# M-Coloring Problem

Given an undirected graph and a number m, the task is to color the given graph with at most m colors such that no two adjacent vertices of the graph are colored with the same color

Note: Here coloring of a graph means the assignment of colors to all vertices

Below is an example of a graph that can be colored with 3 different colors:

Examples:

Input:  graph = {0, 1, 1, 1},
{1, 0, 1, 0},
{1, 1, 0, 1},
{1, 0, 1, 0}
Output: Solution Exists: Following are the assigned colors: 1  2  3  2
Explanation: By coloring the vertices with following colors,
adjacent vertices does not have same colors

Input: graph = {1, 1, 1, 1},
{1, 1, 1, 1},
{1, 1, 1, 1},
{1, 1, 1, 1}

Output: Solution does not exist
Explanation: No solution exits

## Naive Approach for M-Coloring Problem:

Generate all possible configurations of colors. Since each node can be colored using any of the m available colors, the total number of color configurations possible is mV. After generating a configuration of color, check if the adjacent vertices have the same color or not. If the conditions are met, print the combination.

Time Complexity: O(mV). There is a total O(mV) combination of colors
Auxiliary Space: O(V). The Recursive Stack of graph coloring(…) function will require O(V) space.

## M-Coloring Problem using Backtracking:

Assign colors one by one to different vertices, starting from vertex 0. Before assigning a color, check for safety by considering already assigned colors to the adjacent vertices i.e check if the adjacent vertices have the same color or not. If there is any color assignment that does not violate the conditions, mark the color assignment as part of the solution. If no assignment of color is possible then backtrack and return false

Follow the given steps to solve the problem:

• Create a recursive function that takes the graph, current index, number of vertices, and color array.
• If the current index is equal to the number of vertices. Print the color configuration in the color array.
• Assign a color to a vertex from the range (1 to m).
• For every assigned color, check if the configuration is safe, (i.e. check if the adjacent vertices do not have the same color) and recursively call the function with the next index and number of vertices otherwise, return false
• If any recursive function returns true then return true
• If no recursive function returns true then return false

#### Illustration:

• To color the graph, color each node one by one.
• To color the first node there are 3 choices of colors Red, Green and Blue, so lets take the red color for first node.
• After Red color for first node is fixed then we have made choice for second node in similar manner as we did for first node, then for 3rd node and so on.
• There is one important point to remember. while choosing color for the node, it should not be same as the color of the adjacent node.

Choosing Red For Node 1

• As shown in the above diagram, all the solutions are shown by coloring the first node in Red.
• Let’s choose Green color for the first node and explore the options for the remaining nodes.

Choosing Green for Node 1

• As shown in the above diagram, all the solutions are shown by coloring the first node in Green.
• Let’s choose Blue color for the first node and explore the options for the remaining nodes.

Choosing Blue for Node 1

Below is the implementation of the above approach:

