Maximum number of edges that N-vertex graph can have such that graph is Triangle free | Mantel’s Theorem
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.
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready. To complete your preparation from learning a language to DS Algo and many more, please refer Complete Interview Preparation Course.
Input: N = 4
Input: N = 3
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:
Time Complexity: O(1)