Given a number N which is the number of nodes in a graph, the task is to find the maximum number of edges that N-vertex graph can have such that graph is triangle-free (which means there should not be any three edges A, B, C in the graph such that A is connected to B, B is connected to C and C is connected to A). The graph cannot contain a self-loop or multi edges.
Examples:
Input: N = 4
Output: 4
Explanation:
Input: N = 3
Output: 2
Explanation:
If there are three edges in 3-vertex graph then it will have a triangle.
Approach: This Problem can be solved using Mantel’s Theorem which states that the maximum number of edges in a graph without containing any triangle is floor(n2/4). In other words, one must delete nearly half of the edges to obtain a triangle-free graph.
How Mantel’s Theorem Works ?
For any Graph, such that the graph is Triangle free then for any vertex Z can only be connected to any of one vertex from x and y, i.e. For any edge connected between x and y, d(x) + d(y) ? N, where d(x) and d(y) is the degree of the vertex x and y.
- Then, the Degree of all vertex –
- By Cauchy-Schwarz inequality –
- Therefore, 4m2 / n ? mn, which implies m ? n2 / 4
Below is the implementation of above approach:
// C++ implementation to find the maximum // number of edges for triangle free graph #include <bits/stdc++.h> using namespace std;
// Function to find the maximum number of // edges in a N-vertex graph. int solve( int n)
{ // According to the Mantel's theorem
// the maximum number of edges will be
// floor of [(n^2)/4]
int ans = (n * n / 4);
return ans;
} // Driver Function int main()
{ int n = 10;
cout << solve(n) << endl;
return 0;
} |
// Java implementation to find the maximum // number of edges for triangle free graph class GFG
{ // Function to find the maximum number of
// edges in a N-vertex graph.
public static int solve( int n)
{
// According to the Mantel's theorem
// the maximum number of edges will be
// floor of [(n^2)/4]
int ans = (n * n / 4 );
return ans;
}
// Driver code
public static void main(String args[])
{
int n = 10 ;
System.out.println(solve(n));
}
} // This code is contributed by divyamohan123 |
// C# implementation to find the maximum // number of edges for triangle free graph using System;
class GFG
{ // Function to find the maximum number of
// edges in a N-vertex graph.
public static int solve( int n)
{
// According to the Mantel's theorem
// the maximum number of edges will be
// floor of [(n^2)/4]
int ans = (n * n / 4);
return ans;
}
// Driver code
public static void Main()
{
int n = 10;
Console.WriteLine(solve(n));
}
} // This code is contributed by AnkitRai01 |
# Python3 implementation to find the maximum # number of edges for triangle free graph # Function to find the maximum number of # edges in a N-vertex graph. def solve(n):
# According to the Mantel's theorem
# the maximum number of edges will be
# floor of [(n^2)/4]
ans = (n * n / / 4 )
return ans
# Driver Function if __name__ = = '__main__' :
n = 10
print (solve(n))
# This code is contributed by mohit kumar 29 |
<script> // Javascript implementation to find the maximum // number of edges for triangle free graph // Function to find the maximum number of // edges in a N-vertex graph. function solve(n)
{ // According to the Mantel's theorem
// the maximum number of edges will be
// floor of [(n^2)/4]
var ans = (n * n / 4);
return ans;
} // Driver code var n = 10;
document.write(solve(n)); // This code is contributed by aashish1995 </script> |
25
Time Complexity: O(1)