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A Hamiltonian cycle around a network of six vertices Examples of Hamiltonian cycles on a square grid graph 8x8. In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once.
The problems of finding a Hamiltonian path and a Hamiltonian cycle can be related as follows: 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 ...
If the graph is a metric space then there is an efficient approximation algorithm that finds a Hamiltonian cycle with maximum edge weight being no more than twice the optimum. This result follows by Fleischner's theorem , that the square of a 2-vertex-connected graph always contains a Hamiltonian cycle.
Palmer (1997) describes the following simple algorithm for constructing a Hamiltonian cycle in a graph meeting Ore's condition. Arrange the vertices arbitrarily into a cycle, ignoring adjacencies in the graph. While the cycle contains two consecutive vertices v i and v i + 1 that are not adjacent in the graph, perform the following two steps:
A 2-vertex-connected graph, its square, and a Hamiltonian cycle in the square. In graph theory, a branch of mathematics, Fleischner's theorem gives a sufficient condition for a graph to contain a Hamiltonian cycle. It states that, if is a 2-vertex-connected graph, then the square of is Hamiltonian.
A graph is cubic (or 3-regular) if each vertex is the endpoint of exactly three edges. Finally, a graph is Hamiltonian if there exists a cycle that passes through each of its vertices exactly once. Barnette's conjecture states that every cubic bipartite polyhedral graph is Hamiltonian.
Dirac's theorem on Hamiltonian cycles, the statement that an n-vertex graph in which each vertex has degree at least n/2 must have a Hamiltonian cycle; Dirac's theorem on chordal graphs, the characterization of chordal graphs as graphs in which all minimal separators are cliques; Dirac's theorem on cycles in k-connected graphs, the result that ...
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.