- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

A graph is traversable if you can draw a path between all the vertices without retracing the same path. Based on this path, there are some categories like Euler’s path and Euler’s circuit which are described in this chapter.

An Euler’s path contains each edge of ‘G’ exactly once and each vertex of ‘G’ at least once. A connected graph G is said to be traversable if it contains an Euler’s path.

**Euler’s Path** = d-c-a-b-d-e.

In an Euler’s path, if the starting vertex is same as its ending vertex, then it is called an Euler’s circuit.

**Euler’s Path** = a-b-c-d-a-g-f-e-c-a.

A connected graph ‘G’ is traversable if and only if the number of vertices with odd degree in G is exactly 2 or 0. A connected graph G can contain an Euler’s path, but not an Euler’s circuit, if it has exactly two vertices with an odd degree.

**Note** − This Euler path begins with a vertex of odd degree and ends with the other vertex of odd degree.

**Euler’s Path** − b-e-a-b-d-c-a is not an Euler’s circuit, but it is an Euler’s path. Clearly it has exactly 2 odd degree vertices.

**Note** − In a connected graph G, if the number of vertices with odd degree = 0, then Euler’s circuit exists.

A connected graph is said to be Hamiltonian if it contains each vertex of G exactly once. Such a path is called a **Hamiltonian path**.

**Hamiltonian Path** − e-d-b-a-c.

**Note** −

- Euler’s circuit contains each edge of the graph exactly once.
- In a Hamiltonian cycle, some edges of the graph can be skipped.

Take a look at the following graph −

For the graph shown above −

- Euler path exists – false
- Euler circuit exists – false
- Hamiltonian cycle exists – true
- Hamiltonian path exists – true

G has four vertices with odd degree, hence it is not traversable. By skipping the internal edges, the graph has a Hamiltonian cycle passing through all the vertices.

- Related Questions & Answers
- Hamiltonian Cycle
- Hamiltonian Graphs
- Eulerian and Hamiltonian Graphs in Data Structure
- HTML File Paths
- Euler’s criterion in java
- Euler Circuit in a Directed Graph
- C++ Program to Implement Euler Theorem
- Filtering paths and creating html page in express.js
- How to set Temporary and Permanent Paths in Java?
- All-Pairs Shortest Paths
- Unique Paths in Python
- Differences between Data paths.
- Euler Method for solving differential equation in C++
- Euler's Four Square Identity in C++
- Create Shortcuts to Long and Complicated Paths in Linux (Gogo)

Advertisements