KAIST Discrete Math Seminar

Many combinatorial problems and arguments concerning finite point sets in the Euclidean plane (or higher dimensions) often do not use the linear structure. A more general concept is that of an Oriented Matroid (OM). It is well-known that every OM can realized by pseudolines, and in fact most oriented matroids can not be realized by straight lines.
Recently, Alfreod Hubard (Courant Institute) and myself have found a new way to represent an OM by convex sets which retains much more of the “straightness” of the Euclidean plane. Interestingly, in our model the isotopy conjecture holds in a very strong sense, and it unifies several aspects of pseudoline arrangements.

We present a simple geometric description for the set of toroidal triangulations with two odd vertices where each vertex has degree five or more. Each such triangulation is described by a cut-and-glue construction starting from an infinite triangular grid. To achieve that, we define some invariants to study the cycles of toroidal triangulations. The motivation for studying such family comes from Fisk triangulations and Grünbaum coloring, which will be discussed as well.

Let V be a subvariety of the complex projective space. The amoeba of V is the set of all real vectors log|x| where x runs over all points of V in the complex torus. The asymptotic behavior of the amoeba is given by a polyhedral fan called the Bergman fan of V. We use the tropical geometry of the Bergman fan to prove the log-concavity conjecture of Rota and Welsh over any field. This work is joint with Eric Katz and is based on arXiv:1104.2519.

The hard-core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics, a multicast model of calls in communication networks, and as a weighted independent set problem in combinatorics, probability and theoretical computer science.
In this model, each independent set I in a graph G is weighted proportionally to λ^|I|, for a positive real parameter λ. For large λ, computing the partition function (namely, the normalizing constant which makes the weighting a probability distribution on a finite graph) on graphs of maximum degree Δ≥3, is a well known computationally challenging problem. More concretely, let λ_c(T_Δ) denote the critical value for the so-called uniqueness threshold of the hard-core model on the infinite Δ-regular tree; recent breakthrough results of Dror Weitz (2006) and Allan Sly (2010) have identified λ_c(T_Δ) as a threshold where the hardness of estimating the above partition function undergoes a computational transition.
We focus on the well-studied particular case of the square lattice Z², and provide a new lower bound for the uniqueness threshold, in particular taking it well above λ_c(T₄). Our technique refines and builds on the tree of self-avoiding walks approach of Weitz, resulting in a new technical sufficient criterion (of wider applicability) for establishing strong spatial mixing (and hence uniqueness) for the hard-core model. Our new criterion achieves better bounds on strong spatial mixing when the graph has extra structure, improving upon what can be achieved by just using the maximum degree. Applying our technique to Z² we prove that strong spatial mixing holds for all λ<2.3882, improving upon the work of Weitz that held for λ<27/16=1.6875. Our results imply a fully-polynomial deterministic approximation algorithm for estimating the partition function, as well as rapid mixing of the associated Glauber dynamics to sample from the hard-core distribution.
This is joint work with Ricardo Restrepo, Jinwoo Shin, Prasad Tetali, and Linji Yang. A preprint is available from the arXiv at: arxiv:1105.0914

It is very well known that every graph on n vertices and m edges admits a bipartition of size at least m/2. This bound can be improved to m/2 + (n-1)/4 for connected graphs, and m/2 + n/6 for graphs without isolated vertices, as proved by Edwards, and Erdős, Gyárfás, and Kohayakawa, respectively. A bisection of a graph is a bipartition in which the size of the two parts differ by at most 1. We prove that graphs with maximum degree o(n) in fact contain a bisection which asymptotically achieves the above bounds. These results follow from a more general theorem, which can also be used to answer several questions and conjectures of Bollobás and Scott on judicious bisections of graphs.
Joint work with Po-Shen Loh (CMU) and Benny Sudakov (UCLA)