KAIST Discrete Math Seminar

The limits of converging sequences of graphs are natural objects from extremal graph theory that are also representable as measure-theoretic objects (graphons) or as algebraic objects (flag algebra homomorphisms). I will give an introduction to this theory via a geometric relative: its adaptation from graphs to a combinatorial encoding of planar point sets. This is based on joint work with Alfredo Hubard, Rémi de Joannis de Verclos, Jean-Sébastien Sereni and Jan Volec (http://drops.dagstuhl.de/opus/volltexte/2015/5126/). The talk will not assume any specific knowledge.

We show how to compute for n-vertex planar graphs in roughly O(n^11/6) expected time the diameter and the sum of the pairwise distances. These are the first algorithms for these problems using time O(n^c) for some constant c<2, even when restricted to undirected, unweighted planar graphs.

In combinatorial and computational geometry, the complexity of a system of sets is often studied via its shatter function. I will introduce these functions, and discuss how their asymptotic growth rate is governed from a single of its values, in the spirit of the classical notion of “Vapnik-Chernonenkis dimension” of hypergraphs. In particular, I will describe a probabilistic construction that refutes a conjecture of Bondy and Hajnal. This is joint work with Boris Bukh (https://arxiv.org/abs/1701.06632). The talk will start from first principles.

We provide a combinatorial characterization of all testable properties of k-graphs (i.e. k-uniform hypergraphs).
Here, a k-graph property ? is testable if there is a randomized algorithm which quickly distinguishes with high probability between k-graphs that satisfy ? and those that are far from satisfying ?. For the 2-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. This is joint work with Felix Joos, Deryk Osthus and Daniela Kühn.

Let P be a graph property. We look at graph colorings with k colors where each color class induces a graph satisfying P. By a result of Makowsky and Zilber (2005) the number of such colorings ?_P(G;k) is a polynomial in k. We present recent results and open problems on the complexity of evaluating ?_P(G;λ) for various properties P and (not only integer) values of λ.
This is joint work with A. Goodall, M. Hermann, T. Kotek and S. Noble which was initiated during last year’s program “Counting Complexity and Phase Transitions”. See also arXiv:1701.06639v1.