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     There are several ways how to relax the property that a graph possesses a     
  hamiltonian path/cycle. It is usually done if we are not able to show that the   
  graph is hamiltonian, or if we want to measure how close the graph is to being   
   Hamiltonian.\ \In this talk we will present problems and results of the above   
       type. In particular, we will discuss the famous Middle Layer Problem.       
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