---
tag: 062MQGP8M5
kind: definition
name: Trail
msc: [05C38]
defines: 0W50MT1C6X
uses: [0VJZM0F9YM]
sources:
  - url: https://en.wikipedia.org/wiki/Path_(graph_theory)
    locator: Walk, trail, and path
---

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A trail in $G$ is a [[walk|0VJZM0F9YM]] $(w_0, w_1, \dots, w_k)$ whose edges $w_0 w_1, w_1 w_2, \dots, w_{k-1} w_k$ are pairwise distinct.

## Remarks

A trail may visit a vertex more than once, but it uses each edge at most once.
Every [[path|0JDE56HWQ8]] is a trail. In the graph of two triangles that share a vertex $c$, the closed walk $a, b, c, d, e, c, a$ is a trail but not a path.
An [[Eulerian trail|0NNYDD1AVR]] is a trail that uses every edge.

## Explanations

### L1 human

A trail is a walk that never uses the same edge twice.
It is the kind of route that you draw without lifting your pen and without retracing a line.