# M-coloring Problem

## Introduction

To study graphs as mathematical structures, we use graph labeling. Graph labeling is a way of assigning labels to vertices, edges, or faces of a graph. It can be done in various ways.

Graph coloring is one of the prime methods of labeling. It is a process of assigning labels or "colors" to elements of a graph following certain constraints.

In this blog, we'll discuss one such problem, known as the **M-coloring problem**. So, let's get started.

(Also see Data Structures)

## Problem Statement

The m-coloring problem states, **"We are given an undirected graph and m number of different colors. We have to check if we can assign colors to the vertices of the graphs in such a way that no two adjacent vertices have the same color."**

Let’s understand this problem with the help of an example.

**Example: **Given an **adjacency matrix of an undirected graph** and a number M having a value of 3.

We've to check if we can color the above four vertices such that no two adjacent vertices have the same color.

After assigning the colors, the graph will look like this.

In this graph, we can see that nodes 2 and 3 are not adjacent and have the same color; the remaining nodes have different colors. This is one possible solution.

**Note:** The m-coloring problem can have multiple solutions.

There are various ways to solve the M-coloring problem. We’ll be seeing some of them in this article. let’s move on with the brute force approach.

**Recommended** try the Problem yourself before moving on to the solution

## Approach 1: Brute Force

This approach will try to find all the possible combinations in which vertices can be colored. After all the combinations are made, we'll check if it follows the constraints of graph coloring. If the constraints are satisfied, that will be the answer.

The implementation of the above approach is given below:

### C++ Implementation

```
#include<bits/stdc++.h>
using namespace std;
//Number of vertices in the graph
#define V 4
//Function to display
void Display(int color[]) {
cout << "Solution Exists \n The colors given to vertices are: "<<endl;
for (int i = 0; i < V; i++)
cout << "Vertex " << i + 1 << " is given color:" << color[i] << endl;
cout << endl;
}
//Function to check constraints
bool satisfyConstraints(bool Adj_matrix[V][V], int color[]) {
for (int i = 0; i < V; i++) {
for (int j = i + 1; j < V; j++) {
if (Adj_matrix[i][j] && color[j] == color[i]) {
return false;
}
}
}
return true;
}
//Function for m-coloring problem
bool m_Coloring(bool Adj_matrix[V][V], int m, int i, int color[V]) {
// if current index reached end
if (i == V) {
// if constraint is satisfied
if (satisfyConstraints(Adj_matrix, color)) {
Display(color);
return true;
}
return false;
}
// Assign each color from 1 to m
for (int j = 1; j <= m; j++) {
color[i] = j;
// For the rest of the vertices
if (m_Coloring(Adj_matrix, m, i + 1, color))
return true;
color[i] = 0;
}
return false;
}
// Driver code
int main() {
// The adjacency matrix of the graph
bool Adj_matrix[V][V] = {
{0,1,1,1},
{1,0,0,1},
{1,0,0,1},
{1,1,1,0},
};
// Number of colors available
int m = 3;
// Initialising all vertices with the color values as 0
int color[V];
for (int i = 0; i < V; i++) {
color[i] = 0;
}
if (m_Coloring(Adj_matrix, m, 0, color) == 0)
cout << "No such arrangement exists!!" << endl;
return 0;
}
```

