Euler Paths and Euler Circuits

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🌿 What Exactly Is an Euler Path?

An Euler Path is a walk through a graph where:

  • You must use every edge exactly once.
  • You can start and end at different vertices.
  • You are not allowed to repeat any edge.

Think of it like walking through a park and promising yourself that you’ll walk through each pathway one time — no going backward, no re-walking a path.

🌿 What Is an Euler Circuit (or Euler Cycle)?

An Euler Circuit is almost the same, but with one extra condition:

  • You start and end at the same vertex.

So it’s like taking a circular walk:
You cover every path once and end up right where you began.

🎯 The Famous Degree Rules (Super Simple!)

Euler discovered that you don’t even need to trace the paths to know whether such walks are possible.
You just need to look at the degrees (number of edges connected) of each vertex.

Here are the golden rules:

✔ A graph has an Euler Circuit if:

👉 Every vertex has an even degree.

✔ A graph has an Euler Path (but not a circuit) if:

👉 Exactly two vertices have odd degree.

❌ If more than two vertices are odd:

👉 Neither an Euler Path nor an Euler Circuit is possible.

This is incredibly useful — a quick check tells you everything.

🧩 Let’s Look at Some Simple Diagrams

🔷 Example 1: Graph WITH an Euler Circuit

(All vertices have even degree)

     A
    / \
   B---C
    \ /
     D

Degrees:

  • A = 2
  • B = 2
  • C = 2
  • D = 2

All even → ✔ Euler Circuit exists.

One possible Euler Circuit:
A → B → D → C → A

Notice how the walk forms a loop.

🔷 Example 2: Graph WITH an Euler Path

(Exactly two vertices have odd degree)

A ----- B ----- C

Degrees:

  • A = 1 (odd)
  • B = 2 (even)
  • C = 1 (odd)

Two odd vertices → ✔ Euler Path exists.

A possible Euler Path:
A → B → C

But you cannot return to A without repeating edges, so there is no Euler Circuit here.

🪄 A Friendly Analogy

Imagine your graph is a neighborhood map:

  • Vertices = junctions or houses
  • Edges = roads

Now think of yourself as a courier who wants to cover every road exactly once.

  • If you don’t care where you end → You want an Euler Path.
  • If you want to return home at the end → You need an Euler Circuit.

It’s surprisingly similar to real-world route planning.

🌟 Why Should You Care About Euler Paths?

You might wonder, “Why is this important in Data Structures?”

Well, Euler paths appear in:

  • Network routing
  • DNA sequencing
  • Circuit design
  • Solving puzzles
  • Graph-based algorithms
  • Route optimization problems

Even the famous “Seven Bridges of Königsberg” puzzle that started modern graph theory is based on Euler’s ideas.