Table of contents
1.
Introduction
2.
What is a Bipartite Graph?
3.
Example of a bipartite graph
4.
Types of Bipartite Graph
5.
Characteristics of Bipartite Graph
6.
Properties of Bipartite Graphs
7.
How to Check if a Given Graph is Bipartite or Not?
7.1.
Problem Statement
7.2.
Approach 
8.
Algorithm
9.
Dry Run
10.
Implementation
10.1.
Using DFS(Depth First Search)
10.2.
C++
10.3.
Java
10.4.
Python
10.5.
Using BFS(Breadth First Search)
10.6.
C++
10.7.
Java
10.8.
Python
11.
Applications of Bipartite Graphs
12.
Frequently Asked Questions
12.1.
What is the rule of bipartite graph?
12.2.
What are the differences between bipartite graphs and complete graphs?
12.3.
Is a tree a bipartite graph?
12.4.
Can a bipartite graph contain a cycle of odd length?
12.5.
What is bipartite vs tripartite graph?
13.
Conclusion
Last Updated: Jan 7, 2025
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Bipartite Graph

Author Ayushi Goyal
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Introduction

Graphs are non-linear Data Structure composed of nodes and edges. There are different types of graphs like directed graphs, undirected graphs, Euler graphs, hamiltonian graphs, etc. One of them is a Bipartite graph

introduction

In this article, we will learn everything about Bipartite graphs in one of the simplest ways. We will start with a quick definition of bipartite graphs, their properties, and examples. Our primary focus will be to understand an algorithm to check whether a graph is bipartite, along with its analysis in terms of time and space complexity.

What is a Bipartite Graph?

Bipartite graphs or Bi-graphs are a type of graph where all of the vertices are divided into two independent groups, U and V, such that each edge [u,v] in the graph connects a vertex u from set U and a vertex v from set V. In other words, none of the edges connects two vertices from the same set. 

Example of a bipartite graph

Let's see an example of a bipartite graph

bipartite graph example

The above graph is undirected, in which every edge connects one vertex of blue color and the other vertex of green color. The two colors represent two different sets. For instance, see the edge [E, F]E is green while F is blue. Similarly, other edges also like [A, E][A, B], [E, H],[H, G],[G, C], [D, C], [B, C], etc.

We can also see the mapping below to get a clear picture as to why the above graph is an example of a bipartite graph - 

bipartite graph example

The two sets U and V in this graph are:

Types of Bipartite Graph

There are two types of Bipartite Graph

  1. Balanced Bipartite
  2. Complete Bipartite

1. Balanced Bipartite: If both the sets of vertices in a bipartite graph have an equal number of vertices or, in other words, have equal cardinality, then the graph is called balanced bipartite.

2. Complete Bipartite: A bipartite graph is said to be complete if all the vertices in  set-1 are connected to every vertex of set-2. So, if set-1 has m vertices and set-2 has n vertices, then the total number of edges will be - m*n.

Characteristics of Bipartite Graph

There are several characteristics of a Bipartite Graph:

  1. Two Sets of Vertices
    A bipartite graph has two disjoint vertex sets, with edges only between vertices in different sets, never within the same set.
  2. No Odd-Length Cycles
    A graph is bipartite if and only if it contains no odd-length cycles, as such cycles would imply connections within the same set.
  3. Two-Colorable
    Bipartite graphs can be colored using only two colors, assigning one color to each set, without adjacent vertices sharing the same color.
  4. Perfect Matching Possible
    In some bipartite graphs, perfect matching occurs, meaning every vertex in one set is connected to exactly one vertex in the other set.
  5. Applications in Real-World Problems
    Used in tasks like job assignments and network flow where elements of two different types need to be linked without internal conflicts.
  6. Maximum Bipartite Matching
    Algorithms like the Hopcroft–Karp are used to find the maximum number of matching edges, crucial in optimization problems.
  7. Complete Bipartite Graph
    If every vertex in one set is connected to every vertex in the other, it’s called a complete bipartite graph, denoted as Km,nK_{m,n}Km,n​.
  8. Adjacency Matrix Properties
    The adjacency matrix of a bipartite graph can be rearranged into block form, with zero diagonal blocks and two off-diagonal blocks filled.
  9. Can be Multigraph
    Bipartite graphs can be multigraphs, meaning multiple edges between two vertices are allowed, expanding the graph’s application in some models.
  10. Planarity
    Some bipartite graphs are planar, meaning they can be drawn on a plane without edges crossing, often seen in network structures.

