Euleriaanse

Euleriaanse refers to properties or concepts that arise from the work of Leonhard Euler and are used in graph theory to describe trails and circuits that traverse edges of a graph. In Dutch mathematical usage, the term is used to describe Eulerian phenomena, including paths and circuits that cover every edge exactly once.

In an undirected graph, a graph is Euleriaanse if it is connected (ignoring isolated vertices) and every vertex has even degree. Such a graph contains an Eulerian circuit, a closed trail that uses every edge exactly once. If exactly two vertices have odd degree, the graph has an Eulerian trail, a path that uses every edge exactly once but starts and ends at those two odd vertices.

In directed graphs, an Euleriaanse circuit exists when every vertex has equal in-degree and out-degree and all vertices with nonzero degree belong to a single strongly connected component. An Eulerian path is possible if the graph is connected in the underlying undirected sense and exactly one vertex has out-degree one more than in-degree and exactly one has in-degree one more than out-degree; all other vertices have equal in-degree and out-degree.

The concept traces back to Euler's Königsberg bridge problem (1736), which led to the idea of traversing every edge once. The term "Eulerian" and the Dutch adjective "euleriaanse" have since become standard in discussions of such trails and circuits. In Dutch texts, "euleriaanse" may describe either Eulerian graphs, paths, or techniques derived from Euler's methods.

Related notions include semi-Eulerian or quasi-Eulerian cases, referring to graphs that have an Eulerian path but not a circuit. Applications include network design, DNA sequence assembly, puzzle design, and routing problems where a complete edge traversal is required.