## Bipartite Graph in Discrete mathematicsIf 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 graphThe 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.
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
**k**._{4, 3}
## Chromatic Number of Bipartite graphWhen 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.
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 graphThere 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 GraphIn a bipartite graph, there is a formula for perfect matching, which is described as follows: Number of Complete matching for K _{n, 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.
- If there is any bipartite graph that has the n number of vertices, then that graph contains at most (1/4)*n
^{2}number of edges. - In the bipartite graph, maximum possible number of edges on 'n' vertices is equal to the (1/4)*n
^{2}.
- 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 GraphThere are various examples of bipartite graphs, and some of them are described as follows:
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.
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) = (1/4) * 12 * 12 = 1/4 * 144 = 36 Hence, in the bipartite graph, the maximum number of edges on 12 vertices = 36. |