# C++ Program to Find Chromatic Index of Cyclic Graphs

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This is a C++ program to find the chromatic index of cyclic graphs.

Problem Description

1. This algorithm finds the chromatic index of the given cyclic graph.
2. The chromatic index is the maximum number of color needed for the edge coloring of the given graph.

Problem Solution

1. This algorithm takes the input of the number of vertexes and the number of edges in the graph.
2. It takes the input of vertex pairs for the given number of edges.
3. It assigns a color to edges without assigning the same color to two adjacent edges.
4. It prints the number of colors used in edge coloring.
5. Exit.

Program/Source Code

C++ Program to find chromatic index of cyclic graphs.
This program is successfully run on Dev-C++ using TDM-GCC 4.9.2 MinGW compiler on a Windows system.

``` #include<iostream>

using namespace std;

// A function to color edges of the graph.
int EdgeColoring(int edge[], int e)
{
int i, col, j, max = -1;
// Loop to assign a valid color to every edge 'i'.
for(i = 0; i < e; i++)
{
col = 1;
flag:
// Assign a color and then check its validity.
edge[i] = col;
for(j = 0; j < e; j++)
{
if(j == i)
continue;

// Check the colors of the edges adjacent to the edge i.
if(edge[j] == edge[i] || edge[j] == edge[i] || edge[j] == edge[i] || edge[j] == edge[i])
{
// If the color matches then goto line 11 and assign next color to the edge and check again.
if(edge[j] == edge[i])
{
col++;
goto flag;
}
}
}
}

// Find the coloring index and return it, to main.
for(i = 0; i < e; i++)
{
if(max < edge[i])
max = edge[i];
}
return max;
}

int main()
{
int i, v, e, j, max = -1;

// take the input of the number of vertex and edges.
cout<<"Enter the number of vertexes of the graph: ";
cin>>v;
cout<<"Enter the number of edges of the graph: ";
cin>>e;
int edge[e];

// Take the input of the adjacent vertex pairs of the given graph.
for(i = 0; i < e; i++)
{
cout<<"\nEnter the vertex pair for edge "<<i+1;
cout<<"\nV(1): ";
cin>>edge[i];
cout<<"V(2): ";
cin>>edge[i];

edge[i] = -1;
}

// Print the chromatic index.
cout<<"\n\nThe chromatic index of the given graph is: "<<EdgeColoring(edge , e);

// Print the edge color for the edges of the given graph.
for(i = 0; i < e; i++)
cout<<"\nThe color of the edge between vertex V(1):"<<edge[i]<<" and V(2):"<<edge[i]<<" is: color"<<edge[i]<<".";

return 0;
}```
Program Explanation

1. Take the input of the number of vertexes ‘v’ and number of edges ‘e’.
2. Take the input of ‘e’ vertex pairs for the ‘e’ edges in the graph in edge[][].
3. Using EdgeColoring(), Color the graph edges.
4. Inside EdgeColoring(), assign color to current edge as col i.e. 1 initially.
5. If any of the adjacent edges have the same color then discard this color and go to flag again and try next color.
6. Return to main and print the chromatic index of the cyclic graph.
7. Print the color of each edge.
8. Exit.

Runtime Test Cases
```Case 1:
Enter the number of vertexes of the cyclic graph: 5
Enter the number of edges of the cyclic graph: 7

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

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

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

Enter the vertex pair for edge 4
V(1): 4
V(2): 5

Enter the vertex pair for edge 5
V(1): 1
V(2): 3

Enter the vertex pair for edge 6
V(1): 2
V(2): 4

Enter the vertex pair for edge 7
V(1): 5
V(2): 1

The chromatic index of the given graph is: 4
The color of the edge between vertex V(1):1 and V(2):2 is: color1.
The color of the edge between vertex V(1):2 and V(2):3 is: color2.
The color of the edge between vertex V(1):3 and V(2):4 is: color1.
The color of the edge between vertex V(1):4 and V(2):5 is: color2.
The color of the edge between vertex V(1):1 and V(2):3 is: color3.
The color of the edge between vertex V(1):2 and V(2):4 is: color3.
The color of the edge between vertex V(1):5 and V(2):1 is: color4.

Case 2:
Enter the number of vertexes of the cyclic graph: 4
Enter the number of edges of the cyclic graph: 4

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

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

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

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

The chromatic index of the given graph is: 2
The color of the edge between vertex V(1):1 and V(2):2 is: color1.
The color of the edge between vertex V(1):2 and V(2):3 is: color2.
The color of the edge between vertex V(1):3 and V(2):4 is: color1.
The color of the edge between vertex V(1):4 and V(2):1 is: color2.```

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