Graph Coloring Using Greedy Algorithm

We introduced graph coloring and applications in previous post. As discussed in the previous post, graph coloring is widely used. Unfortunately, there is no efficient algorithm available for coloring a graph with minimum number of colors as the problem is a known NP Complete problem. There are approximate algorithms to solve the problem though. Following is the basic Greedy Algorithm to assign colors. It doesn’t guarantee to use minimum colors, but it guarantees an upper bound on the number of colors. The basic algorithm never uses more than d+1 colors where d is the maximum degree of a vertex in the given graph.

Graph Coloring Using Greedy Algorithm:

• Color first vertex with first color. 
• Do following for remaining V-1 vertices. 
  • Consider the currently picked vertex and color it with the lowest numbered color that has not been used on any previously colored vertices adjacent to it. If all previously used colors appear on vertices adjacent to v, assign a new color to it.

Below is the implementation of the above Greedy Algorithm.

C++

// A C++ program to implement greedy algorithm for graph coloring
#include <iostream>
#include <list>
using namespace std;

// A class that represents an undirected graph
class Graph
{
    int V;    // No. of vertices
    list<int> *adj;    // A dynamic array of adjacency lists
public:
    // Constructor and destructor
    Graph(int V)   { this->V = V; adj = new list<int>[V]; }
    ~Graph()       { delete [] adj; }

    // function to add an edge to graph
    void addEdge(int v, int w);

    // Prints greedy coloring of the vertices
    void greedyColoring();
};

void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w);
    adj[w].push_back(v);  // Note: the graph is undirected
}

// Assigns colors (starting from 0) to all vertices and prints
// the assignment of colors
void Graph::greedyColoring()
{
    int result[V];

    // Assign the first color to first vertex
    result[0]  = 0;

    // Initialize remaining V-1 vertices as unassigned
    for (int u = 1; u < V; u++)
        result[u] = -1;  // no color is assigned to u

    // A temporary array to store the available colors. True
    // value of available[cr] would mean that the color cr is
    // assigned to one of its adjacent vertices
    bool available[V];
    for (int cr = 0; cr < V; cr++)
        available[cr] = false;

    // Assign colors to remaining V-1 vertices
    for (int u = 1; u < V; u++)
    {
        // Process all adjacent vertices and flag their colors
        // as unavailable
        list<int>::iterator i;
        for (i = adj[u].begin(); i != adj[u].end(); ++i)
            if (result[*i] != -1)
                available[result[*i]] = true;

        // Find the first available color
        int cr;
        for (cr = 0; cr < V; cr++)
            if (available[cr] == false)
                break;

        result[u] = cr; // Assign the found color

        // Reset the values back to false for the next iteration
        for (i = adj[u].begin(); i != adj[u].end(); ++i)
            if (result[*i] != -1)
                available[result[*i]] = false;
    }

    // print the result
    for (int u = 0; u < V; u++)
        cout << "Vertex " << u << " --->  Color "
             << result[u] << endl;
}

// Driver program to test above function
int main()
{
    Graph g1(5);
    g1.addEdge(0, 1);
    g1.addEdge(0, 2);
    g1.addEdge(1, 2);
    g1.addEdge(1, 3);
    g1.addEdge(2, 3);
    g1.addEdge(3, 4);
    cout << "Coloring of graph 1 \n";
    g1.greedyColoring();

    Graph g2(5);
    g2.addEdge(0, 1);
    g2.addEdge(0, 2);
    g2.addEdge(1, 2);
    g2.addEdge(1, 4);
    g2.addEdge(2, 4);
    g2.addEdge(4, 3);
    cout << "\nColoring of graph 2 \n";
    g2.greedyColoring();

    return 0;
}

Java

// A Java program to implement greedy algorithm for graph coloring
import java.io.*;
import java.util.*;
import java.util.LinkedList;

// This class represents an undirected graph using adjacency list
class Graph
{
    private int V;   // No. of vertices
    private LinkedList<Integer> adj[]; //Adjacency List

