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PRODID:-//Discrete Mathematics Group - ECPv6.15.20//NONSGML v1.0//EN
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X-ORIGINAL-URL:https://dimag.ibs.re.kr
X-WR-CALDESC:Events for Discrete Mathematics Group
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BEGIN:VTIMEZONE
TZID:Asia/Seoul
BEGIN:STANDARD
TZOFFSETFROM:+0900
TZOFFSETTO:+0900
TZNAME:KST
DTSTART:20230101T000000
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BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20241203T163000
DTEND;TZID=Asia/Seoul:20241203T173000
DTSTAMP:20260418T011032
CREATED:20241031T064935Z
LAST-MODIFIED:20241031T065407Z
UID:10022-1733243400-1733247000@dimag.ibs.re.kr
SUMMARY:Yulai Ma\, Pairwise disjoint perfect matchings in regular graphs
DESCRIPTION:An $r$-graph is an $r$-regular graph in which every odd set of vertices is connected to its complement by at least $r$ edges. A central question regarding $r$-graphs is determining the maximum number of pairwise disjoint perfect matchings they can contain. This talk explores how edge connectivity influences this parameter. \nFor ${0 \leq \lambda \leq r}$\, let $m(\lambda\,r)$ denote the maximum number $s$ such that every $\lambda$-edge-connected $r$-graph contains $s$ pairwise disjoint perfect matchings. The values of $m(\lambda\,r)$ are known only in limited cases; for example\, $m(3\,3)=m(4\,r)=1$\, and $m(r\,r) \leq r-2$ for all $r \not = 5$\, with $m(r\,r) \leq r-3$ when $r$ is a multiple of $4$. In this talk\, we present new upper bounds for $m(\lambda\,r)$ and examine connections between $m(5\,5)$ and several well-known conjectures for cubic graphs. \nThis is joint work with Davide Mattiolo\, Eckhard Steffen\, and Isaak H. Wolf.
URL:https://dimag.ibs.re.kr/event/2024-12-03/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20241213T163000
DTEND;TZID=Asia/Seoul:20241213T173000
DTSTAMP:20260418T011032
CREATED:20241115T050831Z
LAST-MODIFIED:20241116T074247Z
UID:10175-1734107400-1734111000@dimag.ibs.re.kr
SUMMARY:Jun Gao (高峻)\, Phase transition of degenerate Turán problems in p-norms
DESCRIPTION:For a positive real number $p$\, the $p$-norm $\|G\|_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $\mathrm{ex}_{p}(n\,F)$ of $F$-free graphs on $n$ vertices\, focusing on the case where $F$ is a bipartite graph. It is natural to conjecture that for every bipartite graph $F$\, there exists a threshold $p_F$ such that for $p< p_{F}$\, the order of $\mathrm{ex}_{p}(n\,F)$ is governed by pseudorandom constructions\, while for $p > p_{F}$\, it is governed by star-like constructions. We determine the exact value of $p_{F}$\, under a mild assumption on the growth rate of $\mathrm{ex}(n\,F)$. Our results extend to $r$-uniform hypergraphs as well. \nWe also prove a general upper bound that is tight up to a $\log n$ factor for $\mathrm{ex}_{p}(n\,F)$ when $p = p_{F}$.\nWe conjecture that this $\log n$ factor is unnecessary and prove this conjecture for several classes of well-studied bipartite graphs\, including one-side degree-bounded graphs and families of short even cycles. \nThis is a joint work with Xizhi Liu\, Jie Ma and Oleg Pikhurko.
URL:https://dimag.ibs.re.kr/event/2024-12-13/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20241217T163000
DTEND;TZID=Asia/Seoul:20241217T173000
DTSTAMP:20260418T011032
CREATED:20241115T062835Z
LAST-MODIFIED:20241115T064748Z
UID:10177-1734453000-1734456600@dimag.ibs.re.kr
SUMMARY:Joonkyung Lee (이준경)\, Counting homomorphisms in antiferromagnetic graphs via Lorentzian polynomials
DESCRIPTION:An edge-weighted graph $G$\, possibly with loops\, is said to be antiferromagnetic if it has nonnegative weights and at most one positive eigenvalue\, counting multiplicities. The number of graph homomorphisms from a graph $H$ to an antiferromagnetic graph $G$ generalises various important parameters in graph theory\, including the number of independent sets and proper vertex colourings. \nWe obtain a number of new homomorphism inequalities for antiferromagnetic target graphs $G$. In particular\, we prove that\, for any antiferromagnetic $G$\, \n$|\mathrm{Hom}(K_d\, G)|^{1/d} ≤ |\mathrm{Hom}(K_{d\,d} \setminus M\, G)|^{1/(2d)}$ \nholds\, where $K_{d\,d} \setminus M$ denotes the complete bipartite graph $K_{d\,d}$ minus a perfect matching $M$. This confirms a conjecture of Sah\, Sawhney\, Stoner and Zhao for complete graphs $K_d$. Our method uses the emerging theory of Lorentzian polynomials due to Brändén and Huh\, which may be of independent interest. \nJoint work with Jaeseong Oh and Jaehyeon Seo.
URL:https://dimag.ibs.re.kr/event/2024-12-17/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20241223T163000
DTEND;TZID=Asia/Seoul:20241223T173000
DTSTAMP:20260418T011032
CREATED:20241115T065846Z
LAST-MODIFIED:20241214T083034Z
UID:10184-1734971400-1734975000@dimag.ibs.re.kr
SUMMARY:Zixiang Xu (徐子翔)\, Multilinear polynomial methods and stability results on set systems
DESCRIPTION:In 1966\, Kleitman established that if \( |A \triangle B| \leq d \) for any \( A\, B \in \mathcal{F} \)\, then \( |\mathcal{F}| \leq \sum_{i=0}^{k} \binom{n}{i} \) for \( d = 2k \)\, and \( |\mathcal{F}| \leq 2 \sum_{i=0}^{k} \binom{n-1}{i} \) for \( d = 2k+1 \). These upper bounds are attained by the radius-\(k\) Hamming ball \( \mathcal{K}(n\, k) := \{ F : F \subseteq [n]\, |F| \leq k \} \) in the even case\, and by the family \( \mathcal{K}_y(n\, k) := \{ F : F \subseteq [n]\, |F \setminus \{y\}| \leq k \} \) in the odd case. In 2017\, Frankl provided a combinatorial proof of a stability result for Kleitman’s theorem\, offering improved upper bounds for \( |\mathcal{F}| \) when \( \mathcal{F} \) is not the extremal structure. \nIn this talk\, I will begin by demonstrating the application of multilinear polynomial methods in extremal set theory\, highlighting some interesting techniques. I will then present an algebraic proof of the stability result for Kleitman’s theorem. Finally\, I will discuss further applications and explore how to employ linear algebra methods more effectively and flexibly. \nThis talk is based on joint work with Jun Gao and Hong Liu.
URL:https://dimag.ibs.re.kr/event/2024-12-23/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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