# Definition:Trail

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## Definition

A **trail** is a walk in which all edges are distinct.

A **trail** between two vertices $u$ and $v$ is called a **$u$-$v$ trail**.

### Subgraph

The set of vertices and edges which go to make up a **trail** in a graph $G$ form a subgraph of $G$.

This subgraph itself is also referred to as a **trail in $G$**.

## Also known as

A **trail** can also be seen referred to as an **Eulerian walk**, for Leonhard Paul Euler.

## Also see

- Definition:Circuit (Graph Theory), otherwise known as a Definition:Closed Trail

- Results about
**trails**can be found**here**.

## Sources

- 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): $\S 2.3$: Connected Graphs - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**walk** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**walk** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**trail**