Assume a weighted graph G and a positive value k are given.  What is the complexity of finding a cycle with total weight k when G is Hamiltonian or Hamiltonian-connected? pointing to papers and books is also welcome!I wish it wouldnt look as a homework!
Complexity of weighted cycle in a hamiltonian graph
cc.complexity theory;graph theory
If we are asking for a simple cycle the problem is NP-complete by a reduction from the Hamiltonian cycle problem. We want to find a Hamiltonian cycle in a graph $G$. We assign weight 1 to all edges of $G$, and add to this graph all the other edges with weight $\infty$. We have thus created a clique, which is obviously Hamiltonian, and we ask whether there exist a cycle with weight $n$ in it.