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A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. 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]
A graph that can be proven non-Hamiltonian using Grinberg's theorem. In graph theory, Grinberg's theorem is a necessary condition for a planar graph to contain a Hamiltonian cycle, based on the lengths of its face cycles. If a graph does not meet this condition, it is not Hamiltonian.
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.
Graph theorists have also studied hypotraceable graphs, graphs that do not contain a Hamiltonian path but such that every subset of n − 1 vertices may be connected by a path. [13] Analogous definitions of hypohamiltonicity and hypotraceability for directed graphs have been considered by several authors. [14]
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]
Euler's recognition that the key information was the number of bridges and the list of their endpoints (rather than their exact positions) presaged the development of topology. The difference between the actual layout and the graph schematic is a good example of the idea that topology is not concerned with the rigid shape of objects.
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 ...
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 ...