## C++

 `// C++ program for solution of M``// Coloring problem using backtracking` `#include ``using` `namespace` `std;` `// Number of vertices in the graph``#define V 4` `void` `printSolution(``int` `color[]);` `/* A utility function to check if``   ``the current color assignment``   ``is safe for vertex v i.e. checks``   ``whether the edge exists or not``   ``(i.e, graph[v][i]==1). If exist``   ``then checks whether the color to``   ``be filled in the new vertex(c is``   ``sent in the parameter) is already``   ``used by its adjacent``   ``vertices(i-->adj vertices) or``   ``not (i.e, color[i]==c) */``bool` `isSafe(``int` `v, ``bool` `graph[V][V], ``int` `color[], ``int` `c)``{``    ``for` `(``int` `i = 0; i < V; i++)``        ``if` `(graph[v][i] && c == color[i])``            ``return` `false``;` `    ``return` `true``;``}` `/* A recursive utility function``to solve m coloring problem */``bool` `graphColoringUtil(``bool` `graph[V][V], ``int` `m, ``int` `color[],``                       ``int` `v)``{` `    ``/* base case: If all vertices are``       ``assigned a color then return true */``    ``if` `(v == V)``        ``return` `true``;` `    ``/* Consider this vertex v and``       ``try different colors */``    ``for` `(``int` `c = 1; c <= m; c++) {` `        ``/* Check if assignment of color``           ``c to v is fine*/``        ``if` `(isSafe(v, graph, color, c)) {``            ``color[v] = c;` `            ``/* recur to assign colors to``               ``rest of the vertices */``            ``if` `(graphColoringUtil(graph, m, color, v + 1)``                ``== ``true``)``                ``return` `true``;` `            ``/* If assigning color c doesn't``               ``lead to a solution then remove it */``            ``color[v] = 0;``        ``}``    ``}` `    ``/* If no color can be assigned to``       ``this vertex then return false */``    ``return` `false``;``}` `/* This function solves the m Coloring``   ``problem using Backtracking. It mainly``   ``uses graphColoringUtil() to solve the``   ``problem. It returns false if the m``   ``colors cannot be assigned, otherwise``   ``return true and prints assignments of``   ``colors to all vertices. Please note``   ``that there may be more than one solutions,``   ``this function prints one of the``   ``feasible solutions.*/``bool` `graphColoring(``bool` `graph[V][V], ``int` `m)``{` `    ``// Initialize all color values as 0.``    ``// This initialization is needed``    ``// correct functioning of isSafe()``    ``int` `color[V];``    ``for` `(``int` `i = 0; i < V; i++)``        ``color[i] = 0;` `    ``// Call graphColoringUtil() for vertex 0``    ``if` `(graphColoringUtil(graph, m, color, 0) == ``false``) {``        ``cout << ``"Solution does not exist"``;``        ``return` `false``;``    ``}` `    ``// Print the solution``    ``printSolution(color);``    ``return` `true``;``}` `/* A utility function to print solution */``void` `printSolution(``int` `color[])``{``    ``cout << ``"Solution Exists:"``         ``<< ``" Following are the assigned colors"``         ``<< ``"\n"``;``    ``for` `(``int` `i = 0; i < V; i++)``        ``cout << ``" "` `<< color[i] << ``" "``;` `    ``cout << ``"\n"``;``}` `// Driver code``int` `main()``{` `    ``/* Create following graph and test``       ``whether it is 3 colorable``      ``(3)---(2)``       ``|   / |``       ``|  /  |``       ``| /   |``      ``(0)---(1)``    ``*/``    ``bool` `graph[V][V] = {``        ``{ 0, 1, 1, 1 },``        ``{ 1, 0, 1, 0 },``        ``{ 1, 1, 0, 1 },``        ``{ 1, 0, 1, 0 },``    ``};` `    ``// Number of colors``    ``int` `m = 3;` `    ``// Function call``    ``graphColoring(graph, m);``    ``return` `0;``}` `// This code is contributed by Shivani`

