Eulerianska

Eulerianska is a term used in mathematics and related fields to describe properties associated with Eulerian ideas and trails.

In graph theory, it refers to graphs that contain an Eulerian trail or Eulerian circuit. An undirected graph is called Eulerian if it is connected and every vertex has even degree, in which case it contains an Eulerian circuit—a closed walk that traverses every edge exactly once. If a connected undirected graph has exactly two vertices of odd degree, it has an Eulerian trail (but not a circuit) that starts and ends at the odd-degree vertices.

For directed graphs, a graph has an Eulerian circuit when every vertex has equal in-degree and out-degree and all of its vertices with nonzero degree belong to a single strongly connected component. An Eulerian trail in a digraph exists if, in addition to the connectivity condition, exactly one vertex has out-degree one higher than in-degree (start) and exactly one vertex has in-degree one higher than out-degree (end); all other vertices have equal in-degree and out-degree.

Historically, the concept is named after Leonhard Euler, who in 1736 solved the Königsberg bridge problem, laying the groundwork for graph theory. Algorithms such as Hierholzer's and Fleury's method can construct Eulerian trails or circuits.

Applications include planning efficient routes for street-sweeping or postal delivery, DNA fragment assembly in computational biology, and network design. See also Eulerian path, Eulerian circuit, degree, and graph connectivity.