    //Constructor
    Graph(int v)
    {
        V = v;
        adj = new LinkedList[v];
        for (int i=0; i<v; ++i)
            adj[i] = new LinkedList();
    }

    //Function to add an edge into the graph
    void addEdge(int v,int w)
    {
        adj[v].add(w);
        adj[w].add(v); //Graph is undirected
    }

    // Assigns colors (starting from 0) to all vertices and
    // prints the assignment of colors
    void greedyColoring()
    {
        int result[] = new int[V];

        // Initialize all vertices as unassigned
        Arrays.fill(result, -1);

        // Assign the first color to first vertex
        result[0]  = 0;

        // A temporary array to store the available colors. False
        // value of available[cr] would mean that the color cr is
        // assigned to one of its adjacent vertices
        boolean available[] = new boolean[V];

        // Initially, all colors are available
        Arrays.fill(available, true);

        // Assign colors to remaining V-1 vertices
        for (int u = 1; u < V; u++)
        {
            // Process all adjacent vertices and flag their colors
            // as unavailable
            Iterator<Integer> it = adj[u].iterator() ;
            while (it.hasNext())
            {
                int i = it.next();
                if (result[i] != -1)
                    available[result[i]] = false;
            }

            // Find the first available color
            int cr;
            for (cr = 0; cr < V; cr++){
                if (available[cr])
                    break;
            }

            result[u] = cr; // Assign the found color

            // Reset the values back to true for the next iteration
            Arrays.fill(available, true);
        }

        // print the result
        for (int u = 0; u < V; u++)
            System.out.println("Vertex " + u + " --->  Color "
                                + result[u]);
    }

    // Driver method
    public static void main(String args[])
    {
        Graph g1 = new Graph(5);
        g1.addEdge(0, 1);
        g1.addEdge(0, 2);
        g1.addEdge(1, 2);
        g1.addEdge(1, 3);
        g1.addEdge(2, 3);
        g1.addEdge(3, 4);
        System.out.println("Coloring of graph 1");
        g1.greedyColoring();

        System.out.println();
        Graph g2 = new Graph(5);
        g2.addEdge(0, 1);
        g2.addEdge(0, 2);
        g2.addEdge(1, 2);
        g2.addEdge(1, 4);
        g2.addEdge(2, 4);
        g2.addEdge(4, 3);
        System.out.println("Coloring of graph 2 ");
        g2.greedyColoring();
    }
}
// This code is contributed by Aakash Hasija

Python3

# Python3 program to implement greedy
# algorithm for graph coloring

def addEdge(adj, v, w):

    adj[v].append(w)

    # Note: the graph is undirected
    adj[w].append(v)
    return adj

# Assigns colors (starting from 0) to all
# vertices and prints the assignment of colors
def greedyColoring(adj, V):

    result = [-1] * V

    # Assign the first color to first vertex
    result[0] = 0;


    # A temporary array to store the available colors.
    # True value of available[cr] would mean that the
    # color cr is assigned to one of its adjacent vertices
    available = [False] * V

    # Assign colors to remaining V-1 vertices
    for u in range(1, V):

        # Process all adjacent vertices and
        # flag their colors as unavailable
        for i in adj[u]:
            if (result[i] != -1):
                available[result[i]] = True

        # Find the first available color
        cr = 0
        while cr < V:
            if (available[cr] == False):
                break

            cr += 1

        # Assign the found color
        result[u] = cr

        # Reset the values back to false
        # for the next iteration
        for i in adj[u]:
            if (result[i] != -1):
                available[result[i]] = False