## C

 `// C program for solution of M``// Coloring problem using backtracking` `#include ``#include ` `// Number of vertices in the graph``#define V 4` `void` `printSolution(``int` `color[]);` `/* A utility function to check if``   ``the current color assignment``   ``is safe for vertex v i.e. checks``   ``whether the edge exists or not``   ``(i.e, graph[v][i]==1). If exist``   ``then checks whether the color to``   ``be filled in the new vertex(c is``   ``sent in the parameter) is already``   ``used by its adjacent``   ``vertices(i-->adj vertices) or``   ``not (i.e, color[i]==c) */``bool` `isSafe(``int` `v, ``bool` `graph[V][V], ``int` `color[], ``int` `c)``{``    ``for` `(``int` `i = 0; i < V; i++)``        ``if` `(graph[v][i] && c == color[i])``            ``return` `false``;``    ``return` `true``;``}` `/* A recursive utility function``to solve m coloring problem */``bool` `graphColoringUtil(``bool` `graph[V][V], ``int` `m, ``int` `color[],``                       ``int` `v)``{``    ``/* base case: If all vertices are``       ``assigned a color then return true */``    ``if` `(v == V)``        ``return` `true``;` `    ``/* Consider this vertex v and``       ``try different colors */``    ``for` `(``int` `c = 1; c <= m; c++) {``        ``/* Check if assignment of color``           ``c to v is fine*/``        ``if` `(isSafe(v, graph, color, c)) {``            ``color[v] = c;` `            ``/* recur to assign colors to``               ``rest of the vertices */``            ``if` `(graphColoringUtil(graph, m, color, v + 1)``                ``== ``true``)``                ``return` `true``;` `            ``/* If assigning color c doesn't``               ``lead to a solution then remove it */``            ``color[v] = 0;``        ``}``    ``}` `    ``/* If no color can be assigned to``       ``this vertex then return false */``    ``return` `false``;``}` `/* This function solves the m Coloring``   ``problem using Backtracking. It mainly``   ``uses graphColoringUtil() to solve the``   ``problem. It returns false if the m``   ``colors cannot be assigned, otherwise``   ``return true and prints assignments of``   ``colors to all vertices. Please note``   ``that there may be more than one solutions,``   ``this function prints one of the``   ``feasible solutions.*/``bool` `graphColoring(``bool` `graph[V][V], ``int` `m)``{``    ``// Initialize all color values as 0.``    ``// This initialization is needed``    ``// correct functioning of isSafe()``    ``int` `color[V];``    ``for` `(``int` `i = 0; i < V; i++)``        ``color[i] = 0;` `    ``// Call graphColoringUtil() for vertex 0``    ``if` `(graphColoringUtil(graph, m, color, 0) == ``false``) {``        ``printf``(``"Solution does not exist"``);``        ``return` `false``;``    ``}` `    ``// Print the solution``    ``printSolution(color);``    ``return` `true``;``}` `/* A utility function to print solution */``void` `printSolution(``int` `color[])``{``    ``printf``(``"Solution Exists:"``           ``" Following are the assigned colors \n"``);``    ``for` `(``int` `i = 0; i < V; i++)``        ``printf``(``" %d "``, color[i]);``    ``printf``(``"\n"``);``}` `// Driver code``int` `main()``{``    ``/* Create following graph and test``       ``whether it is 3 colorable``      ``(3)---(2)``       ``|   / |``       ``|  /  |``       ``| /   |``      ``(0)---(1)``    ``*/``    ``bool` `graph[V][V] = {``        ``{ 0, 1, 1, 1 },``        ``{ 1, 0, 1, 0 },``        ``{ 1, 1, 0, 1 },``        ``{ 1, 0, 1, 0 },``    ``};``    ``int` `m = 3; ``// Number of colors` `    ``// Function call``    ``graphColoring(graph, m);``    ``return` `0;``}`

