Check whether a given graph is Bipartite or not

A Bipartite Graph is a graph whose vertices can be divided into two independent sets, U and V such that every edge (u, v) either connects a vertex from U to V or a vertex from V to U. In other words, for every edge (u, v), either u belongs to U and v to V, or u belongs to V and v to U. We can also say that there is no edge that connects vertices of same set.

A bipartite graph is possible if the graph coloring is possible using two colors such that vertices in a set are colored with the same color. Note that it is possible to color a cycle graph with even cycle using two colors. For example, see the following graph.

It is not possible to color a cycle graph with odd cycle using two colors.

Algorithm to check if a graph is Bipartite:
One approach is to check whether the graph is 2-colorable or not using backtracking algorithm m coloring problem.
Following is a simple algorithm to find out whether a given graph is Bipartite or not using Breadth First Search (BFS).
1. Assign RED color to the source vertex (putting into set U).
2. Color all the neighbors with BLUE color (putting into set V).
3. Color all neighbor’s neighbor with RED color (putting into set U).
4. This way, assign color to all vertices such that it satisfies all the constraints of m way coloring problem where m = 2.
5. While assigning colors, if we find a neighbor which is colored with same color as current vertex, then the graph cannot be colored with 2 vertices (or graph is not Bipartite)

C++

// C++ program to find out whether a  
// given graph is Bipartite or not 
#include <iostream> 
#include <queue> 
#define V 4 
  
using namespace std; 
  
// This function returns true if graph  
// G[V][V] is Bipartite, else false 
bool isBipartite(int G[][V], int src) 
{ 
    // Create a color array to store colors  
    // assigned to all vertices. Vertex  
    // number is used as index in this array.  
    // The value '-1' of colorArr[i]  
    // is used to indicate that no color  
    // is assigned to vertex 'i'. The value 1  
    // is used to indicate first color  
    // is assigned and value 0 indicates  
    // second color is assigned. 
    int colorArr[V]; 
    for (int i = 0; i < V; ++i) 
        colorArr[i] = -1; 
  
    // Assign first color to source 
    colorArr[src] = 1; 
  
    // Create a queue (FIFO) of vertex  
    // numbers and enqueue source vertex 
    // for BFS traversal 
    queue <int> q; 
    q.push(src); 
  
    // Run while there are vertices  
    // in queue (Similar to BFS) 
    while (!q.empty()) 
    { 
        // Dequeue a vertex from queue ( Refer http://goo.gl/35oz8 ) 
        int u = q.front(); 
        q.pop(); 
  
        // Return false if there is a self-loop  
        if (G[u][u] == 1) 
        return false;  
  
        // Find all non-colored adjacent vertices 
        for (int v = 0; v < V; ++v) 
        { 
            // An edge from u to v exists and  
            // destination v is not colored 
            if (G[u][v] && colorArr[v] == -1) 
            { 
                // Assign alternate color to this adjacent v of u 
                colorArr[v] = 1 - colorArr[u]; 
                q.push(v); 
            } 
  
            // An edge from u to v exists and destination  
            // v is colored with same color as u 
            else if (G[u][v] && colorArr[v] == colorArr[u]) 
                return false; 
        } 
    } 
  
    // If we reach here, then all adjacent   
    // vertices can be colored with alternate color 
    return true; 
} 
  
// Driver program to test above function 
int main() 
{ 
    int G[][V] = {{0, 1, 0, 1}, 
        {1, 0, 1, 0}, 
        {0, 1, 0, 1}, 
        {1, 0, 1, 0} 
    }; 
  
    isBipartite(G, 0) ? cout << "Yes" : cout << "No"; 
    return 0; 
} 

Java

// Java program to find out whether  
// a given graph is Bipartite or not 
import java.util.*; 
import java.lang.*; 
import java.io.*; 
  
class Bipartite 
{ 
    final static int V = 4; // No. of Vertices 
  
    // This function returns true if  
    // graph G[V][V] is Bipartite, else false 
    boolean isBipartite(int G[][],int src) 
    { 
        // Create a color array to store  
        // colors assigned to all vertices. 
        // Vertex number is used as index  
        // in this array. The value '-1' 
        // of colorArr[i] is used to indicate  
        // that no color is assigned 
        // to vertex 'i'. The value 1 is  
        // used to indicate first color 
        // is assigned and value 0 indicates  
        // second color is assigned. 
        int colorArr[] = new int[V]; 
        for (int i=0; i<V; ++i) 
            colorArr[i] = -1; 
  
        // Assign first color to source 
        colorArr[src] = 1; 
  
        // Create a queue (FIFO) of vertex numbers  
        // and enqueue source vertex for BFS traversal 
        LinkedList<Integer>q = new LinkedList<Integer>(); 
        q.add(src); 
  
        // Run while there are vertices in queue (Similar to BFS) 
        while (q.size() != 0) 
        { 
            // Dequeue a vertex from queue 
            int u = q.poll(); 
  
            // Return false if there is a self-loop  
            if (G[u][u] == 1) 
                return false;  
  
            // Find all non-colored adjacent vertices 
            for (int v=0; v<V; ++v) 
            { 
                // An edge from u to v exists  
                // and destination v is not colored 
                if (G[u][v]==1 && colorArr[v]==-1) 
                { 
                    // Assign alternate color to this adjacent v of u 
                    colorArr[v] = 1-colorArr[u]; 
                    q.add(v); 
                } 
  
                // An edge from u to v exists and destination 
                //  v is colored with same color as u 
                else if (G[u][v]==1 && colorArr[v]==colorArr[u]) 
                    return false; 
            } 
        } 
        // If we reach here, then all adjacent vertices can 
        // be colored with alternate color 
        return true; 
    } 
  
    // Driver program to test above function 
    public static void main (String[] args) 
    { 
        int G[][] = {{0, 1, 0, 1}, 
            {1, 0, 1, 0}, 
            {0, 1, 0, 1}, 
            {1, 0, 1, 0} 
        }; 
        Bipartite b = new Bipartite(); 
        if (b.isBipartite(G, 0)) 
        System.out.println("Yes"); 
        else
        System.out.println("No"); 
    } 
} 
  
