Finding the flaw in a reduction from Hamiltonian cycle to Hamiltonian cycle on bipartitie graphs I'm trying to solve a problem for class that is stated like so:


  A bipartite graph is an undirected graph in which every cycle has even
  length.  We attempt to show that the Hamiltonian cycle (a cycle that
  passes through each node exactly once) problem polynomially reduces to
  the Hamiltonian cycle problem in bipartite graphs. We need a function
  $T:  \{\text{graphs}\} \to \{\text{bipartite graphs}\}$ such that $T$ can be computed in
  polynomial time and for any graph $G$, $G$ has Hamiltonian cycle iff $T(G)$
  has a Hamiltonian cycle. Let $T(G)$ be the bipartite graph obtained by
  inserting a new vertex on every edge. What is wrong with this
  transformation?


I think the problem with the transformation is that for $T(G)$ you need to insert an edge between each pair of vertices and not just insert a new vertex on every edge. I'm actually a bit stumped by this one. Any advice would be much appreciated!
The problem is that a Hamiltonian cycle has to pass through all vertices: if the cycle in the original graph does not take some edge $e$, then the vertex you split $e$ with will not be taken if you take the same route. This means that the resulting cycle is not Hamiltonian.