199 research outputs found
Empty Rectangles and Graph Dimension
We consider rectangle graphs whose edges are defined by pairs of points in
diagonally opposite corners of empty axis-aligned rectangles. The maximum
number of edges of such a graph on points is shown to be 1/4 n^2 +n -2.
This number also has other interpretations:
* It is the maximum number of edges of a graph of dimension
\bbetween{3}{4}, i.e., of a graph with a realizer of the form
\pi_1,\pi_2,\ol{\pi_1},\ol{\pi_2}.
* It is the number of 1-faces in a special Scarf complex.
The last of these interpretations allows to deduce the maximum number of
empty axis-aligned rectangles spanned by 4-element subsets of a set of
points. Moreover, it follows that the extremal point sets for the two problems
coincide.
We investigate the maximum number of of edges of a graph of dimension
, i.e., of a graph with a realizer of the form
\pi_1,\pi_2,\pi_3,\ol{\pi_3}. This maximum is shown to be .
Box graphs are defined as the 3-dimensional analog of rectangle graphs. The
maximum number of edges of such a graph on points is shown to be
Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs
We study Markov chains for -orientations of plane graphs, these are
orientations where the outdegree of each vertex is prescribed by the value of a
given function . The set of -orientations of a plane graph has
a natural distributive lattice structure. The moves of the up-down Markov chain
on this distributive lattice corresponds to reversals of directed facial cycles
in the -orientation. We have a positive and several negative results
regarding the mixing time of such Markov chains.
A 2-orientation of a plane quadrangulation is an orientation where every
inner vertex has outdegree 2. We show that there is a class of plane
quadrangulations such that the up-down Markov chain on the 2-orientations of
these quadrangulations is slowly mixing. On the other hand the chain is rapidly
mixing on 2-orientations of quadrangulations with maximum degree at most 4.
Regarding examples for slow mixing we also revisit the case of 3-orientations
of triangulations which has been studied before by Miracle et al.. Our examples
for slow mixing are simpler and have a smaller maximum degree, Finally we
present the first example of a function and a class of plane
triangulations of constant maximum degree such that the up-down Markov chain on
the -orientations of these graphs is slowly mixing
Boolean dimension and tree-width
The dimension is a key measure of complexity of partially ordered sets. Small
dimension allows succinct encoding. Indeed if has dimension , then to
know whether in it is enough to check whether in each
of the linear extensions of a witnessing realizer. Focusing on the encoding
aspect Ne\v{s}et\v{r}il and Pudl\'{a}k defined a more expressive version of
dimension. A poset has boolean dimension at most if it is possible to
decide whether in by looking at the relative position of and
in only permutations of the elements of . We prove that posets with
cover graphs of bounded tree-width have bounded boolean dimension. This stays
in contrast with the fact that there are posets with cover graphs of tree-width
three and arbitrarily large dimension. This result might be a step towards a
resolution of the long-standing open problem: Do planar posets have bounded
boolean dimension?Comment: one more reference added; paper revised along the suggestion of three
reviewer
- …