# Eulerian Path **A [[path#^8f9eb6|trail]] that visits every edge exactly once.** ![[EulerianPath.svg|367]] *A traversable graph with a Eulerian path starting from $D$* An undirected [[graph]] has a Eulerian path[^1] or a Eulerian trail if and only if it has *exactly one or two* vertices with an *odd degree*. Additionally, all of its vertices of nonzero degree are part of the same connected component. Such a graph is known as *traversable* or *semi-Eulerian*. ## Eulerian cycle > [!Eulerian cycle] > A *Eulerian cycle* is a [[cycle]] in which every edge is visited exactly once. An undirected [[graph]] has a Eulerian cycle or Eulerian circuit if and only if every vertex has an *even degree* and all vertices of nonzero degree are part of the same connected component. Such a graph is known as *Eulerian*. ![[EulerianCycle.svg|367]] *A Eulerian graph with Eulerian cycle $(D, C, B, E, A, D)$* [^1]: "Eulerian path" is the common term, although it is technically a trail.