// Contributed by Aakash Hasija 

Python3

# Python program to find out whether a  
# given graph is Bipartite or not 
  
class Graph(): 
  
    def __init__(self, V): 
        self.V = V 
        self.graph = [[0 for column in range(V)] \ 
                                for row in range(V)] 
  
    # This function returns true if graph G[V][V]  
    # is Bipartite, else false 
    def isBipartite(self, src): 
  
        # Create a color array to store colors  
        # assigned to all vertices. Vertex 
        # number is used as index in this array.  
        # The value '-1' of  colorArr[i] is used to  
        # indicate that no color is assigned to  
        # vertex 'i'. The value 1 is used to indicate  
        # first color is assigned and value 0 
        # indicates second color is assigned. 
        colorArr = [-1] * self.V 
  
        # Assign first color to source 
        colorArr[src] = 1
  
        # Create a queue (FIFO) of vertex numbers and  
        # enqueue source vertex for BFS traversal 
        queue = [] 
        queue.append(src) 
  
        # Run while there are vertices in queue  
        # (Similar to BFS) 
        while queue: 
  
            u = queue.pop() 
  
            # Return false if there is a self-loop 
            if self.graph[u][u] == 1: 
                return False; 
  
            for v in range(self.V): 
  
                # An edge from u to v exists and destination  
                # v is not colored 
                if self.graph[u][v] == 1 and colorArr[v] == -1: 
  
                    # Assign alternate color to this  
                    # adjacent v of u 
                    colorArr[v] = 1 - colorArr[u] 
                    queue.append(v) 
  
                # An edge from u to v exists and destination  
                # v is colored with same color as u 
                elif self.graph[u][v] == 1 and colorArr[v] == colorArr[u]: 
                    return False
  
        # If we reach here, then all adjacent  
        # vertices can be colored with alternate  
        # color 
        return True
  
# Driver program to test above function 
g = Graph(4) 
g.graph = [[0, 1, 0, 1], 
            [1, 0, 1, 0], 
            [0, 1, 0, 1], 
            [1, 0, 1, 0] 
            ] 
              
print ("Yes" if g.isBipartite(0) else "No") 
  
# This code is contributed by Divyanshu Mehta 

C#

// C# program to find out whether  
// a given graph is Bipartite or not 
using System; 
using System.Collections.Generic; 
  
class GFG 
{ 
    readonly static int V = 4; // No. of Vertices 
  
    // This function returns true if  
    // graph G[V,V] is Bipartite, else false 
    bool isBipartite(int [,]G, int src) 
    { 
        // Create a color array to store  
        // colors assigned to all vertices. 
        // Vertex number is used as index  
        // in this array. The value '-1' 
        // of colorArr[i] is used to indicate  
        // that no color is assigned 
        // to vertex 'i'. The value 1 is  
        // used to indicate first color 
        // is assigned and value 0 indicates  
        // second color is assigned. 
        int []colorArr = new int[V]; 
        for (int i = 0; i < V; ++i) 
            colorArr[i] = -1; 
  
        // Assign first color to source 
        colorArr[src] = 1; 
  
        // Create a queue (FIFO) of vertex numbers  
        // and enqueue source vertex for BFS traversal 
        List<int>q = new List<int>(); 
        q.Add(src); 
  
        // Run while there are vertices 
        // in queue (Similar to BFS) 
        while (q.Count != 0) 
        { 
            // Dequeue a vertex from queue 
            int u = q[0]; 
            q.RemoveAt(0); 
  
            // Return false if there is a self-loop  
            if (G[u, u] == 1) 
                return false;  
  
            // Find all non-colored adjacent vertices 
            for (int v = 0; v < V; ++v) 
            { 
                // An edge from u to v exists  
                // and destination v is not colored 
                if (G[u, v] == 1 && colorArr[v] == -1) 
                { 
                    // Assign alternate color  
                    // to this adjacent v of u 
                    colorArr[v] = 1 - colorArr[u]; 
                    q.Add(v); 
                } 
  
                // An edge from u to v exists and  
                // destination v is colored with 
                // same color as u 
                else if (G[u, v] == 1 &&  
                         colorArr[v] == colorArr[u]) 
                    return false; 
            } 
        } 
          
        // If we reach here, then all adjacent vertices 
        // can be colored with alternate color 
        return true; 
    } 
  
    // Driver Code 
    public static void Main(String[] args) 
    { 
        int [,]G = {{0, 1, 0, 1}, 
                    {1, 0, 1, 0}, 
                    {0, 1, 0, 1}, 
                    {1, 0, 1, 0}}; 
        GFG b = new GFG(); 
        if (b.isBipartite(G, 0)) 
            Console.WriteLine("Yes"); 
        else
            Console.WriteLine("No"); 
    } 
} 
  
// This code is contributed by Rajput-Ji 

Javascript

<script> 
// Javascript program to find out whether 
// a given graph is Bipartite or not 
  
let V = 4; // No. of Vertices 
  
// This function returns true if 
    // graph G[V][V] is Bipartite, else false 
function isBipartite(G,src) 
{ 
    // Create a color array to store 
        // colors assigned to all vertices. 
        // Vertex number is used as index 
        // in this array. The value '-1' 
        // of colorArr[i] is used to indicate 
        // that no color is assigned 
        // to vertex 'i'. The value 1 is 
        // used to indicate first color 
        // is assigned and value 0 indicates 
        // second color is assigned. 
        let colorArr = new Array(V); 
        for (let i=0; i<V; ++i) 
            colorArr[i] = -1; 
   
        // Assign first color to source 
        colorArr[src] = 1; 
   
        // Create a queue (FIFO) of vertex numbers 
        // and enqueue source vertex for BFS traversal 
        let q = []; 
        q.push(src); 
   
        // Run while there are vertices in queue (Similar to BFS) 
        while (q.length != 0) 
        { 
            // Dequeue a vertex from queue 
            let u = q.shift(); 
   
            // Return false if there is a self-loop 
            if (G[u][u] == 1) 
                return false; 
   