### Java Implementation

```
public class Main {
// Number of vertices in the graph
static int V = 4;
/* A utility function to print solution */
static void display(int[] color) {
System.out.println("Solution Exists:\n" +
"The colors given to vertices are: ");
for (int i = 0; i < V; i++) {
System.out.println("Vertex " + (i + 1) + " is given color : " + color[i]);
}
}
// check if the colored
// graph is safe or not
static boolean isSafe(boolean[][] graph, int[] color) {
// check for every edge in a graph
for (int i = 0; i < V; i++)
for (int j = i + 1; j < V; j++)
if (graph[i][j] && color[j] == color[i])
// if matches return false
return false;
// if not return true as it is safe
return true;
}
static boolean graphColoring(boolean[][] graph, int m,
int i, int[] color) {
// if current index reached end
if (i == V) {
// if coloring is safe
if (isSafe(graph, color)) {
// Print the solution
display(color);
return true;
}
return false;
}
// Assign each color from 1 to m
for (int j = 1; j <= m; j++) {
color[i] = j;
// Recur of the rest vertices
if (graphColoring(graph, m, i + 1, color))
return true;
color[i] = 0;
}
return false;
}
public static void main(String[] args) {
boolean[][] graph = {
{
false,
true,
true,
true
},
{
true,
false,
true,
false
},
{
true,
true,
false,
true
},
{
true,
false,
true,
false
},
};
int m = 3; // Number of colors
// Initialize all color values as 0.
// This initialization is needed
// correct functioning of isSafe()
int[] color = new int[V];
for (int i = 0; i < V; i++)
color[i] = 0;
if (!graphColoring(graph, m, 0, color))
System.out.println("Solution does not exist");
}
}
```

### Python Implementation

```
# Number of vertices in the graph
# define 4 4
# check if the colored
# graph is safe or not
def isSafe(graph, color):
# check for every edge in the graph
for i in range(4):
for j in range(i + 1, 4):
if (graph[i][j] and color[j] == color[i]):
return False
return True
def graphColoring(graph, m, i, color):
# if current index reached end
if (i == 4):
# if coloring is safe
if (isSafe(graph, color)):
# Print the solution
display(color)
return True
return False
# Assign each color from 1 to m
for j in range(1, m + 1):
color[i] = j
# Recur of the rest vertices
if (graphColoring(graph, m, i + 1, color)):
return True
color[i] = 0
return False
# /* A utility function to print solution */
def display(color):
print("Solution Exists:" " Following are the assigned colors ")
for i in range(4):
print("Vertex", i+1 ," is given color: ",color[i])
# Driver code
if __name__ == '__main__':
graph = [
[ 0, 1, 1, 1 ],
[ 1, 0, 1, 0 ],
[ 1, 1, 0, 1 ],
[ 1, 0, 1, 0 ],
]
m = 3 # Number of colors
# Initialize all color values as 0.
# This initialization is needed
# correct functioning of isSafe()
color = [0 for i in range(4)]
if (not graphColoring(graph, m, 0, color)):
print ("Solution does not exist")
```

**Output**

```
The colors given to vertices are:
Vertex 1 is given color:1
Vertex 2 is given color:2
Vertex 3 is given color:2
Vertex 4 is given color:3
```

### Complexity Analysis

**Time Complexity:** O(M^V).

**Reason:** Choosing out of M given colors for V vertices will lead to an **O(M^V)** combination. So the time complexity is O(M^V).

**Space Complexity:** O(V).

**Reason:** The m_Coloring function will require O(V) space for the recursive call stack.

## Approach 2: Backtracking

Backtracking is a general algorithm for solving constraint satisfaction problems. It accomplishes this by constructing a solution incrementally, one component at a time, discarding any solutions that fail to satisfy the problem's criteria at any point in time.

In the above approach, we've seen that we are creating combinations after assigning colors to vertices and then checking the constraints. A different solution to this problem is to only assign colors to a particular vertex if it satisfies the constraints.

We'd assign a color to each vertex from 1 to m and see if it has a different color than its next vertex. If we obtain a configuration in which each node is colored from 1 to m, and adjacent vertices are of different colors, this will be the answer.

Let’s see the implementation of this approach.

