On certain Hamiltonian cycles in planar graphs

The problem is considered under which conditions a 4-connected planar or projective planar graph has a Hamiltonian cycle containing certain prescribed edges and missing certain forbidden edges. The results are applied to obtain novel lower bounds on the number of distinct Hamiltonian cycles that must be present in a 5-connected graph that is embedded into the plane or into the projective plane with face-width at least ve. Especially, we show that every 5-connected plane or projective plane triangulation on n vertices with no non-contractible cyles of length less than ve contains at least 2 O(n 1=4) distinct Hamiltonian cycles. c