User:Esequia/Teorema de Mantel

Mantel's Theorem is a particular case of Turán's Theorem for $$r=3$$.

Result
The following was established by Mantel in 1907. It's one of earliest results in Extremal graph theory
 * Mantel's Theorem If $$ G \nsupseteq C_3 $$ and has $n$ vertices, then $$ e(G) \leq \frac{n^2}{4}. $$

Where $e(G)$ is the number of edges on $G$. In other words the theorem provide a bound on the number of edges of a graph that has no triangle. It is easy to see that bigger graph with no triangles is the complete bipartite graph.

Demonstração
For $n$=1 and $n$=2 the result is trivial. Suppose that results holds for $n-1$ and let $G$ be the graph with $n$ vertex. Let $u$ and $v$ be adjacent vertices in $G$, then $$d(u) + d(v) \leq n$$, this is true since each vertex in $G$ is connected with at most one of $u$ or $v$. Then,


 * $$e(G) = e(G-\{u,v\}) + d(u) + d(v) - 1$$


 * $$ e(G-\{u,v\}) + d(u) + d(v) - 1 \leq \frac{(n-2)^2}{4} + n - 1 $$

Finally,
 * $$ e(G) \leq \frac{n^2}{4}$$

Where $$e(G-\{u,v\})$$ follow from hypothesis and -1 comes to the fact that edge $uv$ has being counted twice in $$d(u) + d(v)$$.