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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.
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:
If so, the route is a Hamiltonian cycle. The Hamiltonian path problem and the Hamiltonian cycle problem belong to the class of NP-complete problems, as shown in Michael Garey and David S. Johnson's book Computers and Intractability: A Guide to the Theory of NP-Completeness and Richard Karp's list of 21 NP-complete problems. [2] [3]
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 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.
Barnette's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after David W. Barnette, a professor emeritus at the University of California, Davis; it states that every bipartite polyhedral graph with three edges per vertex has a Hamiltonian cycle.
Walecki's Hamiltonian decomposition of the complete graph . In graph theory, a branch of mathematics, a Hamiltonian decomposition of a given graph is a partition of the edges of the graph into Hamiltonian cycles. Hamiltonian decompositions have been studied both for undirected graphs and for directed graphs.
It implies that computing, up to the -th power of , the Hamiltonian cycle polynomial of a unitary n×n-matrix over the infinite extension of any ring of characteristic q (not necessarily prime) by the formal variable is a # P-complete problem if isn't 2 and computing the Hamiltonian cycle polynomial of a -semi-unitary matrix (i.e. an n×n ...