# Graphs and NetworksPlanar Graphs

Here is another puzzle that is related to graph theory.

In a small village there are three houses and three utility plants that produce water, electricity and gas. We have to connect each of the courses to each of the utility plants, but due to the layout of the village, the different pipes and cables are not allowed to cross.

Try to connect each of the houses to each of the utility companies below, without any of your lines intersecting:

Just like the Königsberg bridges before, you quickly discover that this problem is also impossible. It seems that some graphs can be drawn without overlapping edges – these are called **planar graphs** – but others cannot.

The

The graph in the three utilities puzzle is the *Kuratowski’s theorem*.

### Planarity

This is a planar graph, but the

## Euler’s Formula

All planar graphs divide the plane they are drawn on into a number of areas, called **faces**.

11 Vertices + Faces

15 Vertices + Faces

25 Vertices + Faces

When comparing these numbers, you will notice that the number of edges is always *F* + *V* = *E* + 1. This result is called **Euler’s equation** and is named after the same

Unfortunately, there are infinitely many graphs and we can’t check every one to see if Euler’s equation works. Instead we can try to find a simple

F | V | E |

0 | 1 | 0 |

**0** + **1** = **0** + 1

Any (finite) graph can be constructed by starting with one vertex and adding more vertices one by one. We have shown that, whichever way we add new vertices, Euler’s equation is valid. Therefore it is valid for all graphs.

The process we have used is called **mathematical induction**. It is a very useful technique for proving results in infinitely many cases, simply by starting with the simplest case, and showing that the result holds at every step when constructing more complex cases.

Many planar graphs look very similar to the nets of

This means that we can use Euler’s formula not only for planar graphs but also for all polyhedra – with one small difference. When transforming the polyhedra into graphs, one of the faces disappears: the topmost face of the polyhedra becomes the “outside”; of the graphs.

In other words, if you count the number of **edges**, **faces** and **vertices** of *any* polyhedron, you will find that *F* + *V* = *E* +