Is there anything known about the maximum number of simple-polygonal tours that a planar embedding of a Hamiltonian graph can have?

Remarks.

• I am especially interested in $K_5$ and $K_{3,3}$, mainly, because due to Kuratowski's characterization of planar graphs, one can be sure that permutations of $5$ (resp. $6$) cities, i.e. those permutations which correspond to a Hamilton circuit, can't appear in that order in the optimal tour through all $n$ cities; with $K_4$, this is not so.

• Letting $p$ denote the maximum number of simple polygonal tours and $h$ the number of Hamiltonian tours, $h-p$ permutations of a subset of $m$ cities can be excluded from the optimal tour through all $n$ cities.