Non-hamiltonian Graphs with unique hamiltonian path between exactly 4 pair of vertices
Here's a trivial example class [and now, also generalized slightly in view of Esha's comment]: graphs G(h,k,n) constructed from a path of length n, by adjoining a cycle of length h and a cycle of length k to the two ends. If the two cycles are


  A1&nbsp;&ndash;&nbsp;P0&nbsp;&ndash;&nbsp;A2&nbsp;&ndash;&nbsp;...&nbsp;&ndash;&nbsp;Ah&minus;1&nbsp;&ndash;&nbsp;A1


and


  B1&nbsp;&ndash;&nbsp;Pn&nbsp;&ndash;&nbsp;B2&nbsp;&ndash;&nbsp;...&nbsp;&ndash;&nbsp;Bk&minus;1&nbsp;&ndash;&nbsp;B1&nbsp;,


connected by the path


  P0&nbsp;&ndash;&nbsp;P1&nbsp;&ndash;&nbsp;...&nbsp;&ndash;&nbsp;Pn&nbsp;,


then there are Hamiltonian paths between each vertex pair {Aa&nbsp;,&nbsp;Bb} for a,b&nbsp;&isin;&nbsp;{1,2} but for no others.