What is Mantels Theorem Formula Proof and Examples
We will examine numerous crucial aspects of the Mantel Theorem. Mantel's theorem, established by Willem Mantel in 1907, is the basis of extremal graph theory and implies that any graph on n vertices without a triangle contains at most \[\dfrac{{{n^{\;2}}}}{4}\] edges. By splitting the collection of n vertices into two sets of size \[\left[ {\dfrac{n}{2}} \right]\] and \[\left[ {\dfrac{n}{2}} \right]\], one may build the entire bipartite graph between them, making this the best feasible scenario. There are \[\left[ {\dfrac{{{n^2}}}{4}} \right]\] edges and no triangles in this graph. Let's understand the basics by stating and proving the Mantel theorem.
What is Mantel's Theorem?
If there are no triangles in a graph G on n vertices, then the number of edges is at most \[\dfrac{{{n^{\;2}}}}{4}\].
Triangle Free Graph
Mantel's Theorem Proof
Consider G to have m edges. Let x and y represent two G vertices that are connected by an edge. We can see that d(x) + d(y) ≤ n if d(v) is the degree of a vertex v. This is since each vertex in graph G is only connected to one of x and y. Now observe that
\[\mathop \sum \limits_x {d^{\;2\;}}\left( x \right)\; = \mathop \sum \limits_{\;xy \in E\;} \left( {d\left( x \right)\; + \;d\left( y \right)} \right)\; \le \;mn\]
On the other hand, the Cauchy–Schwarz inequality suggests that because \[\mathop \sum \limits_x \;d\left( x \right)\; = \;2m,\]
\[\mathop \sum \limits_x \;{d^{\;2\;}}\left( x \right)\; \ge \;\dfrac{{{{\left( {\;\mathop \sum \limits_x \;d\left( x \right)} \right)}^2}\;}}{n}\; \ge \dfrac{{\;4{m^2}\;\;}}{n}\]
Therefore,
\[\dfrac{{4{m^2}}}{n} \le mn\]
and the result follows.
Applications of Mantel's Theorem
This theorem can be applied to determine how many edges an N-vertex graph can have while still being free of triangles.
Mantel's theorem graph theory is the basis for the external graph theory.
Mantel’s Theorem Examples
1. Show that there are \[\left[ {\dfrac{{{n^2}}}{4}} \right]\] edges in the n-vertex complete balanced bipartite graph. It implies that some graphs can reach the bound in Mantel's theorem.
Ans: A triangle subgraph does not exist in the n-vertex complete bipartite graph with class sizes \[\left[ {\dfrac{n}{2}} \right]\] and \[\left[ {\dfrac{n}{2}} \right]\], and the graph has exactly \[\left[ {\dfrac{n}{2}} \right]\left[ {\dfrac{n}{2}} \right]\; = \left[ {\dfrac{{{n^2}}}{4}} \right]\] edges. There are no graphs without triangles that have more edges than what is stated by Mantel's theorem.
2. Using an induction method that eliminates two nearby vertices, demonstrate Mantel's theorem.
Ans: Assume n > 2 and that the theorem's statement is true for smaller graphs because if n = 1, 2, we are finished. Let \[xy\;\]be an edge of graph G, which has n vertices and no triangles, and let G be the graph. Since the graph G - xy has n - 2 vertices and is manifestly triangle-free, it can be inferred that it has no more than \[\left[ {\dfrac{{{{\left( {n - 2} \right)}^2}}}{4}} \right]\] edges. At most n edges incident on the edge \(xy\) (otherwise, there is a triangle). Thus, G can only have
\[1\; + \;\left( {n - \;2} \right)\; + \dfrac{{\;{{\left( {n - \;2} \right)}^2}}}{4} = \dfrac{{{n^2}}}{4}\] edges.
3. If G is a triangle-free graph, then there are no common neighbours for neighbouring vertices. Therefore, we have \[d\left( x \right)\; + \;d\left( y \right)\; \le \;n\] n for an edge \[xy\;\] (remember to count the edge \[xy\;\] twice). Use it in the following equation to support your case that it is correct.
\[\mathop \sum \limits_{x \in V\left( G \right)} d{\left( x \right)^2}\; = \mathop \sum \limits_{xy \in E\left( G \right)} \left( {d\left( x \right)d\left( y \right)} \right)\;\] to obtain Mantel's theorem's proof.