            // Find all non-colored adjacent vertices 
            for (let v=0; v<V; ++v) 
            { 
                // An edge from u to v exists 
                // and destination v is not colored 
                if (G[u][v]==1 && colorArr[v]==-1) 
                { 
                    // Assign alternate color to this adjacent v of u 
                    colorArr[v] = 1-colorArr[u]; 
                    q.push(v); 
                } 
   
                // An edge from u to v exists and destination 
                //  v is colored with same color as u 
                else if (G[u][v]==1 && colorArr[v]==colorArr[u]) 
                    return false; 
            } 
        } 
        // If we reach here, then all adjacent vertices can 
        // be colored with alternate color 
        return true; 
} 
  
// Driver program to test above function 
let G=[[0, 1, 0, 1], 
            [1, 0, 1, 0], 
            [0, 1, 0, 1], 
            [1, 0, 1, 0]]; 
  
if (isBipartite(G, 0)) 
    document.write("Yes"); 
else
    document.write("No"); 
  
  
// This code is contributed by avanitrachhadiya2155 
</script> 

Output

Yes

Time Complexity : O(V*V) as adjacency matrix is used for graph but can be made O(V+E) by using adjacency list
Auxiliary Space: O(V) due to queue and color vector.

The above algorithm works only if the graph is connected. In above code, we always start with source 0 and assume that vertices are visited from it. One important observation is a graph with no edges is also Bipartite. Note that the Bipartite condition says all edges should be from one set to another.

We can extend the above code to handle cases when a graph is not connected. The idea is repeatedly called above method for all not yet visited vertices.

C++

// C++ program to find out whether 
// a given graph is Bipartite or not. 
// It works for disconnected graph also. 
#include <bits/stdc++.h> 
  
using namespace std; 
  
const int V = 4; 
  
// This function returns true if 
// graph G[V][V] is Bipartite, else false 
bool isBipartiteUtil(int G[][V], int src, int colorArr[]) 
{ 
    colorArr[src] = 1; 
  
    // Create a queue (FIFO) of vertex numbers a 
    // nd enqueue source vertex for BFS traversal 
    queue<int> q; 
    q.push(src); 
  
    // Run while there are vertices in queue (Similar to 
    // BFS) 
    while (!q.empty()) { 
        // Dequeue a vertex from queue ( Refer 
        // http://goo.gl/35oz8 ) 
        int u = q.front(); 
        q.pop(); 
  
        // Return false if there is a self-loop 
        if (G[u][u] == 1) 
            return false; 
  
        // Find all non-colored adjacent vertices 
        for (int v = 0; v < V; ++v) { 
            // An edge from u to v exists and 
            // destination v is not colored 
            if (G[u][v] && colorArr[v] == -1) { 
                // Assign alternate color to this 
                // adjacent v of u 
                colorArr[v] = 1 - colorArr[u]; 
                q.push(v); 
            } 
  
            // An edge from u to v exists and destination 
            // v is colored with same color as u 
            else if (G[u][v] && colorArr[v] == colorArr[u]) 
                return false; 
        } 
    } 
  
    // If we reach here, then all adjacent vertices can 
    // be colored with alternate color 
    return true; 
} 
  
// Returns true if G[][] is Bipartite, else false 
bool isBipartite(int G[][V]) 
{ 
    // Create a color array to store colors assigned to all 
    // vertices. Vertex/ number is used as index in this 
    // array. The value '-1' of colorArr[i] is used to 
    // indicate that no color is assigned to vertex 'i'. 
    // The value 1 is used to indicate first color is 
    // assigned and value 0 indicates second color is 
    // assigned. 
    int colorArr[V]; 
    for (int i = 0; i < V; ++i) 
        colorArr[i] = -1; 
  
    // This code is to handle disconnected graph 
    for (int i = 0; i < V; i++) 
        if (colorArr[i] == -1) 
            if (isBipartiteUtil(G, i, colorArr) == false) 
                return false; 
  
    return true; 
} 
  
// Driver code 
int main() 
{ 
    int G[][V] = { { 0, 1, 0, 1 }, 
                   { 1, 0, 1, 0 }, 
                   { 0, 1, 0, 1 }, 
                   { 1, 0, 1, 0 } }; 
  
    isBipartite(G) ? cout << "Yes" : cout << "No"; 
    return 0; 
}

Java

// JAVA Code to check whether a given 
// graph is Bipartite or not 
import java.util.*; 
  
class Bipartite { 
  
    public static int V = 4; 
  
    // This function returns true if graph 
    // G[V][V] is Bipartite, else false 
    public static boolean
    isBipartiteUtil(int G[][], int src, int colorArr[]) 
    { 
        colorArr[src] = 1; 
  
        // Create a queue (FIFO) of vertex numbers and 
        // enqueue source vertex for BFS traversal 
        LinkedList<Integer> q = new LinkedList<Integer>(); 
        q.add(src); 
  
        // Run while there are vertices in queue 
        // (Similar to BFS) 
        while (!q.isEmpty()) { 
            // Dequeue a vertex from queue 
            // ( Refer http://goo.gl/35oz8 ) 
            int u = q.getFirst(); 
            q.pop(); 
  
            // Return false if there is a self-loop 
            if (G[u][u] == 1) 
                return false; 
  
            // Find all non-colored adjacent vertices 
            for (int v = 0; v < V; ++v) { 
                // An edge from u to v exists and 
                // destination v is not colored 
                if (G[u][v] == 1 && colorArr[v] == -1) { 
                    // Assign alternate color to this 
                    // adjacent v of u 
                    colorArr[v] = 1 - colorArr[u]; 
                    q.push(v); 
                } 
  
                // An edge from u to v exists and 
                // destination v is colored with same 
                // color as u 
                else if (G[u][v] == 1
                         && colorArr[v] == colorArr[u]) 
                    return false; 
            } 
        } 
  
        // If we reach here, then all adjacent vertices 
        // can be colored with alternate color 
        return true; 
    } 
  
