices of a given graph into two sets.
• The decomposed sets are set X = {A, C}, and set Y = {B, D}.
• The vertices of set X can only make a connection with set Y and vice versa.
• We cannot join the vertices of same set.
• Hence, the given graph is a bipartite graph.

### Complete Bipartite graph

• We can define the bipartite graph in many ways. In other words, we can say that the complete bipartite graph has many definitions, which are described as follows:
• A graph will be known as the complete bipartite graph if every vertex of set X has a connection with every vertex of set Y.
• A graph will be called complete bipartite if it is bipartite and complete both.
• If there is a bipartite graph that is complete, then that graph will be called a complete bipartite graph.

Example of Complete Bipartite graph

The example of a complete bipartite graph is described as follows: In the above graph, we have the following things:

• The above graph is a bipartite graph and also a complete graph.
• Therefore, we can call the above graph a complete bipartite graph.
• We can also call the above graph as k4, 3.

### Chromatic Number of Bipartite graph

When we want to properly color any bipartite graph, then we have to follow the following properties:

• For this, we have required a minimum of two colors.
• We have to use different colors to color the end vertices of every edge.
• In this case, the bipartite graphs will be 2 colorable.

#### Note: If there is no edge in the bipartite graph, then that graph will be 1 colorable.

Example of chromatic number of bipartite graph: In the above bipartite graph, the chromatic number is 2 because the left side vertices are colored with orange color, and the right side vertices are colored with red color. So there are two colors. Hence, chromatic number = 2.

### Properties of Bipartite graph

There are various properties of a bipartite graph, and some important properties are described as follows:

• The bipartite graphs are 2 colorable.
• There are no odd cycles in the bipartite graphs.
• If there is any sub-graph in the bipartite graph, then that subgraph will also be itself bipartite.
• If there is a case where |X| ≠ |Y|, then the bipartite graph will not contain a perfect match with the bipartition X and Y.
• If there is a bipartite graph that contains a bipartition X and Y, then it will have the following relation:

### Perfect matching for a Bipartite Graph

In a bipartite graph, there is a formula for perfect matching, which is described as follows:

Number of Complete matching for Kn, n = n!

To understand this, we will assume that we have a bipartite graph G and its bipartition X and Y. Now we will show the cases for a perfect matching:

• If there is a case where |X| ≠ |Y|, then the bipartite graph will not have a perfect matching.
• If there is a case where a number of elements in the neighborhood of a subset are greater than or equal to all subsets of X and the number of elements in that subsets, in this case, the bipartite graph will contain the perfect matching with bipartition X and Y.

Maximum Number of Edges

• If there is any bipartite graph that has the n number of vertices, then that graph contains at most (1/4)*n2 number of edges.
• In the bipartite graph, maximum possible number of edges on 'n' vertices is equal to the (1/4)*n2.

Explanation:

• Suppose V1 and V2 are the bipartition of the graph where |V1| = k and |V2| = n - k.
• k(n-k) is at most number of edges between bipartition V1 and V2, and the edges can be maximized at k = n/2.
• Thus, there is a maximum 1/4 * n2 number of edges that can be present.
• The graph G will have a triangle if there is a graph G that contains n vertices and also have more than 1/4 * n2 edges.
• The bipartite graph does not have odd cycles. That's why it is not in this graph.

### Examples of Bipartite Graph

There are various examples of bipartite graphs, and some of them are described as follows:

Example 1: In this example, we have to show whether the given graph is a bipartite graph or not. Solution:

We can draw the above graph in another way, which is described as follows: In the above redraw graph, we have the following things:

• In the above graph, we have two sets of vertices, i.e., X and Y.
• The set X = {1, 4, 6, 7} and Y = {2, 3, 5, 8}.
• In his graph, the vertices of set X only have a connection with the vertices of set Y, and the same happened with the set Y and set X.
• We cannot join the vertices of same set.
• All these properties satisfy the bipartite graph definition.

Hence, the above graph is known as the bipartite graph.

Example 2: In this example, we have 12 vertices of the bipartite graph, and we have to determine the maximum number of edges on that graph.

Solution: As we have learned above that, the maximum number of edges in any bipartite graph with n vertices = (1/4) * n2

Now we will put n = 12 in the above formula and get the following:

In a bipartite graph, the maximum number of edges on 12 vertices = (1/4) * (12)2

= (1/4) * 12 * 12

= 1/4 * 144

= 36

Hence, in the bipartite graph, the maximum number of edges on 12 vertices = 36.

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