**1. E ≤ 3V**

Let *G* be a simple planar graph with *V* vertices and *E* edges. It follows from Euler’s theorem that

*E ***≤** 3V

In fact, we have (when V is at least 3,) that E **≤** 3V – 6.

To see this, denote by *F* the number of regions or faces determined by *G *(in other words, the number of connected components in the complement of the embedded graph). Euler’s theorem asserts that

**E – V + F = 2**

**V – E + F = 2**

and now note that every face must have at least three edges and every edge is contained in two faces and therefore , so 6=3V – 3E + 3F ≤ 3V – 3E +2E.

**2. F ****≤** 4E

Now let *K* be a two-dimensional simplicial complex and suppose that *K* can be embedded in . Denote by *E* the number of edges of *K* and by *F* the number of 2-faces of K.

Here is a really great conjecture:

**Conjecture:**

**F ****≤** 4E

A weaker version which is also widely open and very interesting is:

For some absolute constant *C*,

*F ***≤** C E

**Remarks:** The conjecture extends to higher dimensions. If *K* is an *r-*dimensional simplicial complex that can be embedded into then the conjecture is that

Where is a constant depending on *r. *Here* *is the number of* i*-dimensional faces of* K. *A stronger statement is that . The conjecture also extends to polyhedral complexes and more general form of complexes. In the conjecture ‘embed’ refers to a topological embedding.