    // Returns true if G[][] is Bipartite, else false 
    public static boolean isBipartite(int G[][]) 
    { 
        // Create a color array to store colors assigned 
        // to all vertices. Vertex/ number is used as 
        // index in this array. The value '-1' of 
        // colorArr[i] is used to indicate that no color 
        // is assigned to vertex 'i'. The value 1 is used 
        // to indicate first color is assigned and value 
        // 0 indicates second color is assigned. 
        int colorArr[] = new int[V]; 
        for (int i = 0; i < V; ++i) 
            colorArr[i] = -1; 
  
        // This code is to handle disconnected graph 
        for (int i = 0; i < V; i++) 
            if (colorArr[i] == -1) 
                if (isBipartiteUtil(G, i, colorArr) 
                    == false) 
                    return false; 
  
        return true; 
    } 
  
    /* Driver code*/
    public static void main(String[] args) 
    { 
        int G[][] = { { 0, 1, 0, 1 }, 
                      { 1, 0, 1, 0 }, 
                      { 0, 1, 0, 1 }, 
                      { 1, 0, 1, 0 } }; 
  
        if (isBipartite(G)) 
            System.out.println("Yes"); 
        else
            System.out.println("No"); 
    } 
} 
  
// This code is contributed by Arnav Kr. Mandal.

Python3

# Python3 program to find out whether a 
# given graph is Bipartite or not 
  
  
class Graph(): 
  
    def __init__(self, V): 
        self.V = V 
        self.graph = [[0 for column in range(V)] 
                      for row in range(V)] 
  
        self.colorArr = [-1 for i in range(self.V)] 
  
    # This function returns true if graph G[V][V] 
    # is Bipartite, else false 
    def isBipartiteUtil(self, src): 
  
        # Create a color array to store colors 
        # assigned to all vertices. Vertex 
        # number is used as index in this array. 
        # The value '-1' of self.colorArr[i] is used 
        # to indicate that no color is assigned to 
        # vertex 'i'. The value 1 is used to indicate 
        # first color is assigned and value 0 
        # indicates second color is assigned. 
  
        # Assign first color to source 
  
        # Create a queue (FIFO) of vertex numbers and 
        # enqueue source vertex for BFS traversal 
        queue = [] 
        queue.append(src) 
  
        # Run while there are vertices in queue 
        # (Similar to BFS) 
        while queue: 
  
            u = queue.pop() 
  
            # Return false if there is a self-loop 
            if self.graph[u][u] == 1: 
                return False
  
            for v in range(self.V): 
  
                # An edge from u to v exists and 
                # destination v is not colored 
                if (self.graph[u][v] == 1 and
                        self.colorArr[v] == -1): 
  
                    # Assign alternate color to 
                    # this adjacent v of u 
                    self.colorArr[v] = 1 - self.colorArr[u] 
                    queue.append(v) 
  
                # An edge from u to v exists and destination 
                # v is colored with same color as u 
                elif (self.graph[u][v] == 1 and
                      self.colorArr[v] == self.colorArr[u]): 
                    return False
  
        # If we reach here, then all adjacent 
        # vertices can be colored with alternate 
        # color 
        return True
  
    def isBipartite(self): 
        self.colorArr = [-1 for i in range(self.V)] 
        for i in range(self.V): 
            if self.colorArr[i] == -1: 
                if not self.isBipartiteUtil(i): 
                    return False
        return True
  
  
# Driver Code 
g = Graph(4) 
g.graph = [[0, 1, 0, 1], 
           [1, 0, 1, 0], 
           [0, 1, 0, 1], 
           [1, 0, 1, 0]] 
  
print ("Yes" if g.isBipartite() else "No") 
  
# This code is contributed by Anshuman Sharma 

C#

// C# Code to check whether a given 
// graph is Bipartite or not 
using System; 
using System.Collections.Generic; 
  
class GFG { 
    public static int V = 4; 
  
    // This function returns true if graph 
    // G[V,V] is Bipartite, else false 
    public static bool isBipartiteUtil(int[, ] G, int src, 
                                       int[] colorArr) 
    { 
        colorArr[src] = 1; 
  
        // Create a queue (FIFO) of vertex numbers and 
        // enqueue source vertex for BFS traversal 
        Queue<int> q = new Queue<int>(); 
        q.Enqueue(src); 
  
        // Run while there are vertices in queue 
        // (Similar to BFS) 
        while (q.Count != 0) { 
            // Dequeue a vertex from queue 
            // ( Refer http://goo.gl/35oz8 ) 
            int u = q.Peek(); 
            q.Dequeue(); 
  
            // Return false if there is a self-loop 
            if (G[u, u] == 1) 
                return false; 
  
            // Find all non-colored adjacent vertices 
            for (int v = 0; v < V; ++v) { 
  
                // An edge from u to v exists and 
                // destination v is not colored 
                if (G[u, v] == 1 && colorArr[v] == -1) { 
  
                    // Assign alternate color to this 
                    // adjacent v of u 
                    colorArr[v] = 1 - colorArr[u]; 
                    q.Enqueue(v); 
                } 
  
                // An edge from u to v exists and 
                // destination v is colored with same 
                // color as u 
                else if (G[u, v] == 1 
                         && colorArr[v] == colorArr[u]) 
                    return false; 
            } 
        } 
  
        // If we reach here, then all 
        // adjacent vertices can be colored 
        // with alternate color 
        return true; 
    } 
  
    // Returns true if G[,] is Bipartite, 
    // else false 
    public static bool isBipartite(int[, ] G) 
    { 
        // Create a color array to store 
        // colors assigned to all vertices. 
        // Vertex/ number is used as 
        // index in this array. The value '-1' 
        // of colorArr[i] is used to indicate 
        // that no color is assigned to vertex 'i'. 
        // The value 1 is used to indicate 
        // first color is assigned and value 
        // 0 indicates second color is assigned. 
        int[] colorArr = new int[V]; 
        for (int i = 0; i < V; ++i) 
            colorArr[i] = -1; 
  
        // This code is to handle disconnected graph 
        for (int i = 0; i < V; i++) 
            if (colorArr[i] == -1) 
                if (isBipartiteUtil(G, i, colorArr) 
                    == false) 
                    return false; 
  
        return true; 
    } 
  
    // Driver Code 
    public static void Main(String[] args) 
    { 
        int[, ] G = { { 0, 1, 0, 1 }, 
                      { 1, 0, 1, 0 }, 
                      { 0, 1, 0, 1 }, 
                      { 1, 0, 1, 0 } }; 
  
        if (isBipartite(G)) 
            Console.WriteLine("Yes"); 
        else
            Console.WriteLine("No"); 
    } 
} 
  
