On ordering of fl-description trees

Tutte introduced planar maps in the 1960s in connection with what later became the celebrated Four-Color Theorem. A planar map is an embedding of a planar graph in the plane. Description trees, in particular, fldescription trees, were introduced by Cori, Jacquard and Schaeffer in 1997, and they give a powerful tool to study planar maps.In this paper we introduce a relation on fl-description trees and conjecture that this relation is a total order. Towards solving this conjecture, we provide an embedding of fl(a, b) -trees into fl(a - t, b + t) -trees for t <= a <= b + t, which is a far-reaching generalization of an unpublished result of Claesson, Kitaev and Steingrimsson on embedding of fl(1 ,0)-trees into fl(0 ,1)-trees that gives a combinatorial proof of the fact that the number of rooted nonseparable planar maps with n +1 edges is more than the number of bicubic planar maps with 3n edges.