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.

Following are C++ and Java implementations 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

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.

**Analysis of Basic 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.

We will soon be discussing some interesting facts about chromatic number and graph coloring.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above