My research interests lie in discrete mathematics and complexity theory (MSC 05, 52, 60, 68) and my research falls mostly into one or more of the following topics: combinatorial probability, computational complexity, combinatorial geometry, random structures, and extremal problems. After a brief overview of my completed research, I will give a more detailed summary of some of its aspects and fu...

To a large extent the present work is far from being conclusive, instead, new directions of research in combinatorial extremal theory are started. Also questions concerning generalizations are immediately noticeable. The incentive came from problems in several fields such as Algebra, Geometry, Probability, Information and Complexity Theory. Like several basic combinatorial problems they may pla...

What is the maximum number of edges of the d-dimensional hypercube, denoted by S d k , that can be sliced by k hyperplanes? This question on combinatorial properties of Euclidean geometry arising from linear separability considerations in the theory of Perceptrons has become an issue on its own. We use computational and combinatorial methods to obtain new bounds for S d k , d 8. These strengthe...

Over the past twenty years, lecture hall partitions have emerged as fundamental combinatorial structures, leading to new generalizations and interpretations of classical theorems and new results. In recent years, geometric approaches to lecture hall partitions have used polyhedral geometry to discover further properties of these rich combinatorial objects. In this paper we give an overview of s...

If G is a finite graph, a proper coloring of G is a way to color the vertices of the graph using n colors so that no two vertices connected by an edge have the same color. (The celebrated four-color theorem asserts that if G is planar, then there is at least one proper coloring of G with four colors.) By a classical result of Birkhoff, the number of proper colorings of G with n colors is a poly...

In this article, we study relations between the local geometry of planar graphs (combinatorial curvature) and global geometric invariants, namely the Cheeger constants and the exponential growth. We also discuss spectral applications.

The notions of convexity and convex polytopes are introduced in the setting of tropical geometry. Combinatorial types of tropical polytopes are shown to be in bijection with regular triangulations of products of two simplices. Applications to phylogenetic trees are discussed.

a compact formulation of the set of tours neither in a graph nor its complement is presented and illustrates a general methodology proposed for constructing polyhedral models of decision problems based upon permutations, projection and lifting techniques. directed hamilton tours on n vertex graphs are interpreted as (n-1)- permutations. sets of extrema of birkhoff polyhedra are mapped to tours ...

We present a new normal form for bounded-curvature paths that admits a combinatorial discription of such paths and a bound on the algebraic complexity of the underlying geometry.

We introduce and prove some basic results about a combinatorial model which produces embedded polygons in the plane. The model is highly structured and relates to a number of topics in dynamics and geometry.