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DTSTART:20230101T000000
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DTSTART;TZID=Asia/Seoul:20240705T163000
DTEND;TZID=Asia/Seoul:20240705T173000
DTSTAMP:20260417T234125
CREATED:20240613T134309Z
LAST-MODIFIED:20240705T152046Z
UID:8763-1720197000-1720200600@dimag.ibs.re.kr
SUMMARY:Hyunwoo Lee (이현우)\, Random matchings in linear hypergraphs
DESCRIPTION:For a given hypergraph $H$ and a vertex $v\in V(H)$\, consider a random matching $M$ chosen uniformly from the set of all matchings in $H.$ In $1995\,$ Kahn conjectured that if $H$ is a $d$-regular linear $k$-uniform hypergraph\, the probability that $M$ does not cover $v$ is $(1 + o_d(1))d^{-1/k}$ for all vertices $v\in V(H)$. This conjecture was proved for $k = 2$ by Kahn and Kim in 1998. In this paper\, we disprove this conjecture for all $k \geq 3.$ For infinitely many values of $d\,$ we construct $d$-regular linear $k$-uniform hypergraph $H$ containing two vertices $v_1$ and $v_2$ such that $\mathcal{P}(v_1 \notin M) = 1 – \frac{(1 + o_d(1))}{d^{k-2}}$ and $\mathcal{P}(v_2 \notin M) = \frac{(1 + o_d(1))}{d+1}.$ The gap between $\mathcal{P}(v_1 \notin M)$ and $\mathcal{P}(v_2 \notin M)$ in this $H$ is best possible. In the course of proving this\, we also prove a hypergraph analog of Godsil’s result on matching polynomials and paths in graphs\, which is of independent interest.
URL:https://dimag.ibs.re.kr/event/2024-07-05/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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