Real-world examples of Eulerian paths and circuits in graph theory

If you’re looking for the best examples of Eulerian paths and circuits, you pretty much have to start with the Königsberg bridges problem. This is the historical origin story for the field of graph theory.

The city of Königsberg (now Kaliningrad) was built around a river with two islands and several landmasses connected by seven bridges. People wondered whether it was possible to take a walk that crossed each bridge exactly once and returned to the starting point. In modern language, they were asking for an Eulerian circuit.

Leonhard Euler modeled the landmasses as vertices and the bridges as edges. He proved that no such walk exists. Why? Because more than two vertices had odd degree. That makes the Königsberg bridge graph a clean example of a graph with no Eulerian path and no Eulerian circuit.

This is still one of the most referenced examples of Eulerian paths and circuits in textbooks—not because it works, but precisely because it doesn’t. It shows how checking vertex degrees gives you a fast decision rule.

For a readable historical overview, the University of Texas has a nice summary of Euler’s argument and its impact on early graph theory: https://web.ma.utexas.edu/users/m408m/Display12-5.shtml

Street-sweeping and snow-plowing: real examples of Eulerian circuits

Now let’s move from 18th-century bridges to city maintenance in 2024.

Imagine a city wants to schedule a street sweeper or snowplow so that every street is cleaned exactly once and the vehicle returns to the garage. This is a textbook real example of an Eulerian circuit:

• Intersections are vertices.
• Street segments are edges.
• The goal is a closed walk using each edge exactly once.

If every intersection has even degree, the street network (or at least the part you care about) admits an Eulerian circuit. In that case, you can plan a route that never repeats a street and ends where it started.

Most real cities are not that tidy. Some intersections have odd degree, which means a perfect Eulerian circuit doesn’t exist. But the Chinese Postman Problem shows how to modify the network by duplicating a few edges to create an Eulerian circuit with minimal extra travel. Modern routing software for maintenance fleets uses variants of this idea.

This makes municipal street networks one of the most practical examples of examples of Eulerian paths and circuits in action. Even if the raw network is not Eulerian, the concept shapes how we design efficient service routes.

For background on route inspection and the Chinese Postman Problem, you can find a clear discussion in many university lecture notes, for example from the University of Wisconsin–Madison: https://people.math.wisc.edu/~miller/old/m342-09/ChinesePostman.pdf

Postal delivery and network inspection: examples include Eulerian paths

Consider a utility company that needs to inspect every power line in a small region. The crew doesn’t need to end where it started; it just needs a walk that uses each line exactly once. That’s the definition of an Eulerian path (also called an Euler trail), not necessarily a circuit.

Here’s how this turns into one of the clearer examples of Eulerian paths and circuits:

• Poles or substations are vertices.
• Power lines between them are edges.
• The crew’s inspection route is a walk in the graph.

If exactly two vertices have odd degree and all others have even degree, the graph has an Eulerian path but not an Eulerian circuit. The inspection can start at one odd-degree vertex and end at the other, touching every line exactly once.

Real-world examples include:

• Postal workers covering every alley in a neighborhood but finishing at a different post office.
• Inspectors checking every bridge in a park trail system, starting at a visitor center and ending at a parking lot.

These are everyday examples of graphs where an Eulerian path is the natural fit, while a circuit would be either impossible or inefficient.

DNA sequencing: one of the best modern examples of Eulerian circuits

Graph theory is not just for roads and bridges. Modern DNA sequencing pipelines use Eulerian ideas at massive scale.

In a de Bruijn graph built from DNA reads:

• Vertices represent short sequences (k-mers).
• Edges represent overlaps between these sequences.

Reconstructing the original DNA sequence can be modeled as finding an Eulerian path or circuit through this graph: you want to traverse each edge (each observed overlap) exactly once. When the underlying genome is circular (like many bacterial genomes), the problem naturally becomes a search for an Eulerian circuit.

This is one of the best examples of Eulerian paths and circuits in modern computational biology. The mathematics Euler used on seven bridges now drives genome assembly algorithms that power research at institutions like the National Institutes of Health.

For more on how de Bruijn graphs are used in sequencing, see this overview from the National Center for Biotechnology Information (NCBI, part of NIH): https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2943872/

Network maintenance: fiber, pipelines, and inspection routes

Another family of real examples of examples of Eulerian paths and circuits comes from infrastructure inspection:

• Fiber-optic networks: Technicians might need to test every fiber connection in a campus network, visiting each link once.
• Water and gas pipelines: Inspectors may need to traverse each pipeline segment using smart pigs or sensors.
• Railway maintenance: Crews may want to check each track segment exactly once, possibly starting and ending at different depots.

In each case, you model junctions as vertices and segments as edges. Whether you can schedule a perfect one-pass route without repeats is exactly the Eulerian question.

These systems give you three types of behavior:

• All vertices even degree → Eulerian circuit exists.
• Exactly two vertices odd degree → Eulerian path exists but not a circuit.
• More than two vertices odd degree → neither exists; some segments must be repeated.

