What is graph coloring problem?

Graph coloring problem involves assigning colors to certain elements of a graph subject to certain restrictions and constraints. This has found applications in numerous fields in computer science. For example:

• Sudoku: This game is a variation of Graph coloring problem where every cell denotes a node (or vertex) and there exists an edge between two nodes if the nodes are in same row or same column or same block.
• Geographical maps: There can be cases when no two adjacent cities/states can be assigned same color in the maps of countries or states. In this case, only four colors would be sufficient to color any map.

Vertex coloring is the most commonly encountered graph coloring problem. The problem states that given m colors, determine a way of coloring the vertices of a graph such that no two adjacent vertices are assigned same color.

Note: The smallest number of colors needed to color a graph G is called its chromatic number.
For example, the following undirected graph can be colored using minimum of 2 colors.
Hence the chromatic number of the graph is 2. 

Analysis

• Input:
  • A graph represented in 2D array format of size V * V where V is the number of vertices in graph and the 2D array is the adjacency matrix representation and value graph[i][j] is 1 if there is a direct edge from i to j, otherwise the value is 0.
  • An integer m that denotes the maximum number of colors which can be used in graph coloring.
  • Consider the input as shown in the image below:
    • The above graph can be represented as follows:

    graph[4][4] = {
                { 0, 1, 1, 1 },
                { 1, 0, 1, 0 },
                { 1, 1, 0, 1 },
                { 1, 0, 1, 0 },
            };
    

    • Consider m = 3.
• Output:
  • Return array color of size V that has numbers from 1 to m. Note that color[i] represents the color assigned to the i^th vertex.
  • Return false if the graph cannot be colored with m colors.
• Solution:
  • Naive Approach:
    • The brute force approach would be to generate all possible combinations (or configurations) of colors.
    • After generating a configuration, check if the adjacent vertices have the same colour or not. If the conditions are met, add the combination to the result and break the loop.
    • Since each node can be colored by using any of the m colors, the total number of possible color configurations are m^V. The complexity is exponential which is very huge.
    • Pseudo Code:

    public class InterviewBit{
        private static boolean isSafeToColor(int[][] graph, int[] color)
        {
            // check for every edge, if any adjacent edge has same color, return false
            for (int i = 0; i < V; i++)
                for (int j = i + 1; j < V; j++)
                    if (graph[i][j] == 1 && color[j] == color[i])
                        return false;
            return true;
        }
        
        private static void printColorArray(int[] color){
            System.out.println("Solution colors are: ")
            for(int i =0;i < color.length; i++){
                System.out.println(color[i]);
            }
        }
    
        /**
        * returns false if the m colors cannot be assigned, else, 
        * return true and print assignments of colors to all vertices.
        * Note: There may be more than one solutions
        * The function prints only one of the solutions.
        * */
        private static boolean graphColoring(int[][] graph, int m, int i, int[] color)
        {
            // if current index becomes end
            if (i == V) {
                // check whether its safe to color
                if (isSafeToColor(graph, color)) {
                    //If safe, print solution array and return true
                    printColorArray(color);
                    return true;
                }
                return false;
            }
    
            // Assign each color from 1 to m
            for (int j = 1; j <= m; j++) {
                color[i] = j;
    
                // Check for the rest vertices by recursion
                if (graphColoring(graph, m, i + 1, color))
                    return true;
    
                color[i] = 0;
            }
    
            return false;
        }
    }
    

    • Time Complexity: O(m^V).
    • Space Complexity: O(V) which is for storing the output array.
  • Using Backtracking:
    • By using the backtracking method, the main idea is to assign colors one by one to different vertices right from the first vertex (vertex 0).
    • Before color assignment, check if the adjacent vertices have same or different color by considering already assigned colors to the adjacent vertices.
      • If the color assignment does not violate any constraints, then we mark that color as part of the result. If color assignment is not possible then backtrack and return false.
    • Java solution:

        public class InterviewBit {
            final int V = 4;
            int[] color;
            
            boolean isSafeToColor(int v, int[][] graphMatrix, int[] color, int c)
            {
                //check for each edge
                for (int i = 0; i < V; i++)
                    if (graphMatrix[v][i] == 1 && c == color[i])
                        return false;
                return true;
            }
    
            boolean graphColorUtil(int[][] graphMatrix, int m, int[] color, int v)
            {
                // If all vertices are assigned a color then return true
                if (v == V)
                    return true;
    
                // Try different colors for vertex V
                for (int i = 1;i <= m; i++) 
                {
                    // check for assignment safety
                    if (isSafeToColor(v, graphMatrix, color, i))
                    {
                        color[v] =i;
                        // recursion for checking other vertices
                        if (graphColorUtil(graphMatrix, m, color, v + 1))
                            return true;
                        // if color doesnt lead to solution
                        color[v] = 0;
                    }
                }
                // If no color can be assigned to  vertex
                return false;
            }
    
            void printColoringSolution(int color[])
            {
                System.out.println("Color schema for vertices are: ");
                for (int i = 0; i < V; i++)
                    System.out.println(color[i]);
            }
    
            /**
            * It returns false if the m colors cannot be assigned 
            * otherwise return true and 
            * print color assignment result to all vertices.
            */
            boolean graphColoring(int[][] graphMatrix, int m)
            {
                // Initialize all color values as 0. 
                color = new int[V];
                Arrays.fill(color,0);
    
                // Call graphColorUtil() for vertex 0
                if (!graphColorUtil(graphMatrix, m, color, 0)) 
                {
                    System.out.println(
                        "Color schema not possible");
                    return false;
                }
    
                // Print the color schema of vertices
                printColoringSolution(color);
                return true;
            }
    
            
            // Main driver program
            public static void main(String args[])
            {
                InterviewBit interviewBit 
                       = new InterviewBit();
                
                int graphMatrix[][] = {
                    { 0, 1, 1, 1 },
                    { 1, 0, 1, 0 },
                    { 1, 1, 0, 1 },
                    { 1, 0, 1, 0 },
                };
                int m = 3; // Number of colors
                interviewBit.graphColoring(graphMatrix, m);
            }
        }
    

    • Time Complexity: O(m^V). Since backtracking is also a kind of brute force approach, there would be total O(m^V) possible color combinations. It is to be noted that the upperbound time complexity remains the same but the average time taken will be less due to the refined approach.
    • Space Complexity: O(V) for storing the output array in O(V) space