## Java

 `/* Java program for solution of``   ``M Coloring problem using backtracking */` `public` `class` `mColoringProblem {``    ``final` `int` `V = ``4``;``    ``int` `color[];` `    ``/* A utility function to check``       ``if the current color assignment``       ``is safe for vertex v */``    ``boolean` `isSafe(``int` `v, ``int` `graph[][], ``int` `color[], ``int` `c)``    ``{``        ``for` `(``int` `i = ``0``; i < V; i++)``            ``if` `(graph[v][i] == ``1` `&& c == color[i])``                ``return` `false``;``        ``return` `true``;``    ``}` `    ``/* A recursive utility function``       ``to solve m coloring  problem */``    ``boolean` `graphColoringUtil(``int` `graph[][], ``int` `m,``                              ``int` `color[], ``int` `v)``    ``{``        ``/* base case: If all vertices are``           ``assigned a color then return true */``        ``if` `(v == V)``            ``return` `true``;` `        ``/* Consider this vertex v and try``           ``different colors */``        ``for` `(``int` `c = ``1``; c <= m; c++) {``            ``/* Check if assignment of color c to v``               ``is fine*/``            ``if` `(isSafe(v, graph, color, c)) {``                ``color[v] = c;` `                ``/* recur to assign colors to rest``                   ``of the vertices */``                ``if` `(graphColoringUtil(graph, m, color,``                                      ``v + ``1``))``                    ``return` `true``;` `                ``/* If assigning color c doesn't lead``                   ``to a solution then remove it */``                ``color[v] = ``0``;``            ``}``        ``}` `        ``/* If no color can be assigned to``           ``this vertex then return false */``        ``return` `false``;``    ``}` `    ``/* This function solves the m Coloring problem using``       ``Backtracking. It mainly uses graphColoringUtil()``       ``to solve the problem. It returns false if the m``       ``colors cannot be assigned, otherwise return true``       ``and  prints assignments of colors to all vertices.``       ``Please note that there  may be more than one``       ``solutions, this function prints one of the``       ``feasible solutions.*/``    ``boolean` `graphColoring(``int` `graph[][], ``int` `m)``    ``{``        ``// Initialize all color values as 0. This``        ``// initialization is needed correct``        ``// functioning of isSafe()``        ``color = ``new` `int``[V];``        ``for` `(``int` `i = ``0``; i < V; i++)``            ``color[i] = ``0``;` `        ``// Call graphColoringUtil() for vertex 0``        ``if` `(!graphColoringUtil(graph, m, color, ``0``)) {``            ``System.out.println(``"Solution does not exist"``);``            ``return` `false``;``        ``}` `        ``// Print the solution``        ``printSolution(color);``        ``return` `true``;``    ``}` `    ``/* A utility function to print solution */``    ``void` `printSolution(``int` `color[])``    ``{``        ``System.out.println(``"Solution Exists: Following"``                           ``+ ``" are the assigned colors"``);``        ``for` `(``int` `i = ``0``; i < V; i++)``            ``System.out.print(``" "` `+ color[i] + ``" "``);``        ``System.out.println();``    ``}` `    ``// Driver code``    ``public` `static` `void` `main(String args[])``    ``{``        ``mColoringProblem Coloring = ``new` `mColoringProblem();``        ``/* Create following graph and``           ``test whether it is``           ``3 colorable``          ``(3)---(2)``           ``|   / |``           ``|  /  |``           ``| /   |``          ``(0)---(1)``        ``*/``        ``int` `graph[][] = {``            ``{ ``0``, ``1``, ``1``, ``1` `},``            ``{ ``1``, ``0``, ``1``, ``0` `},``            ``{ ``1``, ``1``, ``0``, ``1` `},``            ``{ ``1``, ``0``, ``1``, ``0` `},``        ``};``        ``int` `m = ``3``; ``// Number of colors` `        ``// Function call``        ``Coloring.graphColoring(graph, m);``    ``}``}``// This code is contributed by Abhishek Shankhadhar`

## Python3

 `# Python3 program for solution of M Coloring``# problem using backtracking`  `class` `Graph():` `    ``def` `__init__(``self``, vertices):``        ``self``.V ``=` `vertices``        ``self``.graph ``=` `[[``0` `for` `column ``in` `range``(vertices)]``                      ``for` `row ``in` `range``(vertices)]` `    ``# A utility function to check``    ``# if the current color assignment``    ``# is safe for vertex v``    ``def` `isSafe(``self``, v, colour, c):``        ``for` `i ``in` `range``(``self``.V):``            ``if` `self``.graph[v][i] ``=``=` `1` `and` `colour[i] ``=``=` `c:``                ``return` `False``        ``return` `True` `    ``# A recursive utility function to solve m``    ``# coloring  problem``    ``def` `graphColourUtil(``self``, m, colour, v):``        ``if` `v ``=``=` `self``.V:``            ``return` `True` `        ``for` `c ``in` `range``(``1``, m ``+` `1``):``            ``if` `self``.isSafe(v, colour, c) ``=``=` `True``:``                ``colour[v] ``=` `c``                ``if` `self``.graphColourUtil(m, colour, v ``+` `1``) ``=``=` `True``:``                    ``return` `True``                ``colour[v] ``=` `0` `    ``def` `graphColouring(``self``, m):``        ``colour ``=` `[``0``] ``*` `self``.V``        ``if` `self``.graphColourUtil(m, colour, ``0``) ``=``=` `None``:``            ``return` `False` `        ``# Print the solution``        ``print``(``"Solution exist and Following are the assigned colours:"``)``        ``for` `c ``in` `colour:``            ``print``(c, end``=``' '``)``        ``return` `True`  `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ``g ``=` `Graph(``4``)``    ``g.graph ``=` `[[``0``, ``1``, ``1``, ``1``], [``1``, ``0``, ``1``, ``0``], [``1``, ``1``, ``0``, ``1``], [``1``, ``0``, ``1``, ``0``]]``    ``m ``=` `3` `    ``# Function call``    ``g.graphColouring(m)` `# This code is contributed by Divyanshu Mehta`

