The Order Dimension of Planar Maps

This is a sequel to a previous paper entitled The Order Dimension of Convex Polytopes, by the same authors [SIAM J. Discrete Math., 6 (1993), pp. 230--245]. In that paper, we considered the poset {{\bf P}\protect\boldmath$_M\!\!\!$} formed by taking the vertices, edges, and faces of a 3-connected planar map {\bf M}, ordered by inclusion, and showed that the order dimension of {{\bf P}\protect\boldmath$_M\!\!\!$} is always equal to 4. In this paper, we show that if {\bf M} is any planar map, then the order dimension of {{\bf P}\protect\boldmath$_M\!\!\!$} is still at most 4.