KAIST Discrete Math Seminar

Assume K is a convex body in R^d and X is a (large) finite subset of K. How many convex polytopes are there whose vertices belong toX? Is there a typical shape of such polytopes? How well does the maximal such polytope (which is actually the convex hull of X) approximate K? In this lecture I will talk about these questions mainly in two cases. The first is when X is a random sample of n uniform, independent points from K. In this case motivation comes from Sylvester’s famous four-point problem and from the theory of random polytopes. The second case is when X is the set of lattice points contained in K and the questions come from integer programming and geometry of numbers. Surprisingly (or not so surprisingly), the answers in the two cases are rather similar. The methods are, however, very different.

Wiring diagrams are widely used combinatorial objects that are mainly used to describe reduced words of a permutation. In this talk, I will mention a fun property I recently found about those diagrams, and then introduce other results and problems related to this property.

Robertson and Seymour proved that graphs are well-quasi-ordered by the minor relation and the weak immersion relation. In other words, given infinitely many graphs, one graph contains another as a minor (or a weak immersion, respectively). Unlike the relation of minor and weak immersion, the topological minor relation does not well-quasi-order graphs in general. However, Robertson conjectured in the late 1980’s that for every positive integer k, the topological minor relation well-quasi-orders graphs that do not contain a topological minor isomorphic to the path of length k with each edge duplicated. We will sketch the idea of our recent proof of this conjecture. In addition, we will give a structure theorem for excluding a fixed graph as a topological minor. Such structure theorem were previously obtained by Grohe and Marx and by Dvorak, but we push one of the bounds in their theorems to the optimal value. This improvement is needed for our proof of Robertson’s conjecture. This work is joint with Robin Thomas.

The choosability of a graph G is the minimum k such that having k colors available at each vertex guarantees a proper coloring. Given a toroidal graph G, it is known that , and if and only if G contains . Cai, Wang, and Zhu proved that a toroidal graph G without 7-cycles is 6-choosable, and if and only if G contains . They also prove that a toroidal graph G without 6-cycles is 5-choosable, and conjecture that if and only if G contains . We disprove this conjecture by constructing an infinite family of non-4-colorable toroidal graphs with neither nor cycles of length at least 6; moreover, this family of graphs is embeddable on every surface except the plane and the projective plane. Instead, we prove the following slightly weaker statement suggested by Zhu: toroidal graphs containing neither (a missing one edge) nor 6-cycles are 4-choosable. This is sharp in the sense that forbidding only one of the two structures does not ensure that the graph is 4-choosable.

We introduce a discrete dynamical system on the set of partial orientations of a graph, which generalizes Gioan’s cycle-cocycle reversal system. We explain how this setup allows for a new interpretation of the linear equivalence of divisors on graphs (chip-firing), and a new proof of Baker and Norine’s combinatorial Riemann-Roch formula. Fundamental connections to the max-flow min-cut theorem will be highlighted.