Properties of Bipartite Graphs

  • All the bipartite graphs are 2-colorable, which means that using only two colors, every vertex of the bipartite graph can be assigned a color such that none of the adjacent nodes has the same color.
     
  • Every subgraph of a bipartite graph is also bipartite.
     
  • It does not have odd cycles. This is because a graph having a cycle of odd length is not 2-colorable, so it can't be bipartite.
     
  • In a bipartite graph with the sets of vertices, as set U and set V, the following always holds:
Sum of the degrees of all vertices in set U = Sum of the degrees of all vertices in set V 

How to Check if a Given Graph is Bipartite or Not?

Problem Statement

Given a graph, check whether the graph is bipartite or not. Your function should return true if the given graph's vertices are divided into two independent groups, U and V, such that each edge [u,v] in the graph connects a vertex u from set U and a vertex v from set V

You are given a 2D vector or adjacency matrix edges which contains 0 and 1, where edges[i][j] = 1 denotes a bi-directional edge from i to j. A bi-directional edge implies that the graph is undirected.

Before moving on to the solution approach, please first try to solve the question Check Bipartite Graph

Approach 

We know that all the bipartite graphs are 2-colorable, which means that using only two colors, every vertex of the bipartite graph can be assigned a color such that none of the adjacent nodes has the same color.

Adjacent nodes - Two nodes connected by an edge in a graph are said to be adjacent to each other.

So, for a given graph, we will check if the graph is 2-colorable or not. If it is 2-colorable, then it is a bipartite graph; else, it is not bipartite. We will use both DFS Algorithm(Depth First Search) and BFS(Breadth First Search) to solve this problem.

Algorithm

Following are the steps of the algorithm:

  1. Assign a color (say green) to the source node.
  2. Assign a different color(say blue) to all of the neighbours of the source node.
  3. Iterate over all the neighbours and assign the color green to their neighbours.
  4. If, at any point in time, we encounter a neighbour node with the same color as the current node, the process stops because it means that the graph is not 2-colorable and is not bipartite.

Dry Run

Let’s understand the working before implementation using an example. The graph we are using for this example is displayed below:

graph representation

We will create an array of size = No of vertex and an empty queue. And assign all elements of that array as zero. In this example, the number of vertices = 5.

array and queue

We will start with vertex 0 and assign a color to it (say green). 

vertex 0 iteration

Check all its neighboring vertex and assign a color to them(say blue). 

assign color

After removing 0 from the queue, 1 is the first element, so we will visit all neighboring vertices of 1 and assign the opposite color if uncolored.

vertex 1 iteration

After removing 1 from the queue, 3 is at the top. But all neighboring vertices satisfy the condition of Bipartite Graph. So, no change will is made. After removing 3, vertex 2 will be checked. 

vertex 2 iteration

As the color of vertex 2 and 4 is the same, the loop will go to the else block and return false. So our graph is not a Bipartite graph. 

Implementation

Using DFS(Depth First Search)

The code to check if a given graph is bipartite or not using DFS is given below:

  • C++
  • Java
  • Python

C++

// C++ code using DFS
#include <bits/stdc++.h>
using namespace std;

// Function to add nodes in graph
void storeNodes(vector<int> adj[], int u, int v)
{
adj[u].push_back(v);
adj[v].push_back(u);
}
bool isBipartite(vector<int> edges[], int vtx, vector<bool>& searched, vector<int>& colors)
{
// Start traversing nodes
for (int i : edges[vtx])
{
// Check if vertex i is searched before or not
if (searched[i] == false)
{
// Mark vertex == searched
searched[i] = true;
colors[i] = !colors[vtx];

// Check recursively for subtree
if (!isBipartite(edges, i, searched, colors))
return false;
}

// Check if color is same then return false
else if (colors[i] == colors[vtx])
return false;
}

