What Is Euler Network?

Eulerian networks, also known as Eulerian graphs, are a fundamental concept in graph theory. They were first introduced by the Swiss mathematician Leonhard Euler in the 18th century. In this answer, we will explore the key principles and properties of Eulerian networks, including their definition, characteristics, and how they can be identified.

Definition of Eulerian Networks:
An Eulerian network is a connected graph that contains an Eulerian circuit. An Eulerian circuit is a closed path that traverses each edge of the graph exactly once and returns to the starting vertex. In other words, it is a circuit that visits every edge of the graph without any repeats.

Characteristics of Eulerian Networks:
To determine if a graph is an Eulerian network, we need to consider the degrees of its vertices. The degree of a vertex is the number of edges incident to it. Euler's theorem provides a necessary and sufficient condition for a graph to be an Eulerian network:

A connected graph has an Eulerian circuit if and only if every vertex has an even degree.

Based on this theorem, we can derive the following characteristics of Eulerian networks:

All vertices in an Eulerian network have even degrees.

An Eulerian network must be connected, meaning there is a path between any two vertices.

If a graph has exactly two vertices with odd degrees, it can have an Eulerian path but not an Eulerian circuit. An Eulerian path is a path that traverses each edge exactly once but does not return to the starting vertex.

Identifying Eulerian Networks:
To identify whether a given graph is an Eulerian network, we can follow these steps:

Check if the graph is connected. If it is not connected, it cannot be an Eulerian network.

Count the degree of each vertex in the graph. If any vertex has an odd degree, the graph cannot be an Eulerian network.

If all vertices have even degrees, the graph may be an Eulerian network. Further analysis is required to confirm this.

If the graph is connected and all vertices have even degrees, we can use algorithms such as Fleury's algorithm or Hierholzer's algorithm to find an Eulerian circuit or path.

Fleury's Algorithm:
Fleury's algorithm is an inefficient but straightforward method to find an Eulerian circuit or path in a graph. Here are the steps of Fleury's algorithm:

Start at any vertex in the graph.

Choose an edge from the current vertex that is not a bridge (an edge whose removal would disconnect the graph).

Move to the other endpoint of the chosen edge and remove the edge from the graph.

Repeat steps 2 and 3 until no edges are left in the graph.

If all edges have been traversed, and the final vertex is the same as the starting vertex, it is an Eulerian circuit. If the final vertex is different from the starting vertex, it is an Eulerian path.

Hierholzer's Algorithm:
Hierholzer's algorithm is a more efficient method to find an Eulerian circuit or path. Here are the steps of Hierholzer's algorithm:

Choose any vertex as the starting vertex.

Follow a trail of unused edges from the starting vertex until returning to the starting vertex.

While there are vertices in the current trail that have unused edges, choose one of those vertices and start a new trail from that vertex.

Merge the new trail with the previous trail by inserting it into the appropriate position.

Repeat steps 3 and 4 until all edges have been traversed.

If all edges have been traversed, and the final vertex is the same as the starting vertex, it is an Eulerian circuit. If the final vertex is different from the starting vertex, it is an Eulerian path.