No Simple Graph is Perfect

Theorem
Let $G$ be a simple graph whose order is $2$ or greater.

Then $G$ is not perfect.

Proof
Recall that a perfect graph is one where each vertex is of different degree.

We note in passing that the simple graph consisting of one vertex trivially fulfils the condition for perfection.

Suppose $G$ be a simple graph of order order $n$ where $n \ge 2$ such that $G$ is perfect.

First, suppose that $G$ has no isolated vertices.

By the Pigeonhole Principle, for all vertices to have different degrees, one of them must be of degree at least $n$.

That means it must connect to at least $n$ other vertices.

But there are only $n - 1$ other vertices to connect to.

Therefore $G$ can not be perfect.

Now suppose $G$ has an isolated vertex.

There can be only one, otherwise there would be two vertices of degree zero, and so $G$ would not be perfect.

Again by the Pigeonhole Principle, for all vertices to have different degrees, one of them must be of degree at least $n-1$.

But of the remaining $n-1$ vertices, one of them is of degree zero and so can not be adjacent to any vertex.

So there are only $n-2$ other vertices to connect to.

Therefore $G$ can not be perfect.