기초과학연구원 이산수학그룹
Since the proof of the graph minor structure theorem by Robertson and Seymour in 2004, its underlying ideas have found applications in a much broader range of settings than their original context. They have driven profound progress in areas such as vertex minors, pivot minors, matroids, directed graphs, and 2-dimensional simplicial complexes. In this talk, …
The dimension of a poset is the least integer $d$ such that the poset is isomorphic to a subposet of the product of $d$ linear orders. In 1983, Kelly constructed planar posets of arbitrarily large dimension. Crucially, the posets in his construction involve large standard examples, the canonical structure preventing a poset from having small …
By utilizing the recently developed hypergraph analogue of Godsil's identity by the second author, we prove that for all $n \geq k \geq 2$, one can reconstruct the matching polynomial of an $n$-vertex $k$-uniform hypergraph from the multiset of all induced sub-hypergraphs on $\lfloor \frac{k-1}{k}n \rfloor + 1$ vertices. This generalizes the well-known result of …
In 2015, Kawarabayashi and Kreutzer proved the directed grid theorem. The theorem states the existence of a function f such that every digraph of directed tree-width f(k) contains a cylindrical grid of order k as a butterfly minor, but the given function grows non-elementarily with the size of the grid minor. We present an alternative …
We solve a long-standing open problem posed by Goodman and Pollack in 1988 by establishing a necessary and sufficient condition for a finite family of convex sets in $\mathbb{R}^d$ to admit a $k$-transversal (a $k$-dimensional affine subspace that intersects each set) for any $0 \le k \le d-1$. This result is a common generalization of …
A rooted spanning tree of a graph $G$ is called normal if the endvertices of all edges of $G$ are comparable in the tree order. It is well known that every finite connected graph has a normal spanning tree (also known as depth-first search tree). Also, all countable graphs have normal spanning trees, but uncountable …
What causes a graph to have high chromatic number? One obvious reason is containing a large clique (a set of pairwise adjacent vertices). This naturally leads to investigation of \(\chi\)-bounded classes of graphs --- classes where a large clique is essentially the only reason for large chromatic number. Unfortunately, many interesting graph classes are not …
Modern practical software libraries that are designed for isomorphism tests and symmetry computation rely on combinatorial techniques combined with techniques from algorithmic group theory. The Weisfeiler-Leman algorithm is such a combinatorial technique. When taking a certain view from descriptive complexity theory, the algorithm is universal. After an introduction to problems arising in symmetry computation and …
Let S and T be two sets of points in a metric space with a total of n points. Each point in S and T has an associated value that specifies an upper limit on how many points it can be matched with from the other set. A multimatching between S and T is a way of pairing points such that each point in S is matched with at least as many …
Given a tournament $S$, a tournament is $S$-free if it has no subtournament isomorphic to $S$. Until now, there have been only a small number of tournaments $S$ such that the complete structure of $S$-free tournaments is known. Let $\triangle(1, 2, 2)$ be a tournament obtained from the cyclic triangle by substituting two-vertex tournaments for …