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PRODID:-//Discrete Mathematics Group - ECPv6.0.5//NONSGML v1.0//EN
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METHOD:PUBLISH
X-WR-CALNAME:Discrete Mathematics Group
X-ORIGINAL-URL:https://dimag.ibs.re.kr
X-WR-CALDESC:Events for Discrete Mathematics Group
REFRESH-INTERVAL;VALUE=DURATION:PT1H
X-Robots-Tag:noindex
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BEGIN:VTIMEZONE
TZID:Asia/Seoul
BEGIN:STANDARD
TZOFFSETFROM:+0900
TZOFFSETTO:+0900
TZNAME:KST
DTSTART:20220101T000000
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220425T163000
DTEND;TZID=Asia/Seoul:20220425T173000
DTSTAMP:20221209T152900
CREATED:20220425T073000Z
LAST-MODIFIED:20220410T130608Z
UID:5322-1650904200-1650907800@dimag.ibs.re.kr
SUMMARY:Boram Park (박보람)\, Odd coloring of sparse graphs
DESCRIPTION:We introduce an odd coloring of a graph\, which was introduced very recently\, motivated by parity type colorings of graphs. A proper vertex coloring of graph $G$ is said to be odd if for each non-isolated vertex $x \in V (G)$ there exists a color $c$ such that $c$ is used an odd number of times in the neighborhood of $x$. The recent work on this topic will be presented\, and the work is based on Eun-Kyung Cho\, Ilkyoo Choi\, and Hyemin Kown.
URL:https://dimag.ibs.re.kr/event/2022-04-25/
LOCATION:Room B232\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220502T163000
DTEND;TZID=Asia/Seoul:20220502T173000
DTSTAMP:20221209T152900
CREATED:20220502T073000Z
LAST-MODIFIED:20220426T130131Z
UID:5511-1651509000-1651512600@dimag.ibs.re.kr
SUMMARY:Cheolwon Heo (허철원)\, The complexity of the matroid-homomorphism problems
DESCRIPTION:In this talk\, we introduce homomorphisms between binary matroids that generalize graph homomorphisms. For a binary matroid $N$\, we prove a complexity dichotomy for the problem $\rm{Hom}_\mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. The problem is polynomial-time solvable if $N$ has a loop or has no circuits of odd length\, and is otherwise $\rm{NP}$-complete. We also get dichotomies for the list\, extension\, and retraction versions of the problem.\nThis is joint work with Hyobin Kim and Mark Siggers at Kyungpook National University.
URL:https://dimag.ibs.re.kr/event/2022-05-02/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220509T163000
DTEND;TZID=Asia/Seoul:20220509T173000
DTSTAMP:20221209T152900
CREATED:20220509T073000Z
LAST-MODIFIED:20220425T071741Z
UID:5524-1652113800-1652117400@dimag.ibs.re.kr
SUMMARY:Kyeongsik Nam (남경식)\, Large deviations for subgraph counts in random graphs
DESCRIPTION:The upper tail problem for subgraph counts in the Erdos-Renyi graph\, introduced by Janson-Ruciński\, has attracted a lot of attention. There is a class of Gibbs measures associated with subgraph counts\, called exponential random graph model (ERGM). Despite its importance\, lots of fundamental questions have remained unanswered owing to the lack of exact solvability. In this talk\, I will talk about a brief overview on the upper tail problem and the concentration of measure results for the ERGM. Joint work with Shirshendu Ganguly and Ella Hiesmayr.