### C++ Implementation

```
#include <bits/stdc++.h>
using namespace std;
// Number of vertices in the Adj_matrix
#define V 4
//Function to display
void Display(int color[]) {
cout << "The colors given to vertices are:" << endl;
for (int i = 0; i < V; i++)
cout << "Vertex " << i + 1 << " is given color:" << color[i] << endl;
cout << endl;
}
//Function to check constraints
bool satisfyConstraints(int v, bool Adj_matrix[V][V], int color[], int c) {
for (int i = 0; i < V; i++) {
if (Adj_matrix[v][i] && c == color[i])
return false;
}
return true;
}
/* A recursive utility function
to solve m-coloring problem */
bool m_Coloring_Helper(bool Adj_matrix[V][V], int m, int color[], int v) {
//If all vertices are assigned a color
if (v == V)
return true;
//Try different colors to vertex v
for (int c = 1; c <= m; c++) {
// Check if given color is valid
if (satisfyConstraints(v, Adj_matrix, color, c)) {
color[v] = c;
//Assign colors to rest of the vertices
if (m_Coloring_Helper(Adj_matrix, m, color, v + 1) == true)
return true;
//Backtrack
color[v] = 0;
}
}
// If no color can be assigned
return false;
}
bool m_Coloring(bool Adj_matrix[V][V], int m) {
// Initialize all color values as 0.
int color[V];
for (int i = 0; i < V; i++) {
color[i] = 0;
}
// Call m_Coloring_Helper() for vertex 0
if (m_Coloring_Helper(Adj_matrix, m, color, 0) == false) {
cout << "No such arrangement exists!!";
return false;
}
// Print the solution
Display(color);
return true;
}
int main() {
// The adjacency matrix of the graph
bool Adj_matrix[V][V] = {
{0,1,1,1},
{1,0,0,1},
{1,0,0,1},
{1,1,1,0},
};
// Number of colors
int m = 3;
m_Coloring(Adj_matrix, m);
return 0;
}
```

### Java Implementation

```
class Main {
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;
}
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;
}
// display the solution
display(color);
return true;
}
/* A utility function to print solution */
void display(int color[]) {
System.out.println(
"Solution Exists: Following" +
" are the assigned colors");
for (int i = 0; i < V; i++)
System.out.println(" Vertex " + (i + 1) + " is given color : " + color[i] + " ");
System.out.println();
}
// driver program to test above function
public static void main(String args[]) {
Main Coloring = new Main();
int graph[][] = {{0, 1, 1, 1},{ 1, 0, 1, 0},{1, 1, 0, 1},{1, 0, 1, 0}};
int m = 3; // Number of colors
Coloring.graphColoring(graph, m);
}
}
```

### Python Implementation

```
# Python 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 ("Solution exist and Following are the assigned colours:")
for i in range(4):
print("Vertex", i+1, "is given color: ",colour[i])
return True
# Driver Code
g = Graph(4)
g.graph = [[0, 1, 1, 1], [1, 0, 1, 0], [1, 1, 0, 1], [1, 0, 1, 0]]
m = 3
g.graphColouring(m)
```

**Output**

```
The colors given to vertices are:
Vertex 1 is given color:1
Vertex 2 is given color:2
Vertex 3 is given color:2
Vertex 4 is given color:3
```

### Complexity Analysis

**Time Complexity:** O(M^V).

**Reason:** Choosing out of M given colors for V vertices will lead to an **O(M^V)** combination. So the time complexity is O(M^V).

**Space Complexity:** O(V).

**Reason:** The M_Coloring function will require O(V) space for the recursive call stack.

The above two approaches have the same time complexity regardless of different algorithms. In our next approach, we’ll be discussing an optimal approach to solve the M-coloring problem.

## Approach 3: Breadth-First Search

In this approach, first of all, we’ll color all the vertices of the graph with the first color. Then we’ll traverse the graph in a breadth-first fashion starting from the first unvisited node.

After reaching a node, we’ll check the color of all the adjacent nodes of the current node. If the current and the adjacent node have the same color, we’ll assign the next color to the adjacent node. Mark the current node as visited and move forward.

Let’s see the algorithm of this approach.

Step 1: Start the BFS traversal.

Step 2: Find out the neighboring (adjacent) nodes of the current node.

- If the color of the current node and the adjacent node is the same, assign the next color to the adjacent node.
- Check if the current node has been visited or not. Mark it as visited and add it to a queue if it hasn't been visited yet.

Step 3: Check the condition for color availability. If the condition becomes false, return.

Step 4: Do it for all the given nodes.