// This code is contributed by Rajput-Ji

Javascript

<script> 
// Javascript Code to check whether a given 
// graph is Bipartite or not 
var V = 4; 
  
// This function returns true if graph 
// G[V,V] is Bipartite, else false 
function isBipartiteUtil(G, src, colorArr) 
{ 
    colorArr[src] = 1; 
      
    // Create a queue (FIFO) of vertex numbers and 
    // enqueue source vertex for BFS traversal 
    var q = []; 
    q.push(src); 
      
    // Run while there are vertices in queue 
    // (Similar to BFS) 
    while (q.length != 0) 
    { 
      
        // Dequeue a vertex from queue 
        // ( Refer http://goo.gl/35oz8 ) 
        var u = q[0]; 
        q.shift(); 
          
        // Return false if there is a self-loop 
        if (G[u, u] == 1) 
            return false; 
              
        // Find all non-colored adjacent vertices 
        for (var v = 0; v < V; ++v) 
        { 
          
            // An edge from u to v exists and 
            // destination v is not colored 
            if (G[u][v] == 1 && colorArr[v] == -1)  
            { 
              
                // Assign alternate color to this 
                // adjacent v of u 
                colorArr[v] = 1 - colorArr[u]; 
                q.push(v); 
            } 
              
            // An edge from u to v exists and 
            // destination v is colored with same 
            // color as u 
            else if (G[u, v] == 1 
                     && colorArr[v] == colorArr[u]) 
                return false; 
        } 
    } 
      
    // If we reach here, then all 
    // adjacent vertices can be colored 
    // with alternate color 
    return true; 
} 
  
// Returns true if G[,] is Bipartite, 
// else false 
function isBipartite(G) 
{ 
    // Create a color array to store 
    // colors assigned to all vertices. 
    // Vertex/ number is used as 
    // index in this array. The value '-1' 
    // of colorArr[i] is used to indicate 
    // that no color is assigned to vertex 'i'. 
    // The value 1 is used to indicate 
    // first color is assigned and value 
    // 0 indicates second color is assigned. 
    var colorArr = Array(V); 
    for (var i = 0; i < V; ++i) 
        colorArr[i] = -1; 
    // This code is to handle disconnected graph 
    for (var i = 0; i < V; i++) 
        if (colorArr[i] == -1) 
            if (isBipartiteUtil(G, i, colorArr) 
                == false) 
                return false; 
    return true; 
} 
  
// Driver Code 
var G = [ [ 0, 1, 0, 1 ], 
              [ 1, 0, 1, 0 ], 
              [ 0, 1, 0, 1 ], 
              [ 1, 0, 1, 0 ] ]; 
if (isBipartite(G)) 
    document.write("Yes"); 
else
    document.write("No"); 
  
// This code is contributed by rrrtnx. 
</script>

Output

Yes

Time complexity: O(V+E).
Auxiliary Space: O(V), because we have a V-size array.

If Graph is represented using Adjacency List .Time Complexity will be O(V+E).

Works for connected as well as disconnected graph.

C++

#include <bits/stdc++.h> 
using namespace std; 
  
bool isBipartite(int V, vector<int> adj[]) 
{ 
    // vector to store colour of vertex 
    // assigning all to -1 i.e. uncoloured 
    // colours are either 0 or 1 
      // for understanding take 0 as red and 1 as blue 
    vector<int> col(V, -1); 
  
    // queue for BFS storing {vertex , colour} 
    queue<pair<int, int> > q; 
    
      //loop incase graph is not connected 
    for (int i = 0; i < V; i++) { 
        
      //if not coloured 
        if (col[i] == -1) { 
            
          //colouring with 0 i.e. red 
            q.push({ i, 0 }); 
            col[i] = 0; 
            
            while (!q.empty()) { 
                pair<int, int> p = q.front(); 
                q.pop(); 
                
                  //current vertex 
                int v = p.first; 
                  //colour of current vertex 
                int c = p.second; 
                  
                  //traversing vertexes connected to current vertex 
                for (int j : adj[v]) { 
                    
                      //if already coloured with parent vertex color 
                      //then bipartite graph is not possible 
                    if (col[j] == c) 
                        return 0; 
                    
                      //if uncoloured 
                    if (col[j] == -1) { 
                      //colouring with opposite color to that of parent 
                        col[j] = (c) ? 0 : 1; 
                        q.push({ j, col[j] }); 
                    } 
                } 
            } 
        } 
    } 
    //if all vertexes are coloured such that  
      //no two connected vertex have same colours 
    return 1; 
} 
  
  
// { Driver Code Starts. 
int main() 
{ 
  
    int V, E; 
    V = 4 , E = 8; 
      //adjacency list for storing graph 
    vector<int> adj[V]; 
      adj[0] = {1,3}; 
      adj[1] = {0,2}; 
      adj[2] = {1,3}; 
      adj[3] = {0,2}; 
      
    
    bool ans = isBipartite(V, adj); 
    //returns 1 if bipartite graph is possible 
      if (ans) 
        cout << "Yes\n"; 
    //returns 0 if bipartite graph is not possible 
      else
        cout << "No\n"; 
  
    return 0; 
} 
 // code Contributed By Devendra Kolhe

Java

import java.util.*; 
  
public class GFG{ 
      
    static class Pair{ 
        int first, second; 
          
        Pair(int f, int s){ 
            first = f; 
            second = s; 
        } 
    } 
      
    static boolean isBipartite(int V, ArrayList<ArrayList<Integer>> adj) 
    { 
        
        // vector to store colour of vertex 
        // assigning all to -1 i.e. uncoloured 
        // colours are either 0 or 1 
        // for understanding take 0 as red and 1 as blue 
        int col[] = new int[V]; 
        Arrays.fill(col, -1); 
      
