Euler Graph

If all the vertices of any connected graph have an even degree, then this type of graph will be known as the Euler graph. In other words, we can say that an Euler graph is a type of connected graph which have the Euler circuit.
Imagine a path that traverses every single edge (connection) in the graph exactly once, without ever lifting your pencil or retracing your steps. That’s the essence of an Eulerian graph!

• Connected Graph: The graph must be connected, meaning there’s a path between any two vertices (dots) in the graph.
• Even Degree: Every vertex (dot) in the graph must have an even degree. The degree of a vertex refers to the number of edges (lines) connected to it. So, if a vertex has 2, 4, or 6 edges connected to it, it has an even degree.

Euler Path

• Euler path can be called as Euler walk or Euler Trail.
• A graph will contain the Euler path if each edge of this graph is visited exactly once, and the vertex of the graph can be repeated.
• This path also visits every edge exactly once, but it doesn’t necessarily start and end at the same vertex. It might begin at one vertex and conclude at another.

Euler Circuit

• Euler circuit can be called as Euler tour or Euler Cycle.
• If a connected graph with a circuit or cycle that has all the edges of the graph, then that type of circuit will be known as the Euler circuit.
• It starts and ends at the same vertex, visiting every edge exactly once.
• A closed Euler trail is a Euler circuit.

Even Degree Rule

Verify that every vertex has an even degree. If even one vertex has an odd degree (an odd number of edges connected to it), the graph cannot be Eulerian.

Applications

• Route planning: Optimizing delivery routes to visit every location exactly once.
• Scheduling: Creating a schedule for a salesperson to visit all their clients without backtracking.
• Network analysis: Used in Computer networks to ensure efficient data flow across all connections.

Eulerian vs Hamiltonian Graph

Eulerian Graph:

• An Eulerian graph is a connected graph in which a single continuous path (known as an Eulerian path) traverses each edge exactly once. If this path forms a closed loop by starting and ending at the same vertex, it is called an Eulerian circuit.
• Eulerian graphs are characterized by the property that every vertex has an even degree.

Hamiltonian Graph:

• A Hamiltonian graph is a connected graph that contains a Hamiltonian cycle — a cycle that visits each vertex exactly once and returns to the starting vertex.
• Unlike Eulerian graphs, Hamiltonian graphs are not constrained by the degree of vertices, but rather by the existence of a specific cycle that traverses all vertices exactly once.