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All Eulerian circuits are also Eulerian paths, but not all Eulerian paths are Eulerian circuits. Euler's work was presented to the St. Petersburg Academy on 26 August 1735, and published as Solutio problematis ad geometriam situs pertinentis (The solution of a problem relating to the geometry of position) in the journal Commentarii academiae ...
An Eulerian trail, [note 1] or Euler walk, in an undirected graph is a walk that uses each edge exactly once. If such a walk exists, the graph is called traversable or semi-eulerian. [3] An Eulerian cycle, [note 1] also called an Eulerian circuit or Euler tour, in an undirected graph is a cycle that uses each edge exactly once
When the graph has an Eulerian circuit (a closed walk that covers every edge once), that circuit is an optimal solution. Otherwise, the optimization problem is to find the smallest number of graph edges to duplicate (or the subset of edges with the minimum possible total weight) so that the resulting multigraph does have an Eulerian circuit. [ 1 ]
In 1736, Euler showed that G has an Eulerian circuit if and only if G is connected and the indegree is equal to outdegree at every vertex. In this case G is called Eulerian. We denote the indegree of a vertex v by deg(v). The BEST theorem states that the number ec(G) of Eulerian circuits in a connected Eulerian graph G is given by the formula
The Euler method is explicit, i.e. the solution + is an explicit function of for . Higher-order process ... Nonstiff problems. Berlin, New York: Springer-Verlag.
Euler invented the calculus of variations including its most well-known result, the Euler–Lagrange equation. Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory.
There is no problem with any of this. Consider, however, the observers in the remaining three rooms. Each of these rooms has five walls. If the solution line starts in one of these rooms, its observer will see the line leave (through one wall), re-enter and leave again (two more walls) and enter and leave a second time (the last two walls).
An instance of Guan's route inspection problem (black edges and weights) and its optimal solution (doubling the red edges to produce an Eulerian multigraph). Guan is known for formulating the route inspection problem. [1]