// Otherwise graph is Bipartite
return true;
}

// Main Code
int main()
{
int no_of_vertex = 5;
// Declare adjacency list
vector<int> edges[no_of_vertex];
// Declare required arrays
vector<bool> searched(no_of_vertex);
vector<int> colors(no_of_vertex);
// adding edges to the graph
storeNodes(edges, 0,1);
storeNodes(edges, 2,0);
storeNodes(edges, 0,3);
storeNodes(edges, 3,4);
storeNodes(edges, 2,4);
storeNodes(edges, 4,1);
storeNodes(edges, 3,2);
// Mark start vertex = searched and assign a color
searched[0] = true;
colors[0] = 0;
bool result = isBipartite(edges, 1, searched, colors);
if (result == true)
cout << "The given graph is Bipartite.\n";
else
cout << "The given graph is not Bipartite.\n";

return 0;
}
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Java

import java.util.*;

public class BipartiteGraph {
// Function to add nodes in the graph
static void storeNodes(List<Integer>[] adj, int u, int v) {
adj[u].add(v);
adj[v].add(u);
}

static boolean isBipartite(List<Integer>[] edges, int vtx, boolean[] searched, int[] colors) {
// Traverse nodes
for (int i : edges[vtx]) {
// Check if vertex i is searched before or not
if (!searched[i]) {
// Mark vertex as searched
searched[i] = true;
colors[i] = 1 - colors[vtx];

// Check recursively for subtree
if (!isBipartite(edges, i, searched, colors)) {
return false;
}
}
// If the color is the same, return false
else if (colors[i] == colors[vtx]) {
return false;
}
}
// Otherwise, the graph is bipartite
return true;
}

public static void main(String[] args) {
int no_of_vertex = 5;

// Declare adjacency list
List<Integer>[] edges = new ArrayList[no_of_vertex];
for (int i = 0; i < no_of_vertex; i++) {
edges[i] = new ArrayList<>();
}

// Declare required arrays
boolean[] searched = new boolean[no_of_vertex];
int[] colors = new int[no_of_vertex];

// Adding edges to the graph
storeNodes(edges, 0, 1);
storeNodes(edges, 2, 0);
storeNodes(edges, 0, 3);
storeNodes(edges, 3, 4);
storeNodes(edges, 2, 4);
storeNodes(edges, 4, 1);
storeNodes(edges, 3, 2);

// Mark start vertex as searched and assign a color
searched[0] = true;
colors[0] = 0;

boolean result = isBipartite(edges, 1, searched, colors);
if (result) {
System.out.println("The given graph is Bipartite.");
} else {
System.out.println("The given graph is not Bipartite.");
}
}
}
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Python

from collections import defaultdict

# Function to add nodes in the graph
def store_nodes(adj, u, v):
adj[u].append(v)
adj[v].append(u)

def is_bipartite(edges, vtx, searched, colors):
# Traverse nodes
for i in edges[vtx]:
# Check if vertex i is searched before or not
if not searched[i]:
# Mark vertex as searched
searched[i] = True
colors[i] = not colors[vtx]

# Check recursively for subtree
if not is_bipartite(edges, i, searched, colors):
return False
# If the color is the same, return False
elif colors[i] == colors[vtx]:
return False
# Otherwise, the graph is bipartite
return True

# Main code
if __name__ == "__main__":
no_of_vertex = 5

# Declare adjacency list
edges = defaultdict(list)

# Declare required arrays
searched = [False] * no_of_vertex
colors = [False] * no_of_vertex

# Adding edges to the graph
store_nodes(edges, 0, 1)
store_nodes(edges, 2, 0)
store_nodes(edges, 0, 3)
store_nodes(edges, 3, 4)
store_nodes(edges, 2, 4)
store_nodes(edges, 4, 1)
store_nodes(edges, 3, 2)

# Mark start vertex as searched and assign a color
searched[0] = True
colors[0] = False

result = is_bipartite(edges, 1, searched, colors)
if result:
print("The given graph is Bipartite.")
else:
print("The given graph is not Bipartite.")
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Output

The given graph is not Bipartite.

Complexity

Constraints 

The maximum number of nodes we can use are 10^8. 
 

Time Complexity

It is O(N), where ‘N’ is the number of nodes. As we are traversing all the nodes onces. So, for N number of nodes the number of traversals are also equal to N. Hence, the overall time complexity is O(N).
 

