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The Hamiltonian cycle problem is similar to the Hamiltonian path problem, except it asks if a given graph contains a Hamiltonian cycle. This problem may also specify the start of the cycle. The Hamiltonian cycle problem is a special case of the travelling salesman problem, obtained by setting the distance between two cities to one if they are ...
A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph . Similar notions may be defined for directed graphs , where each edge (arc) of a path or cycle can only be traced in a single direction (i.e., the vertices ...
The Hamiltonian completion problem is to find the minimal number of edges to add to a graph to make it Hamiltonian. The problem is clearly NP-hard in the general case (since its solution gives an answer to the NP-complete problem of determining whether a given graph has a Hamiltonian cycle). The associated decision problem of determining ...
Another version of Lovász conjecture states that . Every finite connected vertex-transitive graph contains a Hamiltonian cycle except the five known counterexamples.. There are 5 known examples of vertex-transitive graphs with no Hamiltonian cycles (but with Hamiltonian paths): the complete graph, the Petersen graph, the Coxeter graph and two graphs derived from the Petersen and Coxeter ...
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.
The problem still remains NP-hard. However, many heuristics work better for it than for other distance functions. The maximum scatter traveling salesman problem is another variation of the traveling salesman problem in which the goal is to find a Hamiltonian cycle that maximizes the minimum edge length rather than minimizing the maximum length ...
Illustration for the proof of Ore's theorem. In a graph with the Hamiltonian path v 1...v n but no Hamiltonian cycle, at most one of the two edges v 1 v i and v i − 1 v n (shown as blue dashed curves) can exist. For, if they both exist, then adding them to the path and removing the (red) edge v i − 1 v i would produce a Hamiltonian cycle.
William Rowan Hamilton, the inventor of the icosian game. At the time of his invention of the icosian game, William Rowan Hamilton was the Andrews Professor of Astronomy at Trinity College Dublin and Royal Astronomer of Ireland, and was already famous for his work on Hamiltonian mechanics and his invention of quaternions. [9]