Editing Eulerian path (section)

== Properties ==
*An undirected graph has an Eulerian cycle if and only if every vertex has even degree, and all of its vertices with nonzero degree belong to a single [[Connected component (graph theory)|connected component]].<ref name=ptw>{{citation
 | last1 = Pólya | first1 = George | author1-link = George Pólya
 | last2 = Tarjan | first2 = Robert E. | author2-link = Robert Tarjan
 | last3 = Woods | first3 = Donald R. | author3-link = Don Woods (programmer)
 | contribution = Hamiltonian and Eulerian Paths
 | date = October 2009
 | doi = 10.1007/978-0-8176-4953-1_13
 | isbn = 9780817649531
 | pages = 157–168
 | publisher = Birkhäuser Boston
 | title = Notes on Introductory Combinatorics}}</ref>
*An undirected graph can be decomposed into edge-disjoint [[cycle (graph theory)|cycle]]s if and only if all of its vertices have even degree. So, a graph has an Eulerian cycle if and only if it can be decomposed into edge-disjoint cycles and its nonzero-degree vertices belong to a single connected component.
*An undirected graph has an Eulerian trail if and only if exactly zero or two vertices have odd degree, and all of its vertices with nonzero degree belong to a single connected component.<ref name=ptw/>
*A directed graph has an Eulerian cycle if and only if every vertex has equal [[in degree (graph theory)|in degree]] and [[out degree (graph theory)|out degree]], and all of its vertices with nonzero degree belong to a single [[strongly connected component]]. Equivalently, a directed graph has an Eulerian cycle if and only if it can be decomposed into edge-disjoint [[cycle (graph theory)|directed cycle]]s and all of its vertices with nonzero degree belong to a single strongly connected component.<ref name=ptw/>
*A directed graph has an Eulerian trail if and only if at most one vertex has {{nowrap|1=([[out degree (graph theory)|out-degree]]) &minus; ([[in degree (graph theory)|in-degree]]) = 1,}} at most one vertex has {{nowrap|1=(in-degree) &minus; (out-degree) = 1,}} every other vertex has equal in-degree and out-degree, and all of its vertices with nonzero degree belong to a single connected component of the underlying undirected graph.<ref name=ptw/>