---
tag: 0NNYDD1AVR
kind: definition
name: Eulerian trail
msc: [05C45]
defines: 05NWGBHKSZ
uses: [062MQGP8M5]
sources:
  - url: https://en.wikipedia.org/wiki/Eulerian_path
    locator: Definition
---

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] or a [[multigraph|01M5T0KDG7]].
An Eulerian trail of $G$ is a [[trail|062MQGP8M5]] in $G$ that uses every edge of $G$ exactly once.
An Eulerian circuit is an Eulerian trail that starts and ends at the same vertex.

## Remarks

A graph with an Eulerian circuit is called Eulerian.
[[Euler's theorem|020422MDQF]] characterizes the connected graphs with an Eulerian circuit by the parity of the degrees.
The [[Seven Bridges of Königsberg|0YCW781Q9D]] form a multigraph with no Eulerian trail.

## Explanations

### L1 human

An Eulerian trail draws the whole graph in one stroke, without lifting the pen and without going over any line twice.
An Eulerian circuit also ends where it starts.