Engineers often modify the network virtually (by allowing certain segments to be traversed twice) to create a near-Eulerian circuit that minimizes repeated travel. That design mindset is one reason Eulerian thinking remains so relevant in 2024–2025 for infrastructure planning and smart-city analytics.

Robotics and warehouse routing: examples include Eulerian paths

Autonomous robots in warehouses provide surprisingly intuitive examples of Eulerian paths and circuits.

Picture a warehouse with a grid of aisles and storage racks:

• Intersection points are vertices.
• Aisle segments are edges.
• A cleaning robot or inventory robot must traverse each aisle segment at least once.

If the layout is designed so that every intersection has even degree, the robot can follow an Eulerian circuit: a closed route that hits each aisle exactly once and returns to its dock. Designers sometimes aim for this kind of layout because it simplifies routing algorithms and reduces energy use.

In practice, warehouse layouts are constrained by space and product flow, so perfect Eulerian circuits are rare. But the example of modeling the layout as a graph and checking for Eulerian properties is standard in robotics and operations research.

This is especially relevant today as companies in the U.S. and worldwide scale up automation; a small tweak to layout that turns a near-Eulerian graph into a fully Eulerian one can translate into real savings in battery life and throughput.

A simple classroom example of an Eulerian path

Let’s strip things down to a small, concrete example of an Eulerian path you can draw on paper.

Take four vertices labeled A, B, C, and D:

• Connect A–B, B–C, C–D, and D–A to form a square.
• Add one more edge from A to C.

Degrees:

• A has degree 3 (edges AB, AD, AC).
• C has degree 3 (edges BC, CD, AC).
• B and D each have degree 2.

Exactly two vertices (A and C) have odd degree, so this graph has an Eulerian path but no Eulerian circuit.

You can verify by drawing a path: start at A, go A–B–C–D–A–C. You’ve used every edge exactly once and ended at C.

This tiny graph is one of the cleanest examples of Eulerian paths and circuits to use when teaching or learning the degree rule.

A small classroom example of an Eulerian circuit

Now adjust that last drawing to get a pure example of an Eulerian circuit.

Take a triangle with vertices X, Y, Z, and connect every pair:

• Edges: X–Y, Y–Z, Z–X.

Degrees:

• Each vertex has degree 2.

All vertices have even degree, so the triangle graph has an Eulerian circuit. For instance, X–Y–Z–X uses each edge exactly once and returns to X.

Extend this by adding another triangle that shares an edge (forming a diamond shape). As long as the degrees of all vertices remain even, the larger graph still has an Eulerian circuit. These small constructions are handy examples of examples of Eulerian paths and circuits that you can scale up for homework or exam questions.

Why Eulerian examples matter for problem solving

So why spend time on all these examples of Eulerian paths and circuits instead of just memorizing the degree rule?

Because when you face real problems—whether it’s designing a delivery route, modeling a lab process, or optimizing a robot’s path—you rarely see a labeled “Eulerian” exercise. You see:

• A map.
• A network diagram.
• A description like “visit each link exactly once.”

Recognizing that this is essentially an Eulerian question is half the battle. The examples include everything from snowplows in Minneapolis to genome assembly at NIH. Once you’ve seen that range, you’re much more likely to spot the pattern in your own work.

For students preparing for contests or standardized tests, being fluent with these patterns turns a potentially messy graph question into a quick degree check and a confident answer.

FAQ: examples of Eulerian paths and circuits

Q1. Can you give a quick example of an Eulerian path in a real city?
Yes. Imagine a park with several footbridges over streams. If exactly two entrances to the park connect to an odd number of bridges and all other junctions connect to an even number, then a walker can start at one entrance, cross every bridge exactly once, and finish at the other entrance. That route is an Eulerian path.

Q2. What are some famous textbook examples of Eulerian circuits?
Classic examples include a perfectly balanced street grid where every intersection has an even number of roads, or a graph formed by the edges of a cube. In both cases, you can start at any vertex, follow a route that uses each edge exactly once, and return to your starting point.

Q3. Are there real examples of Eulerian paths and circuits in computer science?
Absolutely. File system consistency checks, testing network links, and especially DNA sequencing via de Bruijn graphs all provide real examples of Eulerian paths and circuits. In each case, edges represent relationships or overlaps that must be visited exactly once for a correct traversal or reconstruction.

Q4. How do I quickly test whether a graph has an Eulerian path or circuit?
Count vertex degrees in each connected component. If more than two vertices have odd degree, there is no Eulerian path. If exactly two are odd, there is an Eulerian path but not a circuit. If all are even, there is an Eulerian circuit (and also an Eulerian path). These rules are what make all the earlier examples of graphs behave the way they do.

Q5. Why do some street networks almost, but not quite, have Eulerian circuits?
Real cities grow organically. Dead ends, cul-de-sacs, and one-way segments create odd-degree vertices. So while city maps are among the best everyday examples of Eulerian paths and circuits ideas, they often require duplicating a few edges (driving a street twice) to get a workable route. This is exactly what the Chinese Postman Problem formalizes.