What is an Euler Path?

S3-SA5-0391

Grade Level:

Class 10

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

An Euler Path is a path in a graph that visits every edge exactly once. It can start and end at different vertices (corners) or the same vertex. Think of it like drawing a figure without lifting your pen and without drawing over any line twice.

Simple Example

Quick Example

Imagine your school has a network of corridors connecting different classrooms. If you want to walk through every single corridor exactly once, without re-tracing your steps, that's an Euler Path. You might start at the Class 10 room and end up at the library.

Worked Example

Step-by-Step

Let's check if the following graph has an Euler Path. The graph has vertices A, B, C, D and edges AB, BC, CD, DA, AC.---Step 1: Count the degree (number of edges connected) of each vertex.---Vertex A: Edges AB, AD, AC. Degree = 3.---Vertex B: Edges BA, BC. Degree = 2.---Vertex C: Edges CB, CD, CA. Degree = 3.---Vertex D: Edges DA, DC. Degree = 2.---Step 2: For an Euler Path to exist, there can be at most two vertices with an odd degree.---Step 3: In our graph, Vertex A has degree 3 (odd) and Vertex C has degree 3 (odd). Vertices B and D have degree 2 (even).---Step 4: Since there are exactly two vertices with odd degrees (A and C), an Euler Path exists. It must start at one of these odd-degree vertices and end at the other.---Step 5: One possible Euler Path is: A -> B -> C -> D -> A -> C. (Check: All edges AB, BC, CD, DA, AC are used exactly once).---Answer: Yes, an Euler Path exists. One example is A-B-C-D-A-C.

Why It Matters

Euler Paths are crucial in planning efficient routes, like how a delivery person plans to cover all streets exactly once. They are used in computer science for network design and in logistics for optimizing travel. Understanding them helps in building smarter systems for tasks like garbage collection routes or inspecting power lines.

Common Mistakes

MISTAKE: Thinking an Euler Path must start and end at the same vertex. | CORRECTION: An Euler Path can start and end at different vertices. If it starts and ends at the same vertex, it's called an Euler Circuit.

MISTAKE: Confusing an Euler Path with a Hamiltonian Path (which visits every vertex exactly once). | CORRECTION: An Euler Path visits every *edge* exactly once, not necessarily every vertex. A Hamiltonian Path visits every *vertex* exactly once.

MISTAKE: Believing an Euler Path exists even if there are more than two odd-degree vertices. | CORRECTION: An Euler Path exists only if there are zero or exactly two vertices with an odd degree. If there are more than two, an Euler Path is not possible.

Practice Questions

Try It Yourself

QUESTION: A graph has vertices P, Q, R, S and edges PQ, QR, RS, SP. Does this graph have an Euler Path? | ANSWER: Yes, it has an Euler Circuit (which is also an Euler Path that starts and ends at the same point). All vertices have degree 2 (even).

QUESTION: A graph has vertices X, Y, Z, W and edges XY, YZ, ZW, WX, XZ. Does this graph have an Euler Path? If yes, name the start and end vertices. | ANSWER: Yes. Vertex X has degree 3, Y has degree 2, Z has degree 3, W has degree 2. Since there are exactly two odd-degree vertices (X and Z), an Euler Path exists. It would start at X and end at Z, or vice versa.

QUESTION: Consider a map of a small colony with 5 streets: Street 1 (connecting A-B), Street 2 (B-C), Street 3 (C-A), Street 4 (A-D), Street 5 (D-B). Can a postman walk along every street exactly once without repeating any street? If so, what are the possible start and end points? | ANSWER: Let's find degrees: A (Streets 1,3,4) = 3; B (Streets 1,2,5) = 3; C (Streets 2,3) = 2; D (Streets 4,5) = 2. Since there are exactly two odd-degree vertices (A and B), yes, the postman can walk every street exactly once. Possible start/end points are A and B.

MCQ

Quick Quiz

Which of the following conditions must be true for a graph to have an Euler Path?

All vertices must have an even degree.

There must be exactly two vertices with an odd degree.

There can be zero or exactly two vertices with an odd degree.

Every vertex must be visited exactly once.

The Correct Answer Is:

An Euler Path exists if there are zero odd-degree vertices (in which case it's an Euler Circuit) or exactly two odd-degree vertices (in which case it starts at one odd-degree vertex and ends at the other). Option D describes a Hamiltonian Path, not an Euler Path.

Real World Connection

In the Real World

Imagine a drone inspecting power lines in a rural area. The drone needs to fly along every section of the power line exactly once to save battery and time. Engineers use Euler Path concepts to program the most efficient flight path for such drones, similar to how delivery services like Swiggy or Zomato might optimize their routes.

Key Vocabulary

Key Terms

GRAPH: A collection of points (vertices) connected by lines (edges). | VERTEX: A point or corner in a graph. | EDGE: A line connecting two vertices in a graph. | DEGREE OF A VERTEX: The number of edges connected to a vertex. | EULER CIRCUIT: An Euler Path that starts and ends at the same vertex.

What's Next

What to Learn Next

Now that you understand Euler Paths, explore 'Euler Circuits' which are a special type of Euler Path. Then, you can move on to 'Hamiltonian Paths' to see how they differ and where they are used in real-world problems like the 'Traveling Salesperson Problem'.