        // queue for BFS storing {vertex , colour} 
        Queue<Pair> q = new LinkedList<Pair>(); 
      
        //loop incase graph is not connected 
        for (int i = 0; i < V; i++) { 
          
        // if not coloured 
            if (col[i] == -1) { 
              
            // colouring with 0 i.e. red 
                q.add(new Pair(i, 0)); 
                col[i] = 0; 
              
                while (!q.isEmpty()) { 
                    Pair p = q.peek(); 
                    q.poll(); 
                  
                    //current vertex 
                    int v = p.first; 
                    
                    // colour of current vertex 
                    int c = p.second; 
                      
                    // traversing vertexes connected to current vertex 
                    for (int j : adj.get(v))  
                    { 
                      
                        // if already coloured with parent vertex color 
                        // then bipartite graph is not possible 
                        if (col[j] == c) 
                            return false; 
                      
                        // if uncoloured 
                        if (col[j] == -1) 
                        { 
                            
                        // colouring with opposite color to that of parent 
                            col[j] = (c==1) ? 0 : 1; 
                            q.add(new Pair(j, col[j])); 
                        } 
                    } 
                } 
            } 
        } 
        
        // if all vertexes are coloured such that 
        // no two connected vertex have same colours 
        return true; 
    } 
      
    // Driver Code Starts. 
    public static void main(String args[]) 
    { 
      
        int V, E; 
        V = 4 ; 
        E = 8; 
          
        // adjacency list for storing graph 
        ArrayList<ArrayList<Integer>> adj = new ArrayList<ArrayList<Integer>>(); 
          
        for(int i = 0; i < V; i++){ 
            adj.add(new ArrayList<Integer>()); 
        } 
          
        adj.get(0).add(1); 
        adj.get(0).add(3); 
          
        adj.get(1).add(0); 
        adj.get(1).add(2); 
          
        adj.get(2).add(1); 
        adj.get(2).add(3); 
          
        adj.get(3).add(0); 
        adj.get(3).add(2); 
          
        boolean ans = isBipartite(V, adj); 
          
        // returns 1 if bipartite graph is possible 
        if (ans) 
            System.out.println("Yes"); 
        
        // returns 0 if bipartite graph is not possible 
        else
            System.out.println("No"); 
      
    } 
} 
  
// This code is contributed by adityapande88.

Python3

def isBipartite(V, adj): 
    # vector to store colour of vertex 
    # assigning all to -1 i.e. uncoloured 
    # colours are either 0 or 1 
    # for understanding take 0 as red and 1 as blue 
    col = [-1]*(V) 
    
    # queue for BFS storing {vertex , colour} 
    q = [] 
    
    #loop incase graph is not connected 
    for i in range(V): 
        
        # if not coloured 
        if (col[i] == -1): 
            
            # colouring with 0 i.e. red 
            q.append([i, 0]) 
            col[i] = 0
            
            while len(q) != 0: 
                p = q[0] 
                q.pop(0) 
                
                # current vertex 
                v = p[0] 
                  
                # colour of current vertex 
                c = p[1] 
                    
                # traversing vertexes connected to current vertex 
                for j in adj[v]: 
                    
                    # if already coloured with parent vertex color 
                    # then bipartite graph is not possible 
                    if (col[j] == c): 
                        return False
                    
                    # if uncoloured 
                    if (col[j] == -1): 
                        
                        # colouring with opposite color to that of parent 
                        if c == 1: 
                            col[j] = 0
                        else: 
                            col[j] = 1
                        q.append([j, col[j]]) 
      
    # if all vertexes are coloured such that 
    # no two connected vertex have same colours 
    return True
  
V, E = 4, 8
  
# adjacency list for storing graph 
adj = [] 
adj.append([1,3]) 
adj.append([0,2]) 
adj.append([1,3]) 
adj.append([0,2]) 
   
ans = isBipartite(V, adj) 
  
# returns 1 if bipartite graph is possible 
if (ans): 
    print("Yes") 
      
# returns 0 if bipartite graph is not possible 
else: 
    print("No") 
      
    # This code is contributed by divyesh072019.

C#

using System; 
using System.Collections.Generic; 
class GFG { 
      
    static bool isBipartite(int V, List<List<int>> adj) 
    { 
         
        // vector to store colour of vertex 
        // assigning all to -1 i.e. uncoloured 
        // colours are either 0 or 1 
        // for understanding take 0 as red and 1 as blue 
        int[] col = new int[V]; 
        Array.Fill(col, -1); 
       
        // queue for BFS storing {vertex , colour} 
        List<Tuple<int,int>> q = new List<Tuple<int,int>>(); 
       
        //loop incase graph is not connected 
        for (int i = 0; i < V; i++) { 
           
        // if not coloured 
            if (col[i] == -1) { 
               
            // colouring with 0 i.e. red 
                q.Add(new Tuple<int,int>(i, 0)); 
                col[i] = 0; 
               
                while (q.Count > 0) { 
                    Tuple<int,int> p = q[0]; 
                    q.RemoveAt(0); 
                   
                    //current vertex 
                    int v = p.Item1; 
                     
                    // colour of current vertex 
                    int c = p.Item2; 
                       
                    // traversing vertexes connected to current vertex 
                    foreach(int j in adj[v]) 
                    { 
                       
                        // if already coloured with parent vertex color 
                        // then bipartite graph is not possible 
                        if (col[j] == c) 
                            return false; 
                       
                        // if uncoloured 
                        if (col[j] == -1) 
                        { 
                             
                        // colouring with opposite color to that of parent 
                            col[j] = (c==1) ? 0 : 1; 
                            q.Add(new Tuple<int,int>(j, col[j])); 
                        } 
                    } 
                } 
            } 
        } 
         
        // if all vertexes are coloured such that 
        // no two connected vertex have same colours 
        return true; 
    }  
      
  static void Main() { 
    int V; 
    V = 4 ; 
       
    // adjacency list for storing graph 
    List<List<int>> adj = new List<List<int>>(); 
       
    for(int i = 0; i < V; i++){ 
        adj.Add(new List<int>()); 
    } 
       
    adj[0].Add(1); 
    adj[0].Add(3); 
       
    adj[1].Add(0); 
    adj[1].Add(2); 
       
    adj[2].Add(1); 
    adj[2].Add(3); 
       
    adj[3].Add(0); 
    adj[3].Add(2); 
       
    bool ans = isBipartite(V, adj); 
       
    // returns 1 if bipartite graph is possible 
    if (ans) 
        Console.WriteLine("Yes"); 
     
    // returns 0 if bipartite graph is not possible 
    else
        Console.WriteLine("No"); 
  } 
} 
  
// This code is contributed by decode2207.