URL:https://dimag.ibs.re.kr/event/2022-05-09/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220516T163000
DTEND;TZID=Asia/Seoul:20220516T173000
DTSTAMP:20221209T152900
CREATED:20220516T073000Z
LAST-MODIFIED:20220502T220811Z
UID:5553-1652718600-1652722200@dimag.ibs.re.kr
SUMMARY:Andreas Holmsen\, A colorful version of the Goodman-Pollack-Wenger transversal theorem
DESCRIPTION:Hadwiger’s transversal theorem gives necessary and sufficient conditions for the existence of a line transversal to a family of pairwise disjoint convex sets in the plane. These conditions were subsequently generalized to hyperplane transversals in $\mathbb{R}^d$ by Goodman\, Pollack\, and Wenger. Here we establish a colorful extension of their theorem\, which proves a conjecture of Arocha\, Bracho\, and Montejano. The proof uses topological methods\, in particular the Borsuk-Ulam theorem. The same methods also allow us to generalize some colorful transversal theorems of Montejano and Karasev.
URL:https://dimag.ibs.re.kr/event/2022-05-16/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220523T163000
DTEND;TZID=Asia/Seoul:20220523T173000
DTSTAMP:20221209T152900
CREATED:20220523T073000Z
LAST-MODIFIED:20220516T214605Z
UID:5451-1653323400-1653327000@dimag.ibs.re.kr
SUMMARY:Stijn Cambie\, The precise diameter of reconfiguration graphs
DESCRIPTION:Reconfiguration is about changing instances in small steps. For example\, one can perform certain moves on a Rubik’s cube\, each of them changing its configuration a bit. In this case\, in at most 20 steps\, one can end up with the preferred result. One could construct a graph with as nodes the possible configurations of the Rubik’s cube (up to some isomorphism) and connect two nodes if one can be obtained by applying only one move to the other. Finding an optimal solution\, i.e. a minimum number of moves to solve a Rubik’s cube is now equivalent to finding the distance in the graph. \nWe will wonder about similar problems in reconfiguration\, but applied to list- and DP-colouring. In this case\, the small step consists of recolouring precisely one vertex. Now we will be interested in the diameter of the reconfiguration graph and show that sometimes we can determine the precise diameters of these. \nAs such\, during this talk\, we present some main ideas of [arXiv:2204.07928].
URL:https://dimag.ibs.re.kr/event/2022-05-23/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220530T163000
DTEND;TZID=Asia/Seoul:20220530T173000
DTSTAMP:20221209T152900
CREATED:20220530T073000Z
LAST-MODIFIED:20220523T020807Z
UID:5495-1653928200-1653931800@dimag.ibs.re.kr
SUMMARY:Hongseok Yang (양홍석)\, Learning Symmetric Rules with SATNet
DESCRIPTION:SATNet is a differentiable constraint solver with a custom backpropagation algorithm\, which can be used as a layer in a deep-learning system. It is a promising proposal for bridging deep learning and logical reasoning. In fact\, SATNet has been successfully applied to learn\, among others\, the rules of a complex logical puzzle\, such as Sudoku\, just from input and output pairs where inputs are given as images. In this paper\, we show how to improve the learning of SATNet by exploiting symmetries in the target rules of a given but unknown logical puzzle or more generally a logical formula. We present SymSATNet\, a variant of SATNet that translates the given symmetries of the target rules to a condition on the parameters of SATNet and requires that the parameters should have a particular parametric form that guarantees the condition. The requirement dramatically reduces the number of parameters to learn for the rules with enough symmetries\, and makes the parameter learning of SymSATNet much easier than that of SATNet. We also describe a technique for automatically discovering symmetries of the target rules from examples. Our experiments with Sudoku and Rubik’s cube show the substantial improvement of SymSATNet over the baseline SATNet. \nThis is joint work with Sangho Lim and Eungyeol Oh.
