It is mch more difficult than finding an Eulerian path, which contains each edge exactly once.

For example, suppose we follow the following Eulerian path through these nodes:

Such a walk is now called an Eulerian path or Euler walk.

Further, if there are nodes of odd degree, then any Eulerian path will start at one of them and end at the other.

If it has 0 vertices of odd degree, the Eulerian path is an Eulerian circuit.

Since the graph corresponding to historical Königsberg has four nodes of odd degree, it cannot have an Eulerian path.

Therefore, an Eulerian path is now possible, but since it must begin on one island and end on the other, it is impractical for tourists.

An undirected, connected graph has an Eulerian path if and only if it has either 0 or 2 vertices of odd degree.

In graph theory, an Eulerian trail (or Eulerian path) is a trail in a graph which visits every edge exactly once.

All Eulerian circuits are also Eulerian paths, but not all Eulerian paths are Eulerian circuits.

An Eulerian path (a walk which is not closed but uses all edges of G just once) exists if and only if G is connected and exactly two vertices have odd valence.

The identity holds provided that for any two vertices A and B of the graph, the number of odd Eulerian paths from A to B is the same as the number of even ones.

A mathematical game found in West Africa is to draw a certain figure by a line that never ends until it closes the figure by reaching the starting point (in mathematical terminology, this is a Eulerian path on a graph).