Eulerian paths and circuits are essential concepts in graph theory, characterized by traversing every edge in a graph exactly once. An Eulerian path allows for the traversal of each edge without repetition, while an Eulerian circuit returns to the starting point. These paths have numerous applications in various fields, including network design, urban planning, and logistics. Below are three practical examples that illustrate the use of Eulerian paths and circuits in real-world scenarios.
The famous problem of the Seven Bridges of Königsberg, posed by mathematician Leonhard Euler in the 18th century, is a classic example of Eulerian paths. The city had seven bridges connecting four landmasses, and the challenge was to determine whether it was possible to walk through the city and cross each bridge exactly once.
By representing the landmasses as vertices and the bridges as edges, the problem becomes a graph traversal challenge. In this case, the graph derived from the bridges does not have an Eulerian circuit because more than two vertices have an odd degree.
Let’s visualize the situation:
The degrees of the vertices are as follows:
Since all vertices have odd degrees, it’s impossible to traverse all seven bridges without crossing at least one bridge twice. This led to the formulation of Euler’s theorem, establishing the criteria for the existence of Eulerian paths and circuits.
This problem laid the groundwork for graph theory and showed the importance of even and odd degrees in determining the existence of Eulerian paths and circuits.
Consider a postal delivery service that needs to deliver mail along specific streets in a neighborhood. The goal is to minimize the distance traveled while ensuring that every street (edge) is traversed at least once. This scenario can be modeled using Eulerian circuits.
Let’s visualize the streets and intersections:
The degree of each intersection is:
In this case, intersections 2 and 3 have odd degrees, indicating that a Eulerian path exists (but not a circuit). The postal worker can start at either intersection 2 or 3, delivering mail and returning to the starting point while ensuring each street is covered.
The application of Eulerian paths in postal routes helps optimize delivery efficiency, ensuring all streets are serviced while minimizing travel distance.
Cities often use Eulerian circuits to optimize garbage collection routes. By modeling the streets and intersections as a graph, city planners can create efficient routes that ensure every street is covered without unnecessary repetition.
Consider a neighborhood represented as follows:
The degree of each intersection is:
Here, intersections B and C have odd degrees, which indicates the presence of an Eulerian path. The garbage truck can start at either B or C, covering every street once before returning to the starting point.
Utilizing Eulerian circuits in garbage collection scheduling improves operational efficiency, lowers fuel costs, and enhances service delivery to residents.