URL:https://dimag.ibs.re.kr/event/2022-05-30/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220613T163000
DTEND;TZID=Asia/Seoul:20220613T173000
DTSTAMP:20221209T152900
CREATED:20220613T073000Z
LAST-MODIFIED:20220605T122500Z
UID:5578-1655137800-1655141400@dimag.ibs.re.kr
SUMMARY:Amadeus Reinald\, Twin-width and forbidden subdivisions
DESCRIPTION:Twin-width is a recently introduced graph parameter based on vertex contraction sequences. On classes of bounded twin-width\, problems expressible in FO logic can be solved in FPT time when provided with a sequence witnessing the bound. Classes of bounded twin-width are very diverse\, notably including bounded rank-width\, $\Omega ( \log (n) )$-subdivisions of graphs of size $n$\, and proper minor closed classes. In this talk\, we look at developing a structural understanding of twin-width in terms of induced subdivisions. \nStructural characterizations of graph parameters have mostly looked at graph minors\, for instance\, bounded tree-width graphs are exactly those forbidding a large wall minor. An analogue in terms of induced subgraphs could be that\, for sparse graphs\, large treewidth implies the existence of an induced subdivision of a large wall. However\, Sintiari and Trotignon have ruled out such a characterization by showing the existence of graphs with arbitrarily large girth avoiding any induced subdivision of a theta ($K_{2\,3}$). Abrishami\, Chudnovsky\, Hajebi and Spirkl have recently shown that such (theta\, triangle)-free classes have nevertheless logarithmic treewidth. \nAfter an introduction to twin-width and its ties to vertex orderings\, we show that theta-free graphs of girth at least 5 have bounded twin-width. \nJoint work with Édouard Bonnet\, Eun Jung Kim\, Stéphan Thomassé and Rémi Watrigant.
URL:https://dimag.ibs.re.kr/event/2022-06-13/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220627T163000
DTEND;TZID=Asia/Seoul:20220627T173000
DTSTAMP:20221209T152900
CREATED:20220627T073000Z
LAST-MODIFIED:20220627T115421Z
UID:5733-1656347400-1656351000@dimag.ibs.re.kr
SUMMARY:Ben Lund\, Radial projections in finite space
DESCRIPTION:Given a set $E$ and a point $y$ in a vector space over a finite field\, the radial projection $\pi_y(E)$ of $E$ from $y$ is the set of lines that through $y$ and points of $E$. Clearly\, $|\pi_y(E)|$ is at most the minimum of the number of lines through $y$ and $|E|$. I will discuss several results on the general question: For how many points $y$ can $|\pi_y(E)|$ be much smaller than this maximum? \nThis is motivated by an analogous question in fractal geometry. The Hausdorff dimension of a radial projection of a set $E$ in $n$ dimensional real space will typically be the minimum of $n-1$ and the Hausdorff dimension of $E$. Several recent papers by authors including Matilla\, Orponen\, Liu\, Shmerikin\, and Wang consider the question: How large can the set of points with small radial projections be? This body of work has several important applications\, including recent progress on the Falconer distance conjecture. \nThis is joint with Thang Pham and Vu Thi Huong Thu.
URL:https://dimag.ibs.re.kr/event/2022-06-27/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220704T163000
DTEND;TZID=Asia/Seoul:20220704T173000
DTSTAMP:20221209T152900
CREATED:20220704T073000Z
LAST-MODIFIED:20220609T232436Z
UID:5797-1656952200-1656955800@dimag.ibs.re.kr
SUMMARY:Eric Vigoda\, Computational phase transition and MCMC algorithms
DESCRIPTION:This talk will highlight recent results establishing a beautiful computational phase transition for approximate counting/sampling in (binary) undirected graphical models (such as the Ising model or on weighted independent sets). The computational problem is to sample from the equilibrium distribution of the model or equivalently approximate the corresponding normalizing factor known as the partition function. We show that when correlations die off on the infinite D-regular tree then the Gibbs sampler has optimal mixing time of $O(n \log n)$ on any graph of maximum degree D\, whereas when the correlations persist (in the limit) then the sampling/counting problem are NP-hard to approximate. The Gibbs sampler is a simple Markov Chain Monte Carlo (MCMC) algorithm. Key to these mixing results are a new technique known as spectral independence which considers the pairwise influence of vertices. We show that spectral independence implies an optimal convergence rate for a variety of MCMC algorithms.
