C++ Program to Implement Vizing's Theorem

Vizing’s theorem state that the chromatic index of simple graph can be either maxdegree or maxdegree+1. Here, chromatic index means maximum color needed for the edge coloring of the graph.

This is a C++ program to implement the Vizing’s Theorem.

Algorithm

Begin
   Take the number of vertices and edges as input.
   Take the vertex pair for the edges.
   function EdgeColor() : Color the graph edges.
      1) Assign color to current edge as c i.e. 1 initially.
      2) If the same color is occupied by any of the adjacent edges,
      then discard this color and go to flag again and try with the next color.
End

Example

#include<iostream>
using namespace std;
void EdgeColor(int ed[][3], int e) {
   int i, c, j;
   for(i = 0; i < e; i++) {
      c = 1; //Assign color to current edge 1 initially.
      // If the same color is occupied by any of the adjacent edges,
      // then discard this color and go to flag again and try next color.
      flag:
      ed[i][2] = c;
      for(j = 0; j < e; j++) {
         if(j == i)
         continue;
         if(ed[j][0] == ed[i][0] || ed[j][0] == ed[i][1] || ed[j][1] == ed[i][0] || ed[j][1] == ed[i][1]) {
            if(ed[j][2] == ed[i][2]) {
               c++;
               goto flag;
            }
         }
      }
   }
}
int main() {
   int i, n, e, j, max = -1;
   cout<<"Enter the number of vertices of the graph: ";
   cin>>n;
   cout<<"Enter the number of edges of the graph: ";
   cin>>e;
   int ed[e][3], deg[n+1] = {0};
   for(i = 0; i < e; i++) {
      cout<<"\nEnter the vertex pair for edge "<<i+1;
      cout<<"\nN(1): ";
      cin>>ed[i][0];
      cout<<"N(2): ";
      cin>>ed[i][1];
      //calculate the degree of each vertex
      ed[i][2] = -1;
      deg[ed[i][0]]++;
      deg[ed[i][1]]++;
   }
   //find out the maximum degree.
   for(i = 1; i <= n; i++) {
      if(max < deg[i])
      max = deg[i];
   }
   EdgeColor(ed , e);
   cout<<"\nAccording to Vizing's theorem this graph can use maximum of "<<max+1<<" colors to generate a valid edge coloring.\n\n";
   for(i = 0; i < e; i++)
   cout<<"\nThe color of the edge between vertex N(1):"<<ed[i][0]<<" and N(2):"<<ed[i][1]<<" is: color"<<ed[i][2]<<".";
}

Output

Enter the number of vertices of the graph: 4
Enter the number of edges of the graph: 3

Enter the vertex pair for edge 1
N(1): 1
N(2): 2

Enter the vertex pair for edge 2
N(1): 3
N(2): 2

Enter the vertex pair for edge 3
N(1): 4
N(2): 1

According to Vizing's theorem this graph can use maximum of 3 colors to generate a valid edge coloring.

The color of the edge between vertex N(1):1 and N(2):2 is: color1.
The color of the edge between vertex N(1):3 and N(2):2 is: color2.
The color of the edge between vertex N(1):4 and N(2):1 is: color2.