Java Program to Implement the Vizing’s Theorem

Vizing’s Theorem of graph theory states that every simple undirected graph has a chromatic index of one larger than the maximum degree ‘d’ of the graph. In simple meaning, the theorem states that the chromatic index can be either ‘d’ or ‘d’+1. 

Chromatic Index of a graph is the minimum number of colors required to color the edges of the graph such that any two edges that share the same vertex have different colors.

Examples:

Input:

Output:

Chromatic Index = 3

Edge from 1 to 2 : Color 1

Edge from 2 to 3 : Color 2

Edge from 3 to 4 : Color 1

Edge from 4 to 1 : Color 2

Edge from 1 to 3 : Color 3

Algorithm:

Below is the step-by-step approach of the algorithm:-

• Initialize the number of edges and the edge list.
• Color the graph according to the Vizing’s Theorem.
• Assign a color to an edge and check if any adjacent edges have the same color or not.
• If any adjacent edge has the same color, then increment the color to try the next color for that edge.
• Repeat till all the edges get it’s color according to the theorem.
• Once done print the maximum value of color for all the edges and the colors of every edge.

Implementation of the above approach: 

Java

// Java program to Implement
// Vizing's Theorem

import java.util.*;

public class chromaticIndex {

    // Function to find the chromatic index
    public void edgeColoring(int[][] edges, int e)
    {
        // Initialize edge to first
        // edge and color to color 1
        int i = 0, color = 1;

        // Repeat until all edges are done coloring
        while (i < e) {

            // Give the selected edge a color
            edges[i][2] = color;

            boolean flag = false;

            // Iterate through all others edges to check
            for (int j = 0; j < e; j++) {

                // Ignore if same edge
                if (j == i)
                    continue;

                // Check if one vertex is similar
                if ((edges[i][0] == edges[j][0])
                    || (edges[i][1] == edges[j][0])
                    || (edges[i][0] == edges[j][1])
                    || (edges[i][1] == edges[j][1])) {

                    // Check if color is similar
                    if (edges[i][2] == edges[j][2]) {

                        // Increment the color by 1
                        color++;
                        flag = true;
                        break;
                    }
                }
            }

            // If same color faced then repeat again
            if (flag == true) {
                continue;
            }

            // Or else proceed to a new vertex with color 1
            color = 1;
            i++;
        }

        // Check the maximum color from all the edge colors
        int maxColor = -1;
        for (i = 0; i < e; i++) {
            maxColor = Math.max(maxColor, edges[i][2]);
        }

        // Print the chromatic index
        System.out.println("Chromatic Index = " + maxColor);

        for (i = 0; i < e; i++)
        {
            System.out.println("Edge from " + edges[i][0]
                               + " to " + edges[i][1]
                               + " : Color " + edges[i][2]);
        }
    }

    // Driver code
    public static void main(String[] args)
    {

        // Number of edges
        int e = 5;

        // Edge list
        int[][] edges = new int[e][3];

        // Initialize all edge colors to 0
        for (int i = 0; i < e; i++) {
            edges[i][2] = -1;
        }

        // Edges
        edges[0][0] = 1;
        edges[0][1] = 2;

        edges[1][0] = 2;
        edges[1][1] = 3;

        edges[2][0] = 3;
        edges[2][1] = 4;

        edges[3][0] = 4;
        edges[3][1] = 1;

        edges[4][0] = 1;
        edges[4][1] = 3;

        // Run the function
        chromaticIndex c = new chromaticIndex();
        c.edgeColoring(edges, e);
    }
}

Output

Chromatic Index = 3
Edge from 1 to 2 : Color 1
Edge from 2 to 3 : Color 2
Edge from 3 to 4 : Color 1
Edge from 4 to 1 : Color 2
Edge from 1 to 3 : Color 3