Eulerian Paths An Eulerian path is a path that visits each edge of a graph exactly once and starts and ends at the same vertex. The Eulerian paths are a fun...
Eulerian Paths
An Eulerian path is a path that visits each edge of a graph exactly once and starts and ends at the same vertex. The Eulerian paths are a fundamental concept in graph theory and are used in a variety of algorithms and applications.
Formal Definition:
An Eulerian path is a path that starts and ends at a single vertex v, and visits each edge e of V exactly once. This means that the path must visit each edge in the graph exactly once, and it must avoid visiting the same edge more than once.
Examples:
Consider the following graph:
A---B---C---D---E---F---G
A path that visits each edge of this graph exactly once would be:
A---B---C---D---E---F---G
Eulerian paths are closely related to Hamiltonian circuits in a graph. A Hamiltonian circuit is a circuit that consists of a closed path that visits each edge of the graph exactly once. An Eulerian path is a Hamiltonian circuit in which the starting and ending vertices are different.
Applications:
Eulerian paths have a wide range of applications in graph theory, including:
Detecting cycles in a graph.
Finding all connected components in a graph.
Computing the chromatic number of a graph.
Determining the genus of a graph.
Eulerian paths are a powerful tool for understanding the structure and properties of graphs. They are used in a variety of algorithms and applications in computer science and mathematics