Ans: First, we get
\[\mathop \sum \limits_{x \in V\left( G \right)} d{\left( x \right)^2}\; = \mathop \sum \limits_{xy \in E\left( G \right)} \left( {d\left( x \right)d\left( y \right)} \right)\; \le n\left| {E\left( G \right)} \right|\]
Utilising Cauchy-Schwarz,
\[\dfrac{1}{n}{\left( {\mathop \sum \limits_{x \in V\left( G \right)} d\left( x \right)} \right)^2}\; \le \mathop \sum \limits_{x \in V\left( G \right)} d{\left( x \right)^2}\]
The LHS is \[\dfrac{{\;1}}{n}(2|E\left( G \right){|^2}\;\] according to the Handshaking lemma. Thus,
\[\dfrac{{\;1}}{n}(2|E\left( G \right){|^2}\; \le n\left| {E\left( G \right)} \right|\]
Now, the theorem can be obtained by solving for |E(G)|.
Conclusion
In this article, we have discussed about Mantel's theorem and its proof. With Mantel's theorem, we can evaluate how many edges an N-vertex graph can have in total without having any triangles (which means there should not be any three edges A, B, and C in the graph such that A is connected to B, B is connected to C, and C is connected to A). The graph is not allowed to have many edges or a self-loop. To obtain a graph without triangles, it is necessary to remove almost half of the edges.
Important Points From the Theorem
Only if m = \[\dfrac{{{n^{\;2}}}}{4}\] is a graph G maximally triangle-free with regard to edges.
It is necessary to eliminate roughly half of the edges in order to produce a graph without triangles.
If a graph contains no cliques greater than three, then it satisfies Mantel's Theorem that it has few edges.
FAQs on Mantels Theorem in Graph Theory Explained
1. What is Mantel’s Theorem?
Mantel’s Theorem states that a simple graph with n vertices and no triangles has at most ⌊n²/4⌋ edges. This is a fundamental result in extremal graph theory.
- It applies to simple, undirected graphs.
- The graph must contain no 3-cycles (triangles).
- The maximum number of edges is achieved by a complete bipartite graph with parts as equal as possible.
2. What is the formula for the maximum number of edges in a triangle-free graph?
The maximum number of edges in a triangle-free graph with n vertices is ⌊n²/4⌋.
- If n is even: maximum edges = n²/4.
- If n is odd: maximum edges = (n² − 1)/4.
3. How do you prove Mantel’s Theorem?
Mantel’s Theorem is proven by showing that a triangle-free graph has the most edges when it is a balanced complete bipartite graph with ⌊n²/4⌋ edges.
- Assume a triangle-free graph with the maximum number of edges.
- Show that vertices can be partitioned into two sets with no edges inside each set.
- The number of edges is maximized when the partition sizes are as equal as possible.
- This gives a maximum of ⌊n²/4⌋ edges.
4. Can you give an example of Mantel’s Theorem?
Yes, for n = 6, the maximum number of edges in a triangle-free graph is 6²/4 = 9.
- Divide the 6 vertices into two sets of 3 and 3.
- Connect every vertex in one set to every vertex in the other.
- This forms the complete bipartite graph K₃,₃.
- Total edges = 3 × 3 = 9.
5. Why is the complete bipartite graph important in Mantel’s Theorem?
The complete bipartite graph is important because it achieves the maximum number of edges ⌊n²/4⌋ without forming a triangle.
- Bipartite graphs contain no odd cycles, including triangles.
- Making the two parts as equal as possible maximizes cross edges.
- This structure gives the extremal (maximum) configuration.
6. What is the connection between Mantel’s Theorem and Turán’s Theorem?
Mantel’s Theorem is the special case of Turán’s Theorem for r = 2, meaning it forbids triangles (K₃).
- Turán’s Theorem generalizes the result to forbid larger complete graphs K_{r+1}.
- When r = 2, the forbidden subgraph is K₃ (a triangle).
- The extremal graph is the balanced bipartite graph.
7. What happens if a graph has more than n²/4 edges?
If a graph with n vertices has more than n²/4 edges, it must contain at least one triangle.
- This is the contrapositive of Mantel’s Theorem.
- Exceeding the bound forces the existence of a 3-cycle.
- This result is often used in extremal graph arguments.
8. Does Mantel’s Theorem apply to directed or weighted graphs?
Mantel’s Theorem applies only to simple, undirected, unweighted graphs.
- No multiple edges.
- No loops.
- Edges are undirected.
9. How do you calculate ⌊n²/4⌋ for odd and even n?
To calculate ⌊n²/4⌋, compute n²/4 and round down to the nearest integer.
- If n = 8: 8²/4 = 64/4 = 16.
- If n = 7: 49/4 = 12.25 → ⌊12.25⌋ = 12.
10. What are common mistakes when applying Mantel’s Theorem?
A common mistake is forgetting that Mantel’s Theorem applies only to triangle-free simple graphs and gives a maximum of ⌊n²/4⌋ edges.
- Applying it to graphs that already contain triangles.
- Using n(n−1)/2 instead of n²/4.
- Forgetting to round down when n is odd.
- Ignoring that the extremal graph must be balanced bipartite.
