# 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 neighborâ€™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)

Recommended Practice

## C++

 `// C++ program to find out whether a ` `// given graph is Bipartite or not` `#include ` `#include ` `#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``; iq = ``new` `LinkedList();` `        ``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

## 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

 ``

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 `   `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 q = ``new` `LinkedList();` `        ``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

 ``

Output

```Yes
```

Time complexity: O(V*V).
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 ` `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 > 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> 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 q = ``new` `LinkedList();` `    `  `        ``//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> adj = ``new` `ArrayList>();` `        `  `        ``for``(``int` `i = ``0``; i < V; i++){` `            ``adj.add(``new` `ArrayList());` `        ``}` `        `  `        ``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> 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> q = ``new` `List>();` `     `  `        ``//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> adj = ``new` `List>();` `     `  `    ``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

 ``

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 `   `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

## 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

 ``

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