Definition
An Eulerian trail, or Euler walk in an undirected graph is a path that uses each edge exactly once. If such a path exists, the graph is called traversable or semi-eulerian.
An Eulerian cycle, Eulerian circuit or Euler tour in an undirected graph is a cycle that uses each edge exactly once. If such a cycle exists, the graph is called Eulerian or unicursal. The term "Eulerian graph" is also sometimes used in a weaker sense to denote a graph where every vertex has even degree. For connected graphs 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 graphs, "path" has to be replaced with directed path and "cycle" with directed cycle.
The definition and properties of Eulerian trails, cycles and graphs are valid for multigraphs 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 indegree of v equals the 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. Every Eulerian orientation of a connected graph is a strong orientation, an orientation that makes the resulting directed graph strongly connected.
Read more about this topic: Eulerian Path
Famous quotes containing the word definition:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than thisdevoted and obedient. This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (18201910)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“Mothers often are too easily intimidated by their childrens negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)