The Hamiltonian Cycle Problem

Michael Haythorpe has carried out a research of Markov chain based algorithms for Hamiltonian cycle problem. Research has embedded Hamiltonian cycle problem to Markov decision process. Hamiltonian cycle was famous for solving traveling salesmen problem. Optimization problems have applied by Hamiltonian cycle problem and Markov decision process. Through that he has developed new theoretical results with that optimization models. Methods have been conducted a path like branches of tree to find a Hamiltonian cycle. Researcher has considered three graphs. First one was the one or more Hamiltonian cycles which contained by Hamiltonian graphs and Second one is the non-Hamiltonian cycles which conducted by Bridge graphs but it could be found out

Hamiltonian cycle problem has also gained acknowledgment because two special cases: the Knight’s tour and the Icosian game were solved by famous mathematicians Euler and Hamilton, respectively. Finally, Hamiltonian cycle problem is closely related to the even more famous Traveling salesman. The definition of Hamiltonian cycle problem is the following: given a graph Γ containing N nodes determines whether any simple cycles of length N exists in the graph. These are simple cycles of length N are known as Hamiltonian cycles. If Γ contains at least one Hamiltonian cycle, we say that Γ is a Hamiltonian graph. Otherwise, we say that Γ is a non-Hamiltonian graph. It is well known (Robinson and Wormald) that almost all regular graphs are Hamiltonian. It is also well known that all sufficiently dense graphs are Hamiltonian, and so in this thesis we primarily consider sparse graphs. For Hamiltonian graphs, a constructive solution to Hamiltonian cycle problem is to explicitly find a Hamiltonian cycle. Many heuristics have been designed that attempt to find a single Hamiltonian cycle as quickly as possible, solving the HCP quickly in most cases. For non-Hamiltonian graphs, however, no Hamiltonian cycles exist and the heuristics fail. In this thesis we consider three types of

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