Javascript

<script> 
class Pair 
{ 
    constructor(f,s) 
    { 
        this.first = f; 
        this.second = s; 
    } 
} 
  
function isBipartite(V, adj) 
{ 
  
    // vector to store colour of vertex 
        // assigning all to -1 i.e. uncoloured 
        // colours are either 0 or 1 
        // for understanding take 0 as red and 1 as blue 
        let col = new Array(V); 
        for(let i = 0; i < V; i++) 
            col[i] = -1; 
       
        // queue for BFS storing {vertex , colour} 
        let q = []; 
       
        //loop incase graph is not connected 
        for (let i = 0; i < V; i++) { 
           
        // if not coloured 
            if (col[i] == -1) { 
               
            // colouring with 0 i.e. red 
                q.push(new Pair(i, 0)); 
                col[i] = 0; 
               
                while (q.length!=0) { 
                    let p = q[0]; 
                    q.shift(); 
                   
                    //current vertex 
                    let v = p.first; 
                     
                    // colour of current vertex 
                    let c = p.second; 
                       
                    // traversing vertexes connected to current vertex 
                    for (let j of adj[v]) 
                    { 
                       
                        // if already coloured with parent vertex color 
                        // then bipartite graph is not possible 
                        if (col[j] == c) 
                            return false; 
                       
                        // if uncoloured 
                        if (col[j] == -1) 
                        { 
                             
                        // colouring with opposite color to that of parent 
                            col[j] = (c==1) ? 0 : 1; 
                            q.push(new Pair(j, col[j])); 
                        } 
                    } 
                } 
            } 
        } 
         
        // if all vertexes are coloured such that 
        // no two connected vertex have same colours 
        return true; 
} 
  
// Driver Code Starts. 
let V, E; 
        V = 4 ; 
        E = 8; 
           
        // adjacency list for storing graph 
        let adj = []; 
           
        for(let i = 0; i < V; i++){ 
            adj.push([]); 
        } 
           
        adj[0].push(1); 
        adj[0].push(3); 
           
        adj[1].push(0); 
        adj[1].push(2); 
           
        adj[2].push(1); 
        adj[2].push(3); 
           
        adj[3].push(0); 
        adj[3].push(2); 
           
        let ans = isBipartite(V, adj); 
           
        // returns 1 if bipartite graph is possible 
        if (ans) 
            document.write("Yes"); 
         
        // returns 0 if bipartite graph is not possible 
        else
            document.write("No"); 
  
// This code is contributed by patel2127 
</script> 

Output

Yes

Time Complexity: O(V+E)
Auxiliary Space: O(V)

Exercise:
1. Can DFS algorithm be used to check the bipartite-ness of a graph? If yes, how?

Explanation:

The algorithm can be described as follows:

Step 1: To perform a DFS traversal, we require a starting node and an array to track visited nodes.

However, instead of using a visited array, we will utilize a color array where each node is initially

assigned the value -1 to indicate that it has not been colored yet.

Step 2: In the DFS function call, it is important to pass the assigned color value and store it in the color array.

We will attempt to color the nodes using 0 and 1, although alternative colors can also be chosen.

Starting with color 0 is suggested, but starting with color 1 is also acceptable; the key is to ensure that adjacent nodes

have the opposite color to their current node.

Step 3: During the DFS traversal, we explore uncolored neighbors by utilizing the adjacency list.

For each uncolored node encountered, we assign it the opposite color of its current node.

Step 4: If, at any point, we encounter an adjacent node in the adjacency list that has already been colored and shares

the same color as the current node, it implies that coloring is not possible.

Consequently, we return false, indicating that the given graph is not bipartite. Otherwise, we continue returning true.

Solution :

C++

// C++ program to find out whether a given graph is Bipartite or not. 
// Using recursion. 
#include <iostream> 
  
using namespace std; 
#define V 4 
  
  
bool colorGraph(int G[][V],int color[],int pos, int c){ 
      
    if(color[pos] != -1 && color[pos] !=c) 
        return false; 
          
    // color this pos as c and all its neighbours and 1-c 
    color[pos] = c; 
    bool ans = true; 
    for(int i=0;i<V;i++){ 
        if(G[pos][i]){ 
            if(color[i] == -1) 
                ans &= colorGraph(G,color,i,1-c); 
                  
            if(color[i] !=-1 && color[i] != 1-c) 
                return false; 
        } 
        if (!ans) 
            return false; 
    } 
      
    return true; 
} 
  
bool isBipartite(int G[][V]){ 
    int color[V]; 
    for(int i=0;i<V;i++) 
        color[i] = -1; 
          
    //start is vertex 0; 
    int pos = 0; 
    // two colors 1 and 0 
    return colorGraph(G,color,pos,1); 
      
} 
  
int main() 
{ 
    int G[][V] = {{0, 1, 0, 1}, 
        {1, 0, 1, 0}, 
        {0, 1, 0, 1}, 
        {1, 0, 1, 0} 
    }; 
  
    isBipartite(G) ? cout<< "Yes" : cout << "No"; 
    return 0; 
}  
// This code is contributed By Mudit Verma 

Java

// Java program to find out whether 
// a given graph is Bipartite or not. 
// Using recursion. 
class GFG  
{ 
    static final int V = 4; 
  
    static boolean colorGraph(int G[][],  
                              int color[],  
                              int pos, int c) 
    { 
        if (color[pos] != -1 && 
            color[pos] != c) 
            return false; 
  
