Euleriansk

Euleriansk is an adjective relating to the mathematician Leonhard Euler or to concepts associated with his work. In modern English, the standard form is Eulerian; Euleriansk appears occasionally in older texts or translations and may be treated as a variant spelling.

In mathematics, the term is most commonly encountered in graph theory under the related but more common Eulerian. An Eulerian trail (or path) is a walk that uses every edge of a graph exactly once; an Eulerian circuit (or cycle) is such a trail that starts and ends at the same vertex. For undirected graphs, a connected graph has an Eulerian circuit precisely when every vertex has even degree; it has an Eulerian trail (but not a circuit) when exactly two vertices have odd degree. For directed graphs, a closed Eulerian circuit exists when every vertex has equal in-degree and out-degree and the nonzero-degree vertices form a single strongly connected component; an Eulerian trail (not closed) exists if, in addition, the degree conditions are satisfied with at most one start vertex (out-degree = in-degree + 1) and one end vertex (in-degree = out-degree + 1), and the underlying undirected graph is connected.

Algorithms such as Hierholzer's provide efficient methods to construct Eulerian circuits, while Fleury's algorithm offers a simpler, though often less efficient, alternative. The topic originates from Euler's solution to the Königsberg bridge problem, a foundational moment in the development of graph theory.

Outside graph theory, Eulerian is also used in physics and fluid dynamics to describe descriptions of motion in a fixed spatial frame, as opposed to a Lagrangian description that follows individual particles. Euleriansk as a label for such uses is uncommon in English; Eulerian remains the preferred form. See also Eulerian path, Eulerian circuit, Hierholzer's algorithm, Königsberg.