URL:https://dimag.ibs.re.kr/event/2022-07-04/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220711T163000
DTEND;TZID=Asia/Seoul:20220711T173000
DTSTAMP:20221209T152900
CREATED:20220711T073000Z
LAST-MODIFIED:20220706T054910Z
UID:5896-1657557000-1657560600@dimag.ibs.re.kr
SUMMARY:Kevin Hendrey\, Product Structure of Graph Classes with Bounded Treewidth
DESCRIPTION:The strong product $G\boxtimes H$ of graphs $G$ and $H$ is the graph on the cartesian product $V(G)\times V(H)$ such that vertices $(v\,w)$ and $(x\,y)$ are adjacent if and only if $\max\{d_G(v\,x)\,d_H(w\,y)\}=1$. Graph product structure theory aims to describe complicated graphs in terms of subgraphs of strong products of simpler graphs. This area of research was initiated by Dujmović\, Joret\, Micek\, Morin\, Ueckerdt and Wood\, who showed that every planar graph is a subgraph of the strong product of a $H\boxtimes P\boxtimes K_3$ for some path $P$ and some graph $H$ of treewidth at most $3$. In this talk\, I will discuss the product structure of various graph classes of bounded treewidth. As an example\, we show that there is a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that every planar graph of treewidth at most $k$ is a subgraph of $H\boxtimes K_{f(k)}$ for some graph $H$ of treewidth at most $3$. \nThis is based on joint work with Campbell\, Clinch\, Distel\, Gollin\, Hickingbotham\, Huynh\, Illingworth\, Tamitegama\, Tan and Wood.
URL:https://dimag.ibs.re.kr/event/2022-07-11/
LOCATION:Room B232\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220718T163000
DTEND;TZID=Asia/Seoul:20220718T173000
DTSTAMP:20221209T152900
CREATED:20220622T073000Z
LAST-MODIFIED:20220621T152714Z
UID:5878-1658161800-1658165400@dimag.ibs.re.kr
SUMMARY:Jinyoung Park (박진영)\, Thresholds 1/2
DESCRIPTION:Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006\, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set\, its threshold is never far from its “expectation-threshold\,” which is a natural (and often easy to calculate) lower bound on the threshold. \nIn the first talk on Monday\, I will introduce the Kahn-Kalai Conjecture with some motivating examples and then briefly talk about the recent resolution of the Kahn-Kalai Conjecture due to Huy Pham and myself. \nIn the second talk on Tuesday\, I will discuss our proof of the conjecture in detail.
URL:https://dimag.ibs.re.kr/event/2022-07-18/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220719T140000
DTEND;TZID=Asia/Seoul:20220719T160000
DTSTAMP:20221209T152900
CREATED:20220719T050000Z
LAST-MODIFIED:20220621T152553Z
UID:5880-1658239200-1658246400@dimag.ibs.re.kr
SUMMARY:Jinyoung Park (박진영)\, Thresholds 2/2
DESCRIPTION:Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006\, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set\, its threshold is never far from its “expectation-threshold\,” which is a natural (and often easy to calculate) lower bound on the threshold. \nIn the first talk on Monday\, I will introduce the Kahn-Kalai Conjecture with some motivating examples and then briefly talk about the recent resolution of the Kahn-Kalai Conjecture due to Huy Pham and myself. \nIn the second talk on Tuesday\, I will discuss our proof of the conjecture in detail.
