Graph Coloring
Graph coloring is the assignment of labels, known as colors, to the vertices of a graph such that no two adjacent vertices have the same color.
Graph coloring has a wide variety of applications, such as scheduling tasks with dependencies, register allocation in compilers, and solving Sudoku puzzles.
In the context of graph theory, a graph can be represented as an adjacency matrix. An adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not.
To implement graph coloring, we can use a backtracking algorithm. The idea is to recursively try different colors for each vertex until a valid coloring is found.
Here's an example of Java code that performs graph coloring:
1// Example code for graph coloring
2int[][] graph = {
3 {0, 1, 1, 0, 0},
4 {1, 0, 1, 1, 0},
5 {1, 1, 0, 1, 1},
6 {0, 1, 1, 0, 1},
7 {0, 0, 1, 1, 0}
8};
9
10int numOfNodes = graph.length;
11
12int[] colors = new int[numOfNodes];
13// Initialize all nodes with no color
14for (int i = 0; i < numOfNodes; i++) {
15 colors[i] = 0;
16}
17
18int numOfColors = 3; // Number of available colors
19
20// Call the graph coloring function
21if (graphColoring(graph, numOfColors, colors, 0)) {
22 System.out.println("Graph can be colored with " + numOfColors + " colors.");
23 System.out.println("Colors assigned to nodes:");
24 for (int i = 0; i < numOfNodes; i++) {
25 System.out.println("Node " + i + ": Color " + colors[i]);
26 }
27} else {
28 System.out.println("Graph cannot be colored with " + numOfColors + " colors.");
29}
30
31private static boolean graphColoring(int[][] graph, int numOfColors, int[] colors, int node) {
32 int numOfNodes = graph.length;
33
34 // Base case: All nodes have been assigned a color
35 if (node == numOfNodes) {
36 return true;
37 }
38
39 // Try different colors for the current node
40 for (int color = 1; color <= numOfColors; color++) {
41 if (isSafe(graph, colors, node, color)) {
42 // Assign the current color
43 colors[node] = color;
44
45 // Recursively color the remaining nodes
46 if (graphColoring(graph, numOfColors, colors, node + 1)) {
47 return true;
48 }
49
50 // Reset the color if it didn't lead to a solution
51 colors[node] = 0;
52 }
53 }
54
55 return false;
56}
57
58private static boolean isSafe(int[][] graph, int[] colors, int node, int color) {
59 int numOfNodes = graph.length;
60
61 // Check if any adjacent nodes have the same color
62 for (int adjacentNode = 0; adjacentNode < numOfNodes; adjacentNode++) {
63 if (graph[node][adjacentNode] == 1 && colors[adjacentNode] == color) {
64 return false;
65 }
66 }
67
68 return true;
69}xxxxxxxxxx}class Main { public static void main(String[] args) { // replace with your Java logic here // Example code for graph coloring int[][] graph = { {0, 1, 1, 0, 0}, {1, 0, 1, 1, 0}, {1, 1, 0, 1, 1}, {0, 1, 1, 0, 1}, {0, 0, 1, 1, 0} }; int numOfNodes = graph.length; int[] colors = new int[numOfNodes]; // Initialize all nodes with no color for (int i = 0; i < numOfNodes; i++) { colors[i] = 0; } int numOfColors = 3; // Number of available colors // Call the graph coloring function if (graphColoring(graph, numOfColors, colors, 0)) { System.out.println("Graph can be colored with " + numOfColors + " colors."); System.out.println("Colors assigned to nodes:"); for (int i = 0; i < numOfNodes; i++) { System.out.println("Node " + i + ": Color " + colors[i]); }