    # Print the result
    for u in range(V):
        print("Vertex", u, " --->  Color", result[u])

# Driver Code
if __name__ == '__main__':

    g1 = [[] for i in range(5)]
    g1 = addEdge(g1, 0, 1)
    g1 = addEdge(g1, 0, 2)
    g1 = addEdge(g1, 1, 2)
    g1 = addEdge(g1, 1, 3)
    g1 = addEdge(g1, 2, 3)
    g1 = addEdge(g1, 3, 4)
    print("Coloring of graph 1 ")
    greedyColoring(g1, 5)

    g2 = [[] for i in range(5)]
    g2 = addEdge(g2, 0, 1)
    g2 = addEdge(g2, 0, 2)
    g2 = addEdge(g2, 1, 2)
    g2 = addEdge(g2, 1, 4)
    g2 = addEdge(g2, 2, 4)
    g2 = addEdge(g2, 4, 3)
    print("\nColoring of graph 2")
    greedyColoring(g2, 5)

# This code is contributed by mohit kumar 29

C#

// A C# program to implement greedy algorithm for graph coloring
using System;
using System.Collections.Generic;

// This class represents an undirected graph using adjacency list
class Graph
{
  private int V; // No. of vertices
  private List<int>[] adj; //Adjacency List

  //Constructor
  public Graph(int v)
  {
    V = v;
    adj = new List<int>[v];
    for (int i=0; i<v; ++i)
      adj[i] = new List<int>();
  }

  //Function to add an edge into the graph
  public void addEdge(int v,int w)
  {
    adj[v].Add(w);
    adj[w].Add(v); //Graph is undirected
  }

  // Assigns colors (starting from 0) to all vertices and
  // prints the assignment of colors
  public void greedyColoring()
  {
    int[] result = new int[V];

    // Initialize all vertices as unassigned
    for(int i = 0; i < V; i++)
    {
      result[i] = -1;
    }

    // Assign the first color to first vertex
    result[0]  = 0;

    // A temporary array to store the available colors. False
    // value of available[cr] would mean that the color cr is
    // assigned to one of its adjacent vertices
    bool[] available = new bool[V];

    // Initially, all colors are available
    for(int i = 0; i < V; i++)
    {
      available[i] = true;
    }

    // Assign colors to remaining V-1 vertices
    for (int u = 1; u < V; u++)
    {
      // Process all adjacent vertices and flag their colors
      // as unavailable
      foreach (int i in adj[u])
      {
        if (result[i] != -1)
          available[result[i]] = false;
      }

      // Find the first available color
      int cr;
      for (cr = 0; cr < V; cr++)
      {
        if (available[cr])
          break;
      }

      result[u] = cr; // Assign the found color

      // Reset the values back to true for the next iteration
      for(int i = 0; i < V; i++)
      {
        available[i] = true;
      }
    }

    // print the result
    for (int u = 0; u < V; u++)
      Console.WriteLine("Vertex " + u + " --->  Color " + result[u]);
  }

  // Driver method
  public static void Main(string[] args)
  {
    Graph g1 = new Graph(5);
    g1.addEdge(0, 1);
    g1.addEdge(0, 2);
    g1.addEdge(1, 2);
    g1.addEdge(1, 3);
    g1.addEdge(2, 3);
    g1.addEdge(3, 4);
    Console.WriteLine("Coloring of graph 1");
    g1.greedyColoring();


    Graph g2 = new Graph(5);
    g2.addEdge(0, 1);
    g2.addEdge(0, 2);
    g2.addEdge(1, 2);
    g2.addEdge(1, 4);
    g2.addEdge(2, 4);
    g2.addEdge(4, 3);
    Console.WriteLine("\nColoring of graph 2");
    g2.greedyColoring();

  }
}

Javascript

<script>

// Javascript program to implement greedy
// algorithm for graph coloring

// This class represents a directed graph
// using adjacency list representation
class Graph{

// Constructor
constructor(v)
{
    this.V = v;
    this.adj = new Array(v);

    for(let i = 0; i < v; ++i)
        this.adj[i] = [];

    this.Time = 0;
}

// Function to add an edge into the graph
addEdge(v,w)
{
    this.adj[v].push(w);

