On January 4, 2022, Seunghun Lee (이승훈) from the Binghamton University gave a talk at the Discrete Math Seminar on the transversal number and the chromatic number of hypergraphs whose edges are facets of a simplicial sphere. The title of his talk was “Transversals and colorings of simplicial spheres“.

## Seunghun Lee (이승훈), Transversals and colorings of simplicial spheres

Motivated from the surrounding property of a point set in $\mathbb{R}^d$ introduced by Holmsen, Pach and Tverberg, we consider the transversal number and chromatic number of a simplicial sphere. As an attempt to give a lower bound for the maximum transversal ratio of simplicial $d$-spheres, we provide two infinite constructions. The first construction gives infinitely many $(d+1)$-dimensional simplicial polytopes with the transversal ratio exactly $\frac{2}{d+2}$ for every $d\geq 2$. In the case of $d=2$, this meets the previously well-known upper bound $1/2$ tightly. The second gives infinitely many simplicial 3-spheres with the transversal ratio greater than $1/2$. This was unexpected from what was previously known about the surrounding property. Moreover, we show that, for $d\geq 3$, the facet hypergraph $\mathcal{F}(\mathbf{P})$ of a $(d+1)$-dimensional simplicial polytope $\mathbf{P}$ has the chromatic number $\chi(\mathcal{F}(\mathbf{P})) \in O(n^{\frac{\lceil d/2\rceil-1}{d}})$, where $n$ is the number of vertices of $\mathbf{P}$. This slightly improves the upper bound previously obtained by Heise, Panagiotou, Pikhurko, and Taraz. This is a joint work with Joseph Briggs and Michael Gene Dobbins.

## Seunghun Lee (이승훈) gave a talk on the complexes of subgraphs having no large matching at the Discrete Math Seminar

On April 28, 2020, Seunghun Lee (이승훈) from KAIST presented a talk on the topological property of the non-matching complex, that is a simplicial complex consisting of subgraphs on the same vertex set having no matching of size k and its application to the rainbow matching problem of graphs. The title of his talk is “Leray numbers of complexes of graphs with bounded matching number“.

## Seunghun Lee (이승훈), Leray numbers of complexes of graphs with bounded matching number

Given a graph $G$ on the vertex set $V$, the *non-matching complex* of $G$, $\mathsf{NM}_k(G)$, is the family of subgraphs $G’ \subset G$ whose matching number $\nu(G’)$ is strictly less than $k$. As an attempt to generalize the result by Linusson, Shareshian and Welker on the homotopy types of $\mathsf{NM}_k(K_n)$ and $\mathsf{NM}_k(K_{r,s})$ to arbitrary graphs $G$, we show that (i) $\mathsf{NM}_k(G)$ is $(3k-3)$-Leray, and (ii) if $G$ is bipartite, then $\mathsf{NM}_k(G)$ is $(2k-2)$-Leray. This result is obtained by analyzing the homology of the links of non-empty faces of the complex $\mathsf{NM}_k(G)$, which vanishes in all dimensions $d\geq 3k-4$, and all dimensions $d \geq 2k-3$ when $G$ is bipartite. As a corollary, we have the following rainbow matching theorem which generalizes the result by Aharoni et. al. and Drisko’s theorem: Let $E_1, \dots, E_{3k-2}$ be non-empty edge subsets of a graph and suppose that $\nu(E_i\cup E_j)\geq k$ for every $i\ne j$. Then $E=\bigcup E_i$ has a rainbow matching of size $k$. Furthermore, the number of edge sets $E_i$ can be reduced to $2k-1$ when $E$ is the edge set of a bipartite graph.

This is a joint work with Andreas Holmsen.