Editing Eulerian path (section)

==Definition==
An '''Eulerian trail''',<ref name="pathcycle" group="note">Some people reserve the terms ''path'' and ''cycle'' to mean ''non-self-intersecting'' path and cycle.  A (potentially) self-intersecting path is known as a '''trail''' or an '''open walk'''; and a (potentially) self-intersecting cycle, a '''circuit''' or a '''closed walk'''. This ambiguity can be avoided by using the terms Eulerian trail and Eulerian circuit when self-intersection is allowed.</ref>  or '''Euler walk''', in an [[undirected graph]] is a walk that uses each edge exactly once. If such a walk exists, the graph is called '''traversable''' or '''semi-eulerian'''.<ref>Jun-ichi Yamaguchi, [http://jwilson.coe.uga.edu/EMAT6680/Yamaguchi/emat6690/essay1/GT.html Introduction of Graph Theory].</ref>

An '''Eulerian cycle''',<ref name="pathcycle" group="note" /> also called an '''Eulerian circuit''' or '''Euler tour''', in an undirected graph is a [[cycle (graph theory)|cycle]] that uses each edge exactly once.  If such a cycle exists, the graph is called '''Eulerian''' or '''unicursal'''.<ref>Schaum's outline of theory and problems of graph theory By V. K. Balakrishnan [https://books.google.com/books?id=1NTPbSehvWsC&dq=unicursal&pg=PA60].</ref>  The term "Eulerian graph" is also sometimes used in a weaker sense to denote a graph where every vertex has even degree. For finite [[connected graph]]s the two definitions are equivalent, while a possibly unconnected graph is Eulerian in the weaker sense if and only if each connected component has an Eulerian cycle.

For [[directed graph]]s, "path" has to be replaced with ''[[directed path (graph theory)|directed path]]'' and "cycle" with ''[[directed cycle]]''.

The definition and properties of Eulerian trails, cycles and graphs are valid for [[multigraph]]s as well.

An '''Eulerian orientation''' of an undirected graph ''G'' is an assignment of a direction to each edge of ''G'' such that, at each vertex ''v'', the [[Directed graph#Indegree and outdegree|indegree]] of ''v'' equals the [[Directed graph#Indegree and outdegree|outdegree]] of ''v''. Such an orientation exists for any undirected graph in which every vertex has even degree, and may be found by constructing an Euler tour in each connected component of ''G'' and then orienting the edges according to the tour.<ref>{{citation
 | last = Schrijver | first = A. | author-link = Alexander Schrijver
 | doi = 10.1007/BF02579193
 | issue = 3–4
 | journal = Combinatorica
 | mr = 729790
 | pages = 375–380
 | title = Bounds on the number of Eulerian orientations
 | volume = 3
 | year = 1983| s2cid = 13708977 | url = https://ir.cwi.nl/pub/10053 }}.</ref> Every Eulerian orientation of a connected graph is a [[strong orientation]], an orientation that makes the resulting directed graph [[strongly connected]].