### C++ Implementation

```
#include <bits/stdc++.h>
using namespace std;
// Number of vertices in the Adj_matrix
#define V 4
//Function to display
void Display(int color[]) {
cout << "The colors given to vertices are:" << endl;
for (int i = 0; i < V; i++)
cout << "Vertex " << i + 1 << " is given color:" << color[i] << endl;
cout << endl;
}
//Function to check constraints
bool satisfyConstraints(int v, bool Adj_matrix[V][V], int color[], int c) {
for (int i = 0; i < V; i++) {
if (Adj_matrix[v][i] && c == color[i])
return false;
}
return true;
}
/* A recursive utility function
to solve m-coloring problem */
bool m_Coloring_Helper(bool Adj_matrix[V][V], int m, int color[], int v) {
//If all vertices are assigned a color
if (v == V)
return true;
//Try different colors to vertex v
for (int c = 1; c <= m; c++) {
// Check if given color is valid
if (satisfyConstraints(v, Adj_matrix, color, c)) {
color[v] = c;
//Assign colors to rest of the vertices
if (m_Coloring_Helper(Adj_matrix, m, color, v + 1) == true)
return true;
//Backtrack
color[v] = 0;
}
}
// If no color can be assigned
return false;
}
bool m_Coloring(bool Adj_matrix[V][V], int m) {
// Initialize all color values as 0.
int color[V];
for (int i = 0; i < V; i++) {
color[i] = 0;
}
// Call m_Coloring_Helper() for vertex 0
if (m_Coloring_Helper(Adj_matrix, m, color, 0) == false) {
cout << "No such arrangement exists!!";
return false;
}
// Print the solution
Display(color);
return true;
}
int main() {
// The adjacency matrix of the graph
bool Adj_matrix[V][V] = {
{0,1,1,1},
{1,0,0,1},
{1,0,0,1},
{1,1,1,0},
};
// Number of colors
int m = 3;
m_Coloring(Adj_matrix, m);
return 0;
}
```

### Java Implementation

```
// Java Implementation
import java.io.*;
import java.util.*;
class Node {
// A node class which stores the color and the edges connected to the node
int color = 1;
Set < Integer > edges = new HashSet < Integer > ();
}
public class mColoring {
static int possiblePaint(ArrayList < Node > nodes, int n, int m) {
// Create a visited array of n nodes
ArrayList < Integer > visited = new ArrayList < Integer > ();
// initialized to zero
for (int i = 0; i < n + 1; i++) {
visited.add(0);
}
// maxColors used till now are 1 as
// all nodes are painted color 1
int maxColors = 1;
// Do a full BFS traversal from
// all unvisited starting points
for (int sv = 1; sv <= n; sv++) {
if (visited.get(sv) > 0) {
continue;
}
// If the starting point is unvisited,
// mark it visited and push it in queue
visited.set(sv, 1);
Queue < Integer > q = new LinkedList < > ();
q.add(sv);
// BFS Travel starts here
while (q.size() != 0) {
int top = q.peek();
q.remove();
// Checking all adjacent nodes
// to "top" edge in our queue
for (int it: nodes.get(top).edges) {
// IMPORTANT: If the color of the
// adjacent node is same, increase it by 1
if (nodes.get(top).color == nodes.get(it).color) {
nodes.get(it).color += 1;
}
// If number of colors used shoots m, return
// 0
maxColors = Math.max(maxColors,
Math.max(nodes.get(top).color,
nodes.get(it).color));
if (maxColors > m)
return 0;
// If the adjacent node is not visited,
// mark it visited and push it in queue
if (visited.get(it) == 0) {
visited.set(it, 1);
q.remove(it);
}
}
}
}
return 1;
}
// Driver code
public static void main(String[] args) {
int n = 4;
int [][] graph = {{ 0, 1, 1, 1 },{ 1, 0, 1, 0 },{ 1, 1, 0, 1 },{ 1, 0, 1, 0 }};
int m = 3; // Number of colors
// Create a vector of n+1
// nodes of type "node"
// The zeroth position is just
// dummy (1 to n to be used)
ArrayList < Node > nodes = new ArrayList < Node > ();
for (int i = 0; i < n + 1; i++) {
nodes.add(new Node());
}
// Add edges to each node as per given input
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
if (graph[i][j] > 0) {
// Connect the undirected graph
nodes.get(i).edges.add(i);
nodes.get(j).edges.add(j);
}
}
}
// Display answer
int res = possiblePaint(nodes, n, m);
if (res == 1) {
System.out.println("It is possible to color all the nodes!!");
} else {
System.out.println("No solution exists!!");
}
}
}
```

