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X-ORIGINAL-URL:https://dimag.ibs.re.kr
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:20240101T000000
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BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20250604T163000
DTEND;TZID=Asia/Seoul:20250604T173000
DTSTAMP:20260416T181901
CREATED:20250319T134432Z
LAST-MODIFIED:20250603T071758Z
UID:10696-1749054600-1749058200@dimag.ibs.re.kr
SUMMARY:Denys Bulavka\, Strict Erdős-Ko-Rado Theorems for Simplicial Complexes
DESCRIPTION:The now classical theorem of Erdős\, Ko and Rado establishes\nthe size of a maximal uniform family of pairwise-intersecting sets as well as a characterization of the families attaining such upper bound. One natural extension of this theorem is that of restricting the possiblechoices for the sets. That is\, given a simplicial complex\, what is the size of a maximal uniform family of pairwise-intersecting faces. Holroyd and Talbot\, and Borg conjectured that the same phenomena as in the classical case (i.e.\, the simplex) occurs: there is a maximal size pairwise-intersecting family with all its faces having some common vertex. Under stronger hypothesis\, they also conjectured that if a family attains such bound then its members must have a common vertex. In this talk I will present some progress towards the characterization of the maximal families. Concretely I will show that the conjecture is true for near-cones of sufficiently high depth. In particular\, this implies that the characterization of maximal families holds for\, for example\, the independence complex of a chordal graph with an isolated vertex as well as the independence complex of a (large enough) disjoint union of graphs with at least one isolated vertex. Under stronger hypothesis\, i.e.\, more isolated vertices\, we also recover a stability theorem. \nThis talk is based on a joint work with Russ Woodroofe.
URL:https://dimag.ibs.re.kr/event/2025-06-04/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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DTSTART;TZID=Asia/Seoul:20250610T163000
DTEND;TZID=Asia/Seoul:20250610T173000
DTSTAMP:20260416T181901
CREATED:20250407T060939Z
LAST-MODIFIED:20250408T065935Z
UID:10754-1749573000-1749576600@dimag.ibs.re.kr
SUMMARY:On-Hei Solomon Lo\, Minors of non-hamiltonian graphs
DESCRIPTION:A seminal result of Tutte asserts that every 4-connected planar graph is hamiltonian. By Wagner’s theorem\, Tutte’s result can be restated as: every 4-connected graph with no $K_{3\,3}$ minor is hamiltonian. In 2018\, Ding and Marshall posed the problem of characterizing the minor-minimal 3-connected non-hamiltonian graphs. They conjectured that every 3-connected non-hamiltonian graph contains a minor of $K_{3\,4}$\, $\mathfrak{Q}^+$\, or the Herschel graph\, where $\mathfrak{Q}^+$ is obtained from the cube by adding a new vertex and connecting it to three vertices that share a common neighbor in the cube. We recently resolved this conjecture along with some related problems. In this talk\, we review the background and discuss the proof.
URL:https://dimag.ibs.re.kr/event/2025-06-10/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20250617T163000
DTEND;TZID=Asia/Seoul:20250617T173000
DTSTAMP:20260416T181901
CREATED:20250428T140029Z
LAST-MODIFIED:20250428T140029Z
UID:10877-1750177800-1750181400@dimag.ibs.re.kr
SUMMARY:Attila Jung\, The Quantitative Fractional Helly Theorem
DESCRIPTION:Two celebrated extensions of Helly’s theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of Barany\, Katchalski\, and Pach (1982). Improving on several recent works\, we prove an optimal combination of these two results. We show that given a family $F$ of $n$ convex sets in $\mathbb{R}^d$ such that at least $\alpha \binom{n}{d+1}$ of the $(d+1)$-tuples of $F$ have an intersection of volume at least 1\, then one can select $\Omega_{d\,\alpha}(n)$ members of $F$ whose intersection has volume at least $\Omega_d(1)$. Joint work with Nora Frankl and Istvan Tomon.
URL:https://dimag.ibs.re.kr/event/2025-06-17/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20250625T163000
DTEND;TZID=Asia/Seoul:20250625T173000
DTSTAMP:20260416T181901
CREATED:20250606T142052Z
LAST-MODIFIED:20250611T082038Z
UID:10973-1750869000-1750872600@dimag.ibs.re.kr
SUMMARY:Roohani Sharma\, Uniform and Constructive Polynomial Kernel for Deletion to $K_{2\,p}$ Minor-Free Graphs
DESCRIPTION:Let $\mathcal F$ be a fixed\, finite family of graphs. In the $\mathcal F$-Deletion problem\, the input is a graph G and a positive integer k\, and the goal is to determine if there exists a set of at most k vertices whose deletion results in a graph that does not contain any graph of $\mathcal F$ as a minor. The $\mathcal F$-Deletion problem encapsulates a large class of natural and interesting graph problems like Vertex Cover\, Feedback Vertex Set\, Treewidth-$\eta$ Deletion\, Treedepth-$\eta$ Deletion\, Pathwidth-$\eta$ Deletion\, Outerplanar Deletion\, Vertex Planarization and many more. We study the $\mathcal F$-Deletion problem from the kernelization perspective. \nIn a seminal work\, Fomin\, Lokshtanov\, Misra & Saurabh [FOCS 2012] gave a polynomial kernel for this problem when the family F contains at least one planar graph. The asymptotic growth of the size of the kernel is not uniform with respect to the family $\mathcal F$: that is\, the size of the kernel is $k^{f(\mathcal F)}$\, for some function f that depends only on $\mathcal F$. Also the size of the kernel depends on non-constructive constants. \nLater Giannopoulou\, Jansen\, Lokshtanov & Saurabh [TALG 2017] showed that the non-uniformity in the kernel size bound of Fomin et al. is unavoidable as Treewidth-$\eta$ Deletion\, cannot admit a kernel of size $O(k^{(\eta +1)/2 -\epsilon)}$\, for any $\epsilon >0$\, unless NP $\subseteq$ coNP/poly. On the other hand it was also shown that Treedepth-$\eta$ Deletion\, admits a uniform kernel of size $f(\mathcal F) k^6$\, showcasing that there are subclasses of $\mathcal F$ where the asymptotic kernel sizes do not grow as a function of the family $\mathcal F$. This work leads to the natural question of determining classes of $\mathcal F$ where the problem admits uniform polynomial kernels. \nIn this work\, we show that if all the graphs in $\mathcal F$ are connected and $\mathcal F$ contains $K_{2\,p}$ (a bipartite graph with 2 vertices on one side and p vertices on the other)\, then the problem admits a uniform kernel of size $f(\mathcal F) k^{10}$ where the constants in the size bound are also constructive. The graph $K_{2\,p}$ is one natural extension of the graph $\theta_p$\, where $\theta_p$ is a graph on two vertices and p parallel edges. The case when $\mathcal F$ contains $\theta_p$ has been studied earlier and serves as (the only) other example where the problem admits a uniform polynomial kernel. \nThis is joint work with William Lochet.
URL:https://dimag.ibs.re.kr/event/2025-06-25/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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