## C#

 `/* C# program for solution of M Coloring problem``using backtracking */``using` `System;` `class` `GFG {``    ``readonly` `int` `V = 4;``    ``int``[] color;` `    ``/* A utility function to check if the current``    ``color assignment is safe for vertex v */``    ``bool` `isSafe(``int` `v, ``int``[, ] graph, ``int``[] color, ``int` `c)``    ``{``        ``for` `(``int` `i = 0; i < V; i++)``            ``if` `(graph[v, i] == 1 && c == color[i])``                ``return` `false``;``        ``return` `true``;``    ``}` `    ``/* A recursive utility function to solve m``    ``coloring problem */``    ``bool` `graphColoringUtil(``int``[, ] graph, ``int` `m,``                           ``int``[] color, ``int` `v)``    ``{``        ``/* base case: If all vertices are assigned``        ``a color then return true */``        ``if` `(v == V)``            ``return` `true``;` `        ``/* Consider this vertex v and try different``        ``colors */``        ``for` `(``int` `c = 1; c <= m; c++) {``            ``/* Check if assignment of color c to v``            ``is fine*/``            ``if` `(isSafe(v, graph, color, c)) {``                ``color[v] = c;` `                ``/* recur to assign colors to rest``                ``of the vertices */``                ``if` `(graphColoringUtil(graph, m, color,``                                      ``v + 1))``                    ``return` `true``;` `                ``/* If assigning color c doesn't lead``                ``to a solution then remove it */``                ``color[v] = 0;``            ``}``        ``}` `        ``/* If no color can be assigned to this vertex``        ``then return false */``        ``return` `false``;``    ``}` `    ``/* This function solves the m Coloring problem using``    ``Backtracking. It mainly uses graphColoringUtil()``    ``to solve the problem. It returns false if the m``    ``colors cannot be assigned, otherwise return true``    ``and prints assignments of colors to all vertices.``    ``Please note that there may be more than one``    ``solutions, this function prints one of the``    ``feasible solutions.*/``    ``bool` `graphColoring(``int``[, ] graph, ``int` `m)``    ``{``        ``// Initialize all color values as 0. This``        ``// initialization is needed correct functioning``        ``// of isSafe()``        ``color = ``new` `int``[V];``        ``for` `(``int` `i = 0; i < V; i++)``            ``color[i] = 0;` `        ``// Call graphColoringUtil() for vertex 0``        ``if` `(!graphColoringUtil(graph, m, color, 0)) {``            ``Console.WriteLine(``"Solution does not exist"``);``            ``return` `false``;``        ``}` `        ``// Print the solution``        ``printSolution(color);``        ``return` `true``;``    ``}` `    ``/* A utility function to print solution */``    ``void` `printSolution(``int``[] color)``    ``{``        ``Console.WriteLine(``"Solution Exists: Following"``                          ``+ ``" are the assigned colors"``);``        ``for` `(``int` `i = 0; i < V; i++)``            ``Console.Write(``" "` `+ color[i] + ``" "``);``        ``Console.WriteLine();``    ``}` `    ``// Driver Code``    ``public` `static` `void` `Main(String[] args)``    ``{``        ``GFG Coloring = ``new` `GFG();` `        ``/* Create following graph and test whether it is``        ``3 colorable``        ``(3)---(2)``        ``| / |``        ``| / |``        ``| / |``        ``(0)---(1)``        ``*/``        ``int``[, ] graph = { { 0, 1, 1, 1 },``                          ``{ 1, 0, 1, 0 },``                          ``{ 1, 1, 0, 1 },``                          ``{ 1, 0, 1, 0 } };``        ``int` `m = 3; ``// Number of colors` `        ``// Function call``        ``Coloring.graphColoring(graph, m);``    ``}``}` `// This code is contributed by PrinciRaj1992`

## Javascript

 ``

Output
```Solution Exists: Following are the assigned colors
1  2  3  2

```

Time Complexity: O(mV). There is a total of O(mV) combinations of colors. The upper bound time complexity remains the same but the average time taken will be less.
Auxiliary Space: O(V). The recursive Stack of the graph coloring function will require O(V) space.

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