        // color this pos as c and  
        // all its neighbours as 1-c 
        color[pos] = c; 
        boolean ans = true; 
        for (int i = 0; i < V; i++)  
        { 
            if (G[pos][i] == 1)  
            { 
                if (color[i] == -1) 
                    ans &= colorGraph(G, color, i, 1 - c); 
  
                if (color[i] != -1 && color[i] != 1 - c) 
                    return false; 
            } 
            if (!ans) 
                return false; 
        } 
        return true; 
    } 
  
    static boolean isBipartite(int G[][])  
    { 
        int[] color = new int[V]; 
        for (int i = 0; i < V; i++) 
            color[i] = -1; 
  
        // start is vertex 0; 
        int pos = 0; 
      
        // two colors 1 and 0 
        return colorGraph(G, color, pos, 1); 
    } 
  
    // Driver Code 
    public static void main(String[] args)  
    { 
        int G[][] = { { 0, 1, 0, 1 }, 
                      { 1, 0, 1, 0 },  
                      { 0, 1, 0, 1 },  
                      { 1, 0, 1, 0 } }; 
  
        if (isBipartite(G)) 
            System.out.print("Yes"); 
        else
            System.out.print("No"); 
    } 
} 
  
// This code is contributed by Rajput-Ji 

Python3

# Python3 program to find out whether a given  
# graph is Bipartite or not using recursion.  
V = 4 
  
def colorGraph(G, color, pos, c):  
      
    if color[pos] != -1 and color[pos] != c:  
        return False 
          
    # color this pos as c and all its neighbours and 1-c  
    color[pos] = c  
    ans = True 
    for i in range(0, V):  
        if G[pos][i]:  
            if color[i] == -1:  
                ans &= colorGraph(G, color, i, 1-c)  
                  
            if color[i] !=-1 and color[i] != 1-c:  
                return False 
           
        if not ans:  
            return False 
       
    return True 
   
def isBipartite(G):  
      
    color = [-1] * V  
          
    #start is vertex 0  
    pos = 0 
    # two colors 1 and 0  
    return colorGraph(G, color, pos, 1)  
  
if __name__ == "__main__":  
   
    G = [[0, 1, 0, 1],  
         [1, 0, 1, 0],  
         [0, 1, 0, 1],  
         [1, 0, 1, 0]]  
       
    if isBipartite(G): print("Yes")  
    else: print("No")  
  
# This code is contributed by Rituraj Jain 

C#

// C# program to find out whether 
// a given graph is Bipartite or not. 
// Using recursion. 
using System; 
  
class GFG  
{ 
    static readonly int V = 4; 
  
    static bool colorGraph(int [,]G,  
                           int []color,  
                           int pos, int c) 
    { 
        if (color[pos] != -1 && 
            color[pos] != c) 
            return false; 
  
        // color this pos as c and  
        // all its neighbours as 1-c 
        color[pos] = c; 
        bool ans = true; 
        for (int i = 0; i < V; i++)  
        { 
            if (G[pos, i] == 1)  
            { 
                if (color[i] == -1) 
                    ans &= colorGraph(G, color, i, 1 - c); 
  
                if (color[i] != -1 && color[i] != 1 - c) 
                    return false; 
            } 
            if (!ans) 
                return false; 
        } 
        return true; 
    } 
  
    static bool isBipartite(int [,]G)  
    { 
        int[] color = new int[V]; 
        for (int i = 0; i < V; i++) 
            color[i] = -1; 
  
        // start is vertex 0; 
        int pos = 0; 
      
        // two colors 1 and 0 
        return colorGraph(G, color, pos, 1); 
    } 
  
    // Driver Code 
    public static void Main(String[] args)  
    { 
        int [,]G = {{ 0, 1, 0, 1 }, 
                    { 1, 0, 1, 0 },  
                    { 0, 1, 0, 1 },  
                    { 1, 0, 1, 0 }}; 
  
        if (isBipartite(G)) 
            Console.Write("Yes"); 
        else
            Console.Write("No"); 
    } 
} 
  
// This code is contributed by 29AjayKumar 

Javascript

<script> 
      // JavaScript program to find out whether 
      // a given graph is Bipartite or not. 
      // Using recursion. 
      var V = 4; 
  
      function colorGraph(G, color, pos, c) { 
        if (color[pos] != -1 && color[pos] != c) return false; 
  
        // color this pos as c and 
        // all its neighbours as 1-c 
        color[pos] = c; 
        var ans = true; 
        for (var i = 0; i < V; i++) { 
          if (G[pos][i] == 1) { 
            if (color[i] == -1) ans &= colorGraph(G, color, i, 1 - c); 
  
            if (color[i] != -1 && color[i] != 1 - c) return false; 
          } 
          if (!ans) return false; 
        } 
        return true; 
      } 
  
      function isBipartite(G) { 
        var color = new Array(V).fill(0); 
        for (var i = 0; i < V; i++) color[i] = -1; 
  
        // start is vertex 0; 
        var pos = 0; 
  
        // two colors 1 and 0 
        return colorGraph(G, color, pos, 1); 
      } 
  
      // Driver Code 
      var G = [ 
        [0, 1, 0, 1], 
        [1, 0, 1, 0], 
        [0, 1, 0, 1], 
        [1, 0, 1, 0], 
      ]; 
  
      if (isBipartite(G)) document.write("Yes"); 
      else document.write("No"); 
        
      // This code is contributed by rdtank. 
    </script>

Output

Yes

Time Complexity: O(V+E)
Auxiliary Space: O(V)

References:
http://en.wikipedia.org/wiki/Graph_coloring
http://en.wikipedia.org/wiki/Bipartite_graph
This article is compiled by Aashish Barnwal. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

Feeling lost in the world of random DSA topics, wasting time without progress? It’s time for a change! Join our DSA course, where we’ll guide you on an exciting journey to master DSA efficiently and on schedule.
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 neveropen!

Check whether a given graph is Bipartite or not

C++

Java

Python3

C#

Javascript

C++

Java

Python3

C#

Javascript

C++

Java

Python3

C#

Javascript

C++

Java

Python3

C#

Javascript

LEAVE A REPLY Cancel reply

Most Popular

EDITOR PICKS

POPULAR POSTS

POPULAR CATEGORY

ABOUT US

FOLLOW US