## You are here

HomeEuler path

## Primary tabs

# Euler path

An *Euler path* in a graph is a path which traverses each edge of the graph exactly once. An Euler path which is a cycle is called an *Euler cycle*. For loopless graphs without isolated vertices, the existence of an Euler path implies the connectedness of the graph, since traversing every edge of such a graph requires visiting each vertex at least once.

If a connected graph has an Euler path, one can be constructed by applying Fleury’s algorithm. A connected graph has an Euler path if it has exactly zero or two vertices of odd degree. If every vertex has even degree, the graph has an Euler cycle.

This graph has an Euler cycle. All of its vertices are of even degree.

This graph has an Euler path which is not a cycle. It has exactly two vertices of odd degree.

## Mathematics Subject Classification

05C38*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag