Coloring a Cycle Graph

Cycle:- cycle is a path of edges and vertices wherein a vertex is reachable from itself. or in other words, it is a Closed walk. 

Even Cycle:- In which Even number of vertices is present is known as Even Cycle. 

Odd Cycle:- In which Odd number of Vertices is present is known as Odd Cycle.
Given the number of vertices in a Cyclic Graph. The task is to determine the Number of colors required to color the graph so that No two Adjacent vertices have the same color.

Approach: 

If the no. of vertices is Even then it is Even Cycle and to color such graph we require 2 colors. 
If the no. of vertices is Odd then it is Odd Cycle and to color such graph we require 3 colors. 

Examples:  

Input : vertices = 3
Output : No. of colors require is: 3

Input : vertices = 4
Output : No. of colors require is: 2

Example 1: Even Cycle: Number of vertices = 4 

Color required = 2  

Example 2: Odd Cycle: Number of vertices = 5  

Color required = 3  

Implementation:

// CPP program to find number of colors
// required to color a cycle graph
#include <bits/stdc++.h>
using namespace std;

// Function that calculates Color
// require to color a graph.
int Color(int vertices)
{
    int result = 0;

    // Check if number of vertices
    // is odd or even.
    // If number of vertices is even
    // then color require is 2 otherwise 3
    if (vertices % 2 == 0)
        result = 2;
    else
        result = 3;

    return result;
}

// Driver code
int main()
{
    int vertices = 3;
    cout << "No. of colors require is: " << Color(vertices);
    return 0;
}

// Java program to find number of colors
// required to color a cycle graph
import java.io.*; 

class GFG { 
  
    // Function that calculates Color 
    // require to color a graph.  
    static int Color(int vertices) 
    { 
        int result = 0; 
  
        // Check if number of vertices 
        // is odd or even. 
        // If number of vertices is even 
        // then color require is 2 otherwise 3 
        if (vertices % 2 == 0) 
            result = 2; 
        else
            result = 3; 
      
        return result; 
    }  
      
    // Driver program to test above function 
    public static void main (String[] args) 
    { 
        int vertices = 3; 
        
        System.out.println("No. of colors require is: " + Color(vertices));
          
    } 
} 

// this code is contributed by Naman_Garg

# Naive Python3 Program to 
# find the number of colors
# required to color a cycle graph  
  
# Function to find Color required.
def Color(vertices):  
  
    result = 0 
  
    # Check if number of vertices 
    # is odd or even. 
    # If number of vertices is even 
    # then color require is 2 otherwise 3 
    if (vertices % 2 == 0):
        result = 2
    else:
        result = 3 
  
    return result
  
# Driver Code 
if __name__=='__main__':
    vertices = 3
    print ("No. of colors require is:",Color(vertices))

# this code is contributed by Naman_Garg

// C# program to find number of colors
// required to color a cycle graph
using System; 

class GFG
{ 

// Function that calculates Color 
// require to color a graph. 
static int Color(int vertices) 
{ 
    int result = 0; 

    // Check if number of vertices 
    // is odd or even. 
    // If number of vertices is even 
    // then color require is 2 otherwise 3 
    if (vertices % 2 == 0) 
        result = 2; 
    else
        result = 3; 

    return result; 
} 

// Driver Code
public static void Main () 
{ 
    int vertices = 3; 
    
    Console.WriteLine("No. of colors required is: " + 
                                   Color(vertices));
} 
} 

// This code is contributed by anuj_67

<?php
// PHP program to find number of colors
// required to color a cycle graph

// Function that calculates Color
// require to color a graph.
function Color($vertices)
{
    $result = 0;

    // Check if number of vertices
    // is odd or even.
    // If number of vertices is even
    // then color require is 2 otherwise 3
    if ($vertices % 2 == 0)
        $result = 2;
    else
        $result = 3;

    return $result;
}

// Driver code
$vertices = 3;
echo "No. of colors required is: " ,
                  Color($vertices);

// This code is contributed 
// by anuj_67
?>

<script>

// Javascript program to find number of colors
// required to color a cycle graph

// Function that calculates Color
// require to color a graph.
function Color(vertices)
{
    var result = 0;

    // Check if number of vertices
    // is odd or even.
    // If number of vertices is even
    // then color require is 2 otherwise 3
    if (vertices % 2 == 0)
        result = 2;
    else
        result = 3;

    return result;
}

// Driver code
var vertices = 3;
document.write("No. of colors require is: " +
               Color(vertices));
               
// This code is contributed by itsok

</script>

No. of colors require is: 3

Complexity Analysis:

• Time Complexity: O(1)
• Space Complexity: O(1)