%A Isenmann, Lucas
%A GonÃ§alves, Daniel
%D 2019
%T Dushnik-Miller dimension of stair contact complexes
%K
%X 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 R d . We propose two different definitions of stairs in R d 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 R d .
%U http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1293
%J Acta Mathematica Universitatis Comenianae
%0 Journal Article
%P 827-834%V 88
%N 3
%@ 0862-9544
%8 2019-07-30