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.

Basic Greedy Coloring Algorithm:

1. Color first vertex with first color.
2. Do following for remaining V-1 vertices.
          a) 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.
C++ program
// 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;
}
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