Space Complexity

It is O(N), where ‘N’ is the number of nodes. Since we are using an array to store the colour of each node and an array to store visited nodes for DFS traversal, so the overall space complexity will be O(N).

Using BFS(Breadth First Search)

The code to check if a graph is bipartite or not using BFS is given below:

  • C++
  • Java
  • Python

C++

#include <bits/stdc++.h>
using namespace std;

bool isBipartite(vector<vector<int>> &graph)
{
// N is total number of nodes in graph
int n = graph.size();

// initialize an array of size N with all 0 values
vector<int> colors(n, 0);

// Create an empty queue for BFS traversal
queue<int> q;

// Loop over all the nodes one by one
for (int i = 0; i < n; i++)
{
// If color already assigned
if (colors[i] == 1)
continue;

// Assign color 1 to the node
colors[i] = 1;

// Push current node to the queue for bfs
q.push(i);

while (!q.empty())
{
int temp = q.front();

// Loop over all the neighbors of current node
for (auto neighbor : graph[temp])
{
// Check if color is not assigned to the neighbor
if (!colors[neighbor])
{
// Assign an opposite color to the neighbors
colors[neighbor] = -colors[temp];
q.push(neighbor);
}

// If neighbor has the same color return false
else if (colors[neighbor] == colors[temp])
return false;
}

// Pop the front node from the queue
q.pop();
}
}
return true;
}

// Main code to test
int main()
{
int no_of_vertex = 5;

// Declare adjacency list
vector<vector<int>> edges(no_of_vertex, vector<int>(no_of_vertex, 0));

edges[0][1] = edges[1][0] = 1;
edges[0][2] = edges[2][0] = 1;
edges[0][3] = edges[3][0] = 1;
edges[1][2] = edges[2][1] = 1;
edges[1][4] = edges[4][1] = 1;
edges[2][3] = edges[3][2] = 1;
edges[2][4] = edges[4][2] = 1;

// Construct adjacency list from adjacency matrix
vector<vector<int>> graph(no_of_vertex, vector<int>());
for (int i = 0; i < no_of_vertex; i++)
for (int j = 0; j < no_of_vertex; j++)
if (edges[i][j] == 1)
graph[i].push_back(j);

bool result = isBipartite(graph);
if (result == true)
cout << "The given graph is Bipartite.\n";

else
cout << "The given graph is not Bipartite.\n";
}
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Java

import java.util.*;

public class BipartiteCheck {
static boolean isBipartite(List<List<Integer>> graph) {
int n = graph.size(); // Total number of nodes in graph

// Initialize an array to store colors, 0 means uncolored
int[] colors = new int[n];

// Create an empty queue for BFS traversal
Queue<Integer> q = new LinkedList<>();

// Loop over all nodes
for (int i = 0; i < n; i++) {
// If node is already colored, skip it
if (colors[i] != 0) continue;

// Assign color 1 to the node
colors[i] = 1;
q.add(i);

// BFS traversal
while (!q.isEmpty()) {
int temp = q.poll();

// Loop over all neighbors of the current node
for (int neighbor : graph.get(temp)) {
// If the neighbor has no color, assign opposite color
if (colors[neighbor] == 0) {
colors[neighbor] = -colors[temp];
q.add(neighbor);
}
// If the neighbor has the same color, graph is not bipartite
else if (colors[neighbor] == colors[temp]) {
return false;
}
}
}
}
return true; // Graph is bipartite
}

public static void main(String[] args) {
int no_of_vertex = 5;

// Declare adjacency matrix
int[][] edges = new int[no_of_vertex][no_of_vertex];
edges[0][1] = edges[1][0] = 1;
edges[0][2] = edges[2][0] = 1;
edges[0][3] = edges[3][0] = 1;
edges[1][2] = edges[2][1] = 1;
edges[1][4] = edges[4][1] = 1;
edges[2][3] = edges[3][2] = 1;
edges[2][4] = edges[4][2] = 1;

