Hamilton cycles in maximal planar graphs

A classical result of Whitney states that each maximal planar graph without separating triangles is hamiltonian, where a separating triangle is a triangle whose removal separates the graph. We extend Whitneyʹs Theorem by allowing some separating triangles. More precisely we define a tree B whose vertices are the separating triangles of the given maximal planar graph G and we show that if each vertex of the tree B has at most 2 children, then G is hamiltonian for any two boundary edges. Furthermore, it is shown that the conclusion of Whitneyʹs Theorem still holds if there are exactly three separating triangles.