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++
// 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 Functionint main(){ int n = 10; cout << solve(n) << endl; return 0;} |
Java
// 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#
// 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
# 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 Functionif __name__ == '__main__': n = 10 print(solve(n))# This code is contributed by mohit kumar 29 |
Javascript
<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 codevar n = 10;document.write(solve(n));// This code is contributed by aashish1995</script> |
25
Time Complexity: O(1)
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