Torsten Sillke, 20.06.97
The Hamiltonian paths of a special graph
are searched in:

Frits G"obel,
  Solving a Pentomino Problem with Graph Theory,
  CFF 43, (1997), p4-5

I dislike the drawing of the given graph.
First I cut of the lose ends.
It remains a 8 vertex graph and we are
looking for Hamiltonian path from F to Y.
If I discard Y, I am looking for a path from
F to a vertex in {N,P,T}. 

At last I reduce:

    X           X
    |          / \
    A   to    /   \
   / \       /     \
  X   X     X-------X

for A in {V,W,Z}.
Multiple edges can be discarded, as at most
one can be used without double visiting a vertex.

It remains a K_4 with the vertices {F,N,P,T}.
Now there are 6 = 3! Hamiltonian pathes starting 
at F:

      F---N---P---T
      F---N---T---P
      F---P---N---T
      F---P---T---N
      F---T---N---P
      F---T---P---N

Now try to extend each path by inserting the 
vertices {V,W,Z} into the three edges (and add Y).
This need not be possible, but in this case each
reduced solution extends uniquely (unique matching).

      F-Z-N-W-P-V-T-Y
      F-W-N-Z-T-V-P-Y
      F-W-P-V-N-Z-T-Y
      F-W-P-V-T-Z-N-Y
      F-Z-T-V-N-W-P-Y
      F-Z-T-V-P-W-N-Y

Now connect the lose ends to get all solutions:

      X-F-Z-N-W-P-V-T-Y-U-L-I
      X-F-W-N-Z-T-V-P-Y-U-L-I
      X-F-W-P-V-N-Z-T-Y-U-L-I
      X-F-W-P-V-T-Z-N-Y-U-L-I
      X-F-Z-T-V-N-W-P-Y-U-L-I
      X-F-Z-T-V-P-W-N-Y-U-L-I

2nd approach:
-------------
The alternation between vertices from {F,N,P,T} and
{V,W,Y,Z} is easily seen, as {V,W,Y,Z} is an independent
set. Therefore a Hamiltonian path between F and Y must
be alternating. So wie can discard all edges between
vertices from {F,N,P,T}. After ommitting Y the resulting
graph is very sparse:

               F
              / \
             Z   W
            / \ / \
           T   N   P
            \  |  /
             \ | /
              \|/
               V

All Hamiltonian pathes starting at F end in {N,P,T}
and therefore at Y. Again we find 6 pathes.