### Python Implementation

```
# Python Implementation
from queue import Queue
class node:
color = 1
edges = set()
def possiblePaint(nodes, n, m):
# Create a visited array of n
# nodes, initialized to zero
visited = [0 for _ in range(n+1)]
# maxColors used till now are 1 as
# all nodes are painted color 1
maxColors = 1
# Do a full BFS traversal from
# all unvisited starting points
for _ in range(1, n + 1):
if visited[_]:
continue
# If the starting point is unvisited,
# mark it visited and push it in queue
visited[_] = 1
q = Queue()
q.put(_)
# BFS Travel starts here
while not q.empty():
top = q.get()
# Checking all adjacent nodes
# to "top" edge in our queue
for _ in nodes[top].edges:
# IMPORTANT: If the color of the
# adjacent node is same, increase it by 1
if nodes[top].color == nodes[_].color:
nodes[_].color += 1
# If number of colors used shoots m,
# return 0
maxColors = max(maxColors, max(
nodes[top].color, nodes[_].color))
if maxColors > m:
print(maxColors)
return 0
# If the adjacent node is not visited,
# mark it visited and push it in queue
if not visited[_]:
visited[_] = 1
q.put(_)
return 1
# Driver code
if __name__ == "__main__":
n = 4
graph = [ [ 0, 1, 1, 1 ],
[ 1, 0, 1, 0 ],
[ 1, 1, 0, 1 ],
[ 1, 0, 1, 0 ] ]
m = 3
nodes = []
for _ in range(n+1):
nodes.append(node())
# Add edges to each node as
# per given input
for _ in range(n):
for __ in range(n):
if graph[_][__]:
# Connect the undirected graph
nodes[_].edges.add(_)
nodes[__].edges.add(__)
# Display the final answer
ans = possiblePaint(nodes, n, m)
if(ans == 1):
print("It is possible to color all the nodes!!")
else:
print("No solution exists!!")
```

**Output**

`It is possible to color all the nodes!!`

### Complexity Analysis

**Time Complexity:** O(V + E).

**Reason:** The time complexity of BFS traversal is O(V). For every node, we are checking all its adjacent nodes, this will lead to the time complexity of O(V + E).

**Space Complexity:** O(V).

**Reason:** The M_coloring function will require O(V) space for the visited array.

## Frequently Asked Questions

### What is Graph Coloring?

Graph coloring is an important problem of graph theory. In this problem, we'll be given a graph to color all of the vertices with the 'm' colors provided, ensuring that no two neighboring vertices have the same color.

### What do you mean by the M-coloring decision problem?

The smallest number m for which the graph G can be colored is the M – colorability optimization problem. The integer is referred to as the graph's chromatic Number.

### What are the applications of graph coloring?

There are various graph coloring applications, such as creating a schedule, register allocation, and Mobile Radio Frequency Assignment.

## Conclusion

In this article, we discussed the graph coloring problem and some approaches to how to solve it in detail. Graph coloring is one of the most frequently asked and important concepts in graph theory.

To get a clear insight into graph theory fundamentals, we suggest you go through __this__ fantastic article. If you have any difficulty understanding the graph representation in the above-given approaches, __here's__ a blog to get you acquainted with the types and representation of graphs.

#### Here are some recommended problems to test your understanding of graphs

#### Recommended Reading

Also, check out some more Problems, Courses for topics like DSA in C++, Guided Paths for topics like Data Structures and Algorithms, Interview Experiences, Interview Bundles, and Contests on Codestudio to give yourself an edge.

**Happy Learning!**

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