    // Graph is undirected
    this.adj[w].push(v);
}

// Assigns colors (starting from 0) to all
// vertices and prints the assignment of colors
greedyColoring()
{
    let result = new Array(this.V);

    // Initialize all vertices as unassigned
    for(let i = 0; i < this.V; i++)
        result[i] = -1;

    // Assign the first color to first vertex
    result[0]  = 0;

    // A temporary array to store the available
    // colors. False value of available[cr] would
    // mean that the color cr is assigned to one
    // of its adjacent vertices
    let available = new Array(this.V);

    // Initially, all colors are available
    for(let i = 0; i < this.V; i++)
        available[i] = true;

    // Assign colors to remaining V-1 vertices
    for(let u = 1; u < this.V; u++)
    {

        // Process all adjacent vertices and
        // flag their colors as unavailable
        for(let it of this.adj[u])
        {
            let i = it;
            if (result[i] != -1)
                available[result[i]] = false;
        }

        // Find the first available color
        let cr;
        for(cr = 0; cr < this.V; cr++)
        {
            if (available[cr])
                break;
        }

        // Assign the found color
        result[u] = cr;

        // Reset the values back to true
        // for the next iteration
        for(let i = 0; i < this.V; i++)
            available[i] = true;
    }

    // print the result
    for(let u = 0; u < this.V; u++)
        document.write("Vertex " + u + " --->  Color " +
                       result[u] + "<br>");
}
}

// Driver code
let g1 = new Graph(5);
g1.addEdge(0, 1);
g1.addEdge(0, 2);
g1.addEdge(1, 2);
g1.addEdge(1, 3);
g1.addEdge(2, 3);
g1.addEdge(3, 4);
document.write("Coloring of graph 1<br>");
g1.greedyColoring();

document.write("<br>");
let g2 = new Graph(5);
g2.addEdge(0, 1);
g2.addEdge(0, 2);
g2.addEdge(1, 2);
g2.addEdge(1, 4);
g2.addEdge(2, 4);
g2.addEdge(4, 3);
document.write("Coloring of graph 2<br> ");
g2.greedyColoring();

// This code is contributed by avanitrachhadiya2155

</script>

Output: 

Coloring of graph 1
Vertex 0 --->  Color 0
Vertex 1 --->  Color 1
Vertex 2 --->  Color 2
Vertex 3 --->  Color 0
Vertex 4 --->  Color 1

Coloring of graph 2
Vertex 0 --->  Color 0
Vertex 1 --->  Color 1
Vertex 2 --->  Color 2
Vertex 3 --->  Color 0
Vertex 4 --->  Color 3

Time Complexity: O(V^2 + E), in worst case.
Auxiliary Space: O(1), as we are not using any extra space.

Analysis of Graph Coloring Using Greedy Algorithm:

The above algorithm doesn’t always use minimum number of colors. Also, the number of colors used sometime depend on the order in which vertices are processed. For example, consider the following two graphs. Note that in graph on right side, vertices 3 and 4 are swapped. If we consider the vertices 0, 1, 2, 3, 4 in left graph, we can color the graph using 3 colors. But if we consider the vertices 0, 1, 2, 3, 4 in right graph, we need 4 colors. 

So the order in which the vertices are picked is important. Many people have suggested different ways to find an ordering that work better than the basic algorithm on average. The most common is Welsh–Powell Algorithm which considers vertices in descending order of degrees. 

How does the basic algorithm guarantee an upper bound of d+1?

Here d is the maximum degree in the given graph. Since d is maximum degree, a vertex cannot be attached to more than d vertices. When we color a vertex, at most d colors could have already been used by its adjacent. To color this vertex, we need to pick the smallest numbered color that is not used by the adjacent vertices. If colors are numbered like 1, 2, …., then the value of such smallest number must be between 1 to d+1 (Note that d numbers are already picked by adjacent vertices). 
This can also be proved using induction. See this video lecture for proof.