# Introduction to Graph Coloring

Last Updated : 02 Apr, 2024

Graph coloring refers to the problem of coloring vertices of a graph in such a way that no two adjacent vertices have the same color. This is also called the vertex coloring problem. If coloring is done using at most m colors, it is called m-coloring.

## Chromatic Number:

The minimum number of colors needed to color a graph is called its chromatic number. For example, the following can be colored a minimum of 2 colors.

Example of Chromatic Number

The problem of finding a chromatic number of a given graph is NP-complete.

Graph coloring problem is both, a decision problem as well as an optimization problem.

• A decision problem is stated as, “With given M colors and graph G, whether a such color scheme is possible or not?”.
• The optimization problem is stated as, “Given M colors and graph G, find the minimum number of colors required for graph coloring.”

## Algorithm of Graph ColoringÂ using Backtracking:

Assign colors one by one to different vertices, starting from vertex 0. Before assigning a color, 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 else return false
• If any recursive function returns true then break the loop and return true
• If no recursive function returns true then return false

Below is the implementation of the above approach:

C++ ```// C++ program for solution of M // Coloring problem using backtracking #include <bits/stdc++.h> 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; } ``` Java ```// Nikunj Sonigara public class Main { static final int V = 4; // A utility function to check if the current color assignment is safe for vertex v static boolean isSafe(int v, boolean[][] graph, 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 static boolean graphColoringUtil(boolean[][] graph, int m, int[] color, int v) { if (v == V) return true; for (int c = 1; c <= m; c++) { if (isSafe(v, graph, color, c)) { color[v] = c; if (graphColoringUtil(graph, m, color, v + 1)) return true; color[v] = 0; } } return false; } // This function solves the m Coloring problem using Backtracking. // It returns false if the m colors cannot be assigned, otherwise, return true // and prints assignments of colors to all vertices. static boolean graphColoring(boolean[][] graph, int m) { int[] color = new int[V]; for (int i = 0; i < V; i++) color[i] = 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 the solution static void printSolution(int[] color) { System.out.print("Solution Exists: Following are the assigned colors\n"); for (int i = 0; i < V; i++) System.out.print(" " + color[i] + " "); System.out.println(); } // Driver code public static void main(String[] args) { // Create following graph and test whether it is 3 colorable // (3)---(2) // | / | // | / | // | / | // (0)---(1) boolean[][] graph = { { false, true, true, true }, { true, false, true, false }, { true, true, false, true }, { true, false, true, false } }; // Number of colors int m = 3; // Function call graphColoring(graph, m); } } ``` Python3 ```V = 4 def print_solution(color): print("Solution Exists: Following are the assigned colors") print(" ".join(map(str, color))) def is_safe(v, graph, color, c): # Check if the color 'c' is safe for the vertex 'v' for i in range(V): if graph[v][i] and c == color[i]: return False return True def graph_coloring_util(graph, m, color, v): # Base case: If all vertices are assigned a color, return true if v == V: return True # Try different colors for the current vertex 'v' for c in range(1, m + 1): # Check if assignment of color 'c' to 'v' is fine if is_safe(v, graph, color, c): color[v] = c # Recur to assign colors to the rest of the vertices if graph_coloring_util(graph, m, color, v + 1): return True # If assigning color 'c' doesn't lead to a solution, remove it color[v] = 0 # If no color can be assigned to this vertex, return false return False def graph_coloring(graph, m): color = [0] * V # Call graph_coloring_util() for vertex 0 if not graph_coloring_util(graph, m, color, 0): print("Solution does not exist") return False # Print the solution print_solution(color) return True # Driver code if __name__ == "__main__": graph = [ [0, 1, 1, 1], [1, 0, 1, 0], [1, 1, 0, 1], [1, 0, 1, 0], ] m = 3 # Function call graph_coloring(graph, m) #This code is contrubting by Raja Ramakrishna ``` C# ```using System; class GraphColoringProblem { // Number of vertices in the graph const int V = 4; // A utility function to check if the current color assignment is safe for vertex v static bool IsSafe(int v, bool[,] graph, 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 static bool GraphColoringUtil(bool[,] graph, int m, int[] color, int v) { if (v == V) return true; for (int c = 1; c <= m; c++) { if (IsSafe(v, graph, color, c)) { color[v] = c; if (GraphColoringUtil(graph, m, color, v + 1)) return true; color[v] = 0; } } return false; } // This function solves the m Coloring problem using Backtracking static bool SolveGraphColoring(bool[,] graph, int m) { int[] color = new int[V]; for (int i = 0; i < V; i++) color[i] = 0; if (!GraphColoringUtil(graph, m, color, 0)) { Console.WriteLine("Solution does not exist"); return false; } PrintSolution(color); return true; } // A utility function to print solution static 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 static void Main(string[] args) { /* Create following graph and test whether it is 3 colorable (3)---(2) | / | | / | | / | (0)---(1) */ bool[,] graph = { { false, true, true, true }, { true, false, true, false }, { true, true, false, true }, { true, false, true, false } }; // Number of colors int m = 3; // Function call SolveGraphColoring(graph, m); } } // This code is contributed by shivamgupta310570 ``` Javascript ```// Equivalent JavaScript program for M Coloring problem using backtracking // Number of vertices in the graph const V = 4; // Function to print the solution function printSolution(color) { console.log("Solution Exists: Following are the assigned colors"); for (let i = 0; i < V; i++) { console.log(color[i] + " "); } console.log("\n"); } // Utility function to check if the current color assignment is safe for the vertex function isSafe(v, graph, color, c) { for (let i = 0; i < V; i++) { if (graph[v][i] && c == color[i]) { return false; } } return true; } // Recursive utility function to solve the M coloring problem function graphColoringUtil(graph, m, color, v) { // Base case: If all vertices are assigned a color, return true if (v === V) { return true; } // Consider the vertex v and try different colors for (let 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 the rest of the vertices if (graphColoringUtil(graph, m, color, v + 1)) { return true; } // If assigning color c doesn't lead to a solution, remove it color[v] = 0; } } // If no color can be assigned to this vertex, return false return false; } // Function to solve the M Coloring problem using backtracking function graphColoring(graph, m) { // Initialize all color values as 0 const color = new Array(V).fill(0); // Call graphColoringUtil() for vertex 0 if (!graphColoringUtil(graph, m, color, 0)) { console.log("Solution does not exist"); return false; } // Print the solution printSolution(color); return true; } // Driver code const graph = [ [0, 1, 1, 1], [1, 0, 1, 0], [1, 1, 0, 1], [1, 0, 1, 0], ]; const m = 3; // Function call graphColoring(graph, m); // This code is contributed by shivamgupta310570 ```

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

## Applications of Graph Coloring:

• Design a timetable.
• Sudoku.
• Register allocation in the compiler.
• Map coloring.