Dushnik-Miller dimension of stair contact complexes

Main Article Content

Lucas Isenmann Daniel Gonçalves


The theorem of Schnyder asserts that a graph is planar if and only if the Dushnik-Miller dimension of its poset of incidence is at most 3. Trotter aksed how this can be generalized to higher dimensions. Towards this goal, Dushnik-Miller dimension has been studied in terms of TD-Delaunay complexes, in terms of orthogonal surfaces, and in terms of polytopes. Here we consider the relation between the Dushnik-Miller dimension and contact systems of stairs in Rd.We propose two different definitions of stairs in Rd which are connected to the Dushnik-Miller dimension as follows.  The first definition allows us to characterize supremum sections, which are simplicial complexes related to the Dushnik-Miller dimension, in two different ways.  The second definition provides for any Dushnik-Miller dimension at most d+1 complex a representation as a contact system of stairs in Rd.

Article Details

How to Cite
Isenmann, L., & Gonçalves, D. (2019). Dushnik-Miller dimension of stair contact complexes. Acta Mathematica Universitatis Comenianae, 88(3), 827-834. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1293/733