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  2. Eulerian path - Wikipedia

    en.wikipedia.org/wiki/Eulerian_path

    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 .

  3. Eulerian matroid - Wikipedia

    en.wikipedia.org/wiki/Eulerian_matroid

    Eulerian matroids were defined by Welsh (1969) as a generalization of the Eulerian graphs, graphs in which every vertex has even degree.By Veblen's theorem the edges of every such graph may be partitioned into simple cycles, from which it follows that the graphic matroids of Eulerian graphs are examples of Eulerian matroids.

  4. Euler tour technique - Wikipedia

    en.wikipedia.org/wiki/Euler_tour_technique

    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

  5. Dual graph - Wikipedia

    en.wikipedia.org/wiki/Dual_graph

    In particular, Barnette's conjecture on the Hamiltonicity of cubic bipartite polyhedral graphs is equivalent to the conjecture that every Eulerian maximal planar graph can be partitioned into two induced trees. [33] If a planar graph G has Tutte polynomial T G (x,y), then the Tutte polynomial of its dual graph is obtained by swapping x and y.

  6. Seven Bridges of Königsberg - Wikipedia

    en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg

    Since the graph corresponding to historical Königsberg has four nodes of odd degree, it cannot have an Eulerian path. An alternative form of the problem asks for a path that traverses all bridges and also has the same starting and ending point. Such a walk is called an Eulerian circuit or an Euler tour. Such a circuit exists if, and only if ...

  7. Euler diagram - Wikipedia

    en.wikipedia.org/wiki/Euler_diagram

    For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the adjacent diagram, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations; some of the intermediate diagrams have concurrency of curves.

  8. BEST theorem - Wikipedia

    en.wikipedia.org/wiki/BEST_theorem

    In graph theory, a part of discrete mathematics, the BEST theorem gives a product formula for the number of Eulerian circuits in directed (oriented) graphs. The name is an acronym of the names of people who discovered it: N. G. de Bruijn , Tatyana Ehrenfest , Cedric Smith and W. T. Tutte .

  9. Travelling salesman problem - Wikipedia

    en.wikipedia.org/wiki/Travelling_salesman_problem

    To improve the lower bound, a better way of creating an Eulerian graph is needed. By the triangle inequality, the best Eulerian graph must have the same cost as the best travelling salesman tour; hence, finding optimal Eulerian graphs is at least as hard as TSP. One way of doing this is by minimum weight matching using algorithms with a ...