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DTSTART;TZID=Asia/Seoul:20200428T163000
DTEND;TZID=Asia/Seoul:20200428T173000
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SUMMARY:Seunghun Lee (이승훈)\, Leray numbers of complexes of graphs with bounded matching number
DESCRIPTION: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. \nThis is a joint work with Andreas Holmsen.
URL:https://dimag.ibs.re.kr/event/2020-04-28/
LOCATION:Room B232\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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