// Construct adjacency list from adjacency matrix
List<List<Integer>> graph = new ArrayList<>();
for (int i = 0; i < no_of_vertex; i++) {
graph.add(new ArrayList<>());
for (int j = 0; j < no_of_vertex; j++) {
if (edges[i][j] == 1) {
graph.get(i).add(j);
}
}
}

boolean result = isBipartite(graph);
if (result) {
System.out.println("The given graph is Bipartite.");
} else {
System.out.println("The given graph is not Bipartite.");
}
}
}
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Python

from collections import deque

# Function to check if the graph is bipartite
def is_bipartite(graph):
n = len(graph) # Total number of nodes in graph

# Initialize an array to store colors, 0 means uncolored
colors = [0] * n

# BFS traversal using a queue
q = deque()

# Loop over all nodes
for i in range(n):
# If the node is already colored, skip it
if colors[i] != 0:
continue

# Assign color 1 to the node
colors[i] = 1
q.append(i)

while q:
temp = q.popleft()

# Loop over all neighbors of the current node
for neighbor in graph[temp]:
# If the neighbor has no color, assign opposite color
if colors[neighbor] == 0:
colors[neighbor] = -colors[temp]
q.append(neighbor)
# If the neighbor has the same color, graph is not bipartite
elif colors[neighbor] == colors[temp]:
return False
return True # Graph is bipartite

# Main code to test
if __name__ == "__main__":
no_of_vertex = 5

# Declare adjacency matrix
edges = [[0] * no_of_vertex for _ in range(no_of_vertex)]
edges[0][1] = edges[1][0] = 1
edges[0][2] = edges[2][0] = 1
edges[0][3] = edges[3][0] = 1
edges[1][2] = edges[2][1] = 1
edges[1][4] = edges[4][1] = 1
edges[2][3] = edges[3][2] = 1
edges[2][4] = edges[4][2] = 1

# Construct adjacency list from adjacency matrix
graph = [[] for _ in range(no_of_vertex)]
for i in range(no_of_vertex):
for j in range(no_of_vertex):
if edges[i][j] == 1:
graph[i].append(j)

result = is_bipartite(graph)
if result:
print("The given graph is Bipartite.")
else:
print("The given graph is not Bipartite.")
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Output

The given graph is not Bipartite.

Complexity Analysis

Constraints 

The maximum number of nodes we can use are 10^5. 

Time Complexity 

It is O(N ^ 2), where ‘N’ is the number of nodes. As we have to color all the nodes present, and for each node, we have to traverse its neighbours. In the worst cases, there can be at most ‘N’ neighbours. Hence, the overall time complexity is O(N ^ 2).

Space Complexity 

It is O(N), where ‘N’ is the number of nodes. Since we are using an array to store the color of each node and a queue to do BFS, the overall space complexity will be O(N).

Applications of Bipartite Graphs

  • Bipartite graphs are used for similarity ranking in search advertising and e-commerce.
  • They are used to predict the movie preferences of a person.
  • It is extensively used in coding theory. Bipartite graphs are used for decoding codewords. 
  • Bipartite graphs are also used to mathematically model real-world problems like big data, cloud computing, etc.

Frequently Asked Questions

What is the rule of bipartite graph?

A bipartite graph consists of two disjoint sets of vertices, with edges only between vertices from different sets. No edge connects vertices within the same set, and it must not contain odd-length cycles.

What are the differences between bipartite graphs and complete graphs?

A bipartite graph has two disjoint sets of vertices with edges between them, while a complete graph connects every vertex to every other vertex. Bipartite graphs don't require all vertices to be connected, but complete graphs do.

Is a tree a bipartite graph?

Yes, a tree is always a bipartite graph because it does not contain any cycles, and any acyclic graph (tree) can be colored with two colors without adjacent vertices having the same color.

Can a bipartite graph contain a cycle of odd length?

No, a bipartite graph cannot contain a cycle of odd length. The presence of an odd-length cycle would force some adjacent vertices to share the same color, violating the bipartite property.

What is bipartite vs tripartite graph?

A bipartite graph has two disjoint vertex sets with edges between them, while a tripartite graph has three disjoint sets of vertices. In a tripartite graph, edges only connect vertices from different sets, similar to bipartite graphs but with three sets.

Conclusion

In this article, we learnt all about bipartite graphs. Bipartite graphs play a crucial role in graph theory and have significant applications in real-world scenarios like matching problems, network design, and resource allocation. Their unique structure, defined by two disjoint vertex sets and the absence of odd-length cycles, makes them both simple and versatile.

If you want to get confident about the various algorithms in graphs, do check out:

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