Eulerian

Eulerian describes properties and concepts named after the Swiss mathematician Leonhard Euler. In mathematics, the term is most commonly encountered in graph theory and combinatorics, where it denotes traversal-related or counting properties arising from Euler’s work.

In graph theory, an Eulerian path (or Eulerian trail) is a walk that uses every edge of a graph exactly once. If such a walk starts and ends at the same vertex, it is an Eulerian circuit (also called an Euler tour). For undirected graphs, a connected graph has an Eulerian circuit if and only if every vertex has even degree; it has an Eulerian path but not a circuit if exactly two vertices have odd degree (the remaining vertices have even degree). For directed graphs, a circuit exists when in-degree equals out-degree at every vertex and the nonzero-degree vertices form a single strongly connected component; an Eulerian trail exists if at most one vertex has out-degree minus in-degree equal to one, at most one vertex has in-degree minus out-degree equal to one, all other vertices are balanced, and the underlying undirected graph is connected when orientation is ignored.

In combinatorics, Eulerian numbers count permutations with a given number of ascents. The Eulerian number A(n, k) counts permutations of n elements with exactly k ascents, and these numbers give rise to Eulerian polynomials. These concepts originate from Euler’s studies of orderings and statistics on permutations.

The term Eulerian can also appear in other contexts where traversal, symmetry, or combinatorial structure is emphasized, always in reference to ideas introduced or studied by Euler.