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A Travelling salesman problem is : there is a sales man a n number of cities and the distance between them, he has to visit each sity travelling the minimum possible distance such that each city is visited only once.
NP- complete problem : its a nondeterministic polynomial time problem in which it is hard to find a solution.
hamiltonian cycle: you need to find a hamiltonian path between vertices such that each vertice is crosses only once and there are no vertices left...plus since its a cycle you need to move onto the next neighbour and in the end no vertice should be left unvisited.