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X-WR-CALNAME:Discrete Mathematics Group
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X-WR-CALDESC:Events for Discrete Mathematics Group
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TZID:Asia/Seoul
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TZOFFSETFROM:+0900
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DTSTART:20210101T000000
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BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20211207T163000
DTEND;TZID=Asia/Seoul:20211207T173000
DTSTAMP:20221206T093542
CREATED:20211207T073000Z
LAST-MODIFIED:20211204T004721Z
UID:4804-1638894600-1638898200@dimag.ibs.re.kr
SUMMARY:Eun-Kyung Cho (조은경)\, Independent domination of graphs with bounded maximum degree
DESCRIPTION:The independent domination number of a graph $G$\, denoted $i(G)$\, is the minimum size of an independent dominating set of $G$. In this talk\, we prove a series of results regarding independent domination of graphs with bounded maximum degree. \nLet $G$ be a graph with maximum degree at most $k$ where $k \ge 1$. We prove that if $k = 4$\, then $i(G) \le \frac{5}{9}|V(G)|$\, which is tight. Generalizing this result and a result by Akbari et al.\, we suggest a conjecture on the upper bound of $i(G)$ for $k \ge 1$\, which is tight if true. \nLet $G’$ be a connected $k$-regular graph that is not $K_{k\, k}$ where $k\geq 3$. We prove that $i(G’)\le \frac{k-1}{2k-1}|V(G’)|$\, which is tight for $k \in \{3\, 4\}$\, generalizing a result by Lam\, Shiu\, and Sun. This result also answers a question by Goddard et al. in the affirmative. \nIn addition\, we show that $\frac{i(G’)}{\gamma(G’)} \le \frac{k^3-3k^2+2}{2k^2-6k+2}$\, strengthening upon a result of Knor\, Škrekovski\, and Tepeh\, where $\gamma(G’)$ is the domination number of $G’$. \nMoreover\, if we restrict $G’$ to be a cubic graph without $4$-cycles\, then we prove that $i(G’) \le \frac{4}{11}|V(G’)|$\, which improves a result by Abrishami and Henning. \nThis talk is based on joint work with Ilkyoo Choi\, Hyemin Kwon\, and Boram Park.
URL:https://dimag.ibs.re.kr/event/2021-12-07/
LOCATION:Room B232\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20211214T163000
DTEND;TZID=Asia/Seoul:20211214T173000
DTSTAMP:20221206T093542
CREATED:20211214T073000Z
LAST-MODIFIED:20211201T235610Z
UID:4913-1639499400-1639503000@dimag.ibs.re.kr
SUMMARY:Tuan Tran\, Exponential decay of intersection volume with applications on list-decodability and sphere-covering bounds
DESCRIPTION:We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking\, this happens when the metric space is (i) expanding and (ii) well-spread\, and (iii) certain random variable on the boundary of a ball has a small tail. As applications\, we show that the volume of intersection of balls in Hamming space and symmetric groups decays exponentially as their centers drift apart. To verify condition (iii)\, we prove some deviation inequalities `on the slice’ for functions with Lipschitz conditions. \nWe then use these estimates on intersection volumes to \n\nobtain a sharp lower bound on list-decodability of random q-ary codes\, confirming a conjecture of Li and Wootters [IEEE Trans. Inf. Theory 2021]; and\nimprove sphere-covering bound from the 70s on constant weight codes by a factor linear in dimension\, resolving a problem raised by Jiang and Vardy [IEEE Trans. Inf. Theory 2004].\n\nOur probabilistic point of view also offers a unified framework to obtain improvements on other sphere-covering bounds\, giving conceptually simple and calculation-free proofs for q-ary codes\, permutation codes\, and spherical codes. \nThis is joint work with Jaehoon Kim and Hong Liu.
URL:https://dimag.ibs.re.kr/event/2021-12-14/
LOCATION:Room B232\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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