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 Bipartite Graphs.
Bipartite Graph
A bipartite graph is a special kind of graph with the following properties
 It consists of two sets of vertices X and Y.
 The vertices of set X join only with the vertices of set Y.
 The vertices within the same set do not join.
Bipartite Graph Example
The following graph is an example of a bipartite graph
Here,
 The vertices of the graph can be decomposed into two sets.
 The two sets are X = {A, C} and Y = {B, D}.
 The vertices of set X join only with the vertices of set Y and viceversa.
 The vertices within the same set do not join.
 Therefore, it is a bipartite graph.
Also Read Planar Graph
Complete Bipartite Graph
A complete bipartite graph may be defined as follows
A bipartite graph where every vertex of set X is joined to every vertex of set Y
is called as complete bipartite graph. OR Complete bipartite graph is a bipartite graph which is complete. OR Complete bipartite graph is a graph which is bipartite as well as complete. 
Complete Bipartite Graph Example
The following graph is an example of a complete bipartite graph
Here,
 This graph is a bipartite graph as well as a complete graph.
 Therefore, it is a complete bipartite graph.
 This graph is called as K_{4,3}.
Bipartite Graph Chromatic Number
To properly color any bipartite graph,
 Minimum 2 colors are required.
 This ensures that the end vertices of every edge are colored with different colors.
 Thus, bipartite graphs are 2colorable.
NoteIf graph is bipartite with no edges, then it is 1colorable. 
Example
The chromatic number of the following bipartite graph is 2
Bipartite Graph Properties
Few important properties of bipartite graph are
 Bipartite graphs are 2colorable.
 Bipartite graphs contain no odd cycles.
 Every sub graph of a bipartite graph is itself bipartite.
 There does not exist a perfect matching for a bipartite graph with bipartition X and Y if X ≠ Y.
 In any bipartite graph with bipartition X and Y,
Sum of degree of vertices of set X = Sum of degree of vertices of set Y
Bipartite Graph Perfect Matching
Number of complete matchings for K_{n,n} = n! 
Given a bipartite graph G with bipartition X and Y,
 There does not exist a perfect matching for G if X ≠ Y.
 A perfect matching exists on a bipartite graph G with bipartition X and Y if and only if for all the subsets of X, the number of elements in the subset is less than or equal to the number of elements in the neighborhood of the subset.
Maximum Number Of Edges
 Any bipartite graph consisting of ‘n’ vertices can have at most (1/4) x n^{2} edges.
 Maximum possible number of edges in a bipartite graph on ‘n’ vertices = (1/4) x n^{2}.
Explanation

Also Read Euler Graph & Hamiltonian Graph
PRACTICE PROBLEMS BASED ON BIPARTITE GRAPH IN GRAPH THEORY
Problem01:
Is the following graph a bipartite graph?
Solution
The given graph may be redrawn as
Here,
 This graph consists of two sets of vertices.
 The two sets are X = {1, 4, 6, 7} and Y = {2, 3, 5, 8}.
 The vertices of set X are joined only with the vertices of set Y and viceversa.
 Also, any two vertices within the same set are not joined.
 This satisfies the definition of a bipartite graph.
Therefore, Given graph is a bipartite graph.
Problem02:
The maximum number of edges in a bipartite graph on 12 vertices is _________?
Solution
We know, Maximum possible number of edges in a bipartite graph on ‘n’ vertices = (1/4) x n^{2}.
Substituting n = 12, we get
Maximum number of edges in a bipartite graph on 12 vertices
= (1/4) x (12)^{2}
= (1/4) x 12 x 12
= 36
Therefore, Maximum number of edges in a bipartite graph on 12 vertices = 36.
To gain better understanding about Bipartite Graphs in Graph Theory,
Next Article Konigsberg Bridge Problem
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