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A connected graph has an Euler cycle if and only if every vertex has even degree. The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. [2]
All Hamiltonian graphs are biconnected, but a biconnected graph need not be Hamiltonian (see, for example, the Petersen graph). [9] An Eulerian graph G (a connected graph in which every vertex has even degree) necessarily has an Euler tour, a closed walk passing through each edge of G exactly once.
In graph theory, a Harris graph is defined as an Eulerian, tough, non-Hamiltonian graph. [1] [2] Harris graphs were introduced in 2013 when, at the University of Michigan, Harris Spungen conjectured that a tough, Eulerian graph would be sufficient to be Hamiltonian. [3]
In one direction, the Hamiltonian path problem for graph G can be related to the Hamiltonian cycle problem in a graph H obtained from G by adding a new universal vertex x, connecting x to all vertices of G. Thus, finding a Hamiltonian path cannot be significantly slower (in the worst case, as a function of the number of vertices) than finding a ...
For example, the teardrop orbifold has Euler characteristic 1 + 1 / p , where p is a prime number corresponding to the cone angle 2 π / p . The concept of Euler characteristic of the reduced homology of a bounded finite poset is another generalization, important in combinatorics. A poset is "bounded" if it has smallest and ...
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 path or Euler walk in his honor ...
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.
A complete bipartite graph K m,n has a maximum matching of size min{m,n}. A complete bipartite graph K n,n has a proper n-edge-coloring corresponding to a Latin square. [14] Every complete bipartite graph is a modular graph: every triple of vertices has a median that belongs to shortest paths between each pair of vertices. [15]