How Does Mantel's Theorem Apply in Graph Theory Problems?
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 Mantel's Theorem Explained: Proof, Examples & Uses
1. What is Mantel's Theorem in the context of graph theory?
Mantel's Theorem is a foundational result in extremal graph theory. It states that the maximum number of edges a graph with 'n' vertices can have without containing a triangle (a cycle of length three, or K₃) is ⌊n²/4⌋. If a simple graph on 'n' vertices has more than ⌊n²/4⌋ edges, it is guaranteed to have at least one triangle.
2. Can you provide a simple example of how to apply Mantel's Theorem?
Certainly. Consider a graph with n = 5 vertices. According to Mantel's Theorem, the maximum number of edges it can have without a triangle is ⌊5²/4⌋ = ⌊6.25⌋ = 6 edges. A complete bipartite graph K₂,₃ is an example of such a graph with 5 vertices and 2 * 3 = 6 edges, and it contains no triangles. However, if you add just one more edge (for a total of 7 edges), Mantel's Theorem guarantees that a triangle must be formed, regardless of how the edges are arranged.
3. What is the importance of a 'triangle-free' graph in Mantel's Theorem?
A triangle-free graph is the central condition of the theorem. Mantel's Theorem explores the limit of how 'dense' a graph can become before this condition is violated. A triangle represents the smallest possible cycle in a simple graph and often signifies a basic cluster or a tightly-knit group. The theorem provides a precise threshold (⌊n²/4⌋ edges) that separates all triangle-free graphs from graphs that are dense enough to necessarily contain such a cluster.
4. What is a high-level explanation of the proof of Mantel's Theorem?
A common proof method involves considering the degrees of the vertices. Here is a simplified overview:
- Assume a graph G is triangle-free.
- Pick an edge {u, v}. Since there are no triangles, the set of neighbours of u, N(u), and the set of neighbours of v, N(v), are disjoint.
- Therefore, the sum of the degrees of u and v, d(u) + d(v), must be less than or equal to the total number of vertices, n.
- By summing this inequality over all edges in the graph and using the handshaking lemma (sum of degrees = 2|E|), we can derive the inequality |E| ≤ n²/4, proving the theorem.
5. How does a complete bipartite graph relate to the maximum edge limit in Mantel's Theorem?
A complete bipartite graph is the key to understanding why the limit is ⌊n²/4⌋. This type of graph, by its very definition, is triangle-free because edges only exist between two separate sets of vertices, not within them. The graph with the maximum number of edges that is still triangle-free (the 'extremal graph') is a complete bipartite graph where the vertices are split as evenly as possible: K⌊n/2⌋, ⌈n/2⌉. The number of edges in this specific graph is exactly ⌊n²/4⌋, showing that the bound given by Mantel's Theorem is tight and achievable.
6. What is the difference between Mantel's Theorem and Turan's Theorem?
Mantel's Theorem is a special case of the more general Turan's Theorem.
- Mantel's Theorem specifies the maximum number of edges in a graph that is K₃-free (triangle-free).
- Turan's Theorem generalizes this concept for any complete graph, Kᵣ₊₁. It gives the maximum number of edges a graph can have without containing a clique of size r+1.
Essentially, Mantel's Theorem is Turan's Theorem with r=2, focusing specifically on avoiding triangles.
7. Why is the bound in Mantel's Theorem specifically ⌊n²/4⌋? What does this imply about network structures?
The bound ⌊n²/4⌋ arises because a graph structure that is maximally connected without forming triangles is one that is 'bipartite' or 'polarized'. By dividing vertices into two groups and only allowing connections between the groups, you eliminate any possibility of a 3-vertex cycle. The edge count is maximized when these two groups are as equal in size as possible. This implies that networks designed to avoid small, closed-loop clusters (like triangles) are most efficient when they have a balanced, bipartite structure, preventing any three nodes from being mutually connected.
8. What are some real-world applications or the importance of Mantel's Theorem?
While highly theoretical, the principles of Mantel's Theorem are fundamental to network science and combinatorics. Its importance lies in:
- Extremal Graph Theory: It is the starting point for this field, which asks questions about the maximum or minimum possible size of a graph with certain properties.
- Algorithm Design: Concepts from extremal graph theory influence algorithms for finding cliques and independent sets in networks.
- Social Network Analysis: It provides a baseline for understanding how dense a social network can be before small, tight-knit groups (triangles of friends) are guaranteed to emerge.
