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In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex .
Euler's argument shows that a necessary condition for the walk of the desired form is that the graph be connected and have exactly zero or two nodes of odd degree. This condition turns out also to be sufficient—a result stated by Euler and later proved by Carl Hierholzer. Such a walk is now called an Eulerian trail or Euler walk in his honor ...
The Euler tour technique (ETT), named after Leonhard Euler, is a method in graph theory for representing trees. The tree is viewed as a directed graph that contains two directed edges for each edge in the tree. The tree can then be represented as a Eulerian circuit of the directed graph, known as the Euler tour representation (ETR) of the tree
In either case, the resulting closed trail is known as an Eulerian trail. If a finite undirected graph has even degree at each of its vertices, regardless of whether it is connected, then it is possible to find a set of simple cycles that together cover each edge exactly once: this is Veblen's theorem. [10]
The first long-distance hiking trail in Europe was the National Blue Trail of Hungary, established in 1938. The formation of the European Union made transnational hiking trails possible. Today, the network consists of 12 paths and covers more than 65,000 kilometres (40,000 mi), crisscrossing Europe.
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph.A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or (in weighted graphs) by the sum of the weights of its edges.
English: Using Eulerian paths to draw shapes with a continuous stroke by CMG Lee. 1. As the Haus vom Nikolaus puzzle has two odd vertices (orange), the path must start at one and end at the other. 2. The Annie Pope one with four odd vertices has no solution. 3. If there are no odd vertices, the path can start anywhere and forms a closed circuit. 4.
In order to prove this generalized form of the theorem, Petersen first proved that a 4-regular graph can be factorized into two 2-factors by taking alternate edges in a Eulerian trail. He noted that the same technique used for the 4-regular graph yields a factorization of a 2 k {\displaystyle 2k} -regular graph into two k {\displaystyle k ...