Dirac's Theorem/Proof 2

Theorem

Let $G$ be a connected simple graph with $n$ vertices such that $n > 3$.

Let the degree of each vertex be at least $\dfrac n 2$.

Then $G$ is Hamiltonian.

Proof

Take any two non-adjacent vertices $u, v \in G$.

Then:

$\deg u + \deg v \ge \dfrac n 2 + \dfrac n 2 = n$

The result follows by a direct application of Ore's Theorem.

$\blacksquare$