URL:https://dimag.ibs.re.kr/event/2022-07-19/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220801T163000
DTEND;TZID=Asia/Seoul:20220801T173000
DTSTAMP:20221209T152900
CREATED:20220801T073000Z
LAST-MODIFIED:20220620T165417Z
UID:5867-1659371400-1659375000@dimag.ibs.re.kr
SUMMARY:Seunghun Lee (이승훈)\, Inscribable order types
DESCRIPTION:We call an order type inscribable if it is realized by a point configuration where all extreme points are all on a circle. In this talk\, we investigate inscribability of order types. We first show that every simple order type with at most 2 interior points is inscribable\, and that the number of such order types is $\Theta(\frac{4^n}{n^{3/2}})$. We further construct an infinite family of minimally uninscribable order types. The proof of uninscribability mainly uses Möbius transformations. We also suggest open problems around inscribability. This is a joint work with Michael Gene Dobbins.
URL:https://dimag.ibs.re.kr/event/2022-08-01/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220809T163000
DTEND;TZID=Asia/Seoul:20220809T173000
DTSTAMP:20221209T152900
CREATED:20220808T073000Z
LAST-MODIFIED:20220715T132940Z
UID:5821-1660062600-1660066200@dimag.ibs.re.kr
SUMMARY:Eun Jung Kim (김은정)\, Directed flow-augmentation
DESCRIPTION:We show a flow-augmentation algorithm in directed graphs: There exists a polynomial-time algorithm that\, given a directed graph G\, two integers $s\,t\in V(G)$\, and an integer $k$\, adds (randomly) to $G$ a number of arcs such that for every minimal st-cut $Z$ in $G$ of size at most $k$\, with probability $2^{−\operatorname{poly}(k)}$ the set $Z$ becomes a minimum $st$-cut in the resulting graph.\nThe directed flow-augmentation tool allows us to prove fixed-parameter tractability of a number of problems parameterized by the cardinality of the deletion set\, whose parameterized complexity status was repeatedly posed as open problems:\n(1) Chain SAT\, defined by Chitnis\, Egri\, and Marx [ESA’13\, Algorithmica’17]\,\n(2) a number of weighted variants of classic directed cut problems\, such as Weighted st-Cut\, Weighted Directed Feedback Vertex Set\, or Weighted Almost 2-SAT.\nBy proving that Chain SAT is FPT\, we confirm a conjecture of Chitnis\, Egri\, and Marx that\, for any graph H\, if the List H-Coloring problem is polynomial-time solvable\, then the corresponding vertex-deletion problem is fixed-parameter tractable. \nJoint work with Stefan Kratsch\, Marcin Pilipczuk\, Magnus Wahlström.
URL:https://dimag.ibs.re.kr/event/2022-08-09/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220816T163000
DTEND;TZID=Asia/Seoul:20220816T173000
DTSTAMP:20221209T152900
CREATED:20220718T235006Z
LAST-MODIFIED:20220718T235006Z
UID:5967-1660667400-1660671000@dimag.ibs.re.kr
SUMMARY:Noleen Köhler\, Testing first-order definable properties on bounded degree graphs
DESCRIPTION:Property testers are probabilistic algorithms aiming to solve a decision problem efficiently in the context of big-data. A property tester for a property P has to decide (with high probability correctly) whether a given input graph has property P or is far from having property P while having local access to the graph. We study property testing of properties that are definable in first-order logic (FO) in the bounded-degree model. We show that any FO property that is defined by a formula with quantifier prefix ∃*∀* is testable\, while there exists an FO property that is expressible by a formula with quantifier prefix ∀*∃* that is not testable. In the dense graph model\, a similar picture is long known (Alon\, Fischer\, Krivelevich\, Szegedy\, Combinatorica 2000)\, despite the very different nature of the two models. In particular\, we obtain our lower bound by a first-order formula that defines a class of bounded-degree expanders\, based on zig-zag products of graphs. \nThis is joint work with Isolde Adler and Pan Peng.
URL:https://dimag.ibs.re.kr/event/2022-08-16/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220823T163000
DTEND;TZID=Asia/Seoul:20220823T173000
DTSTAMP:20221209T152900
CREATED:20220823T073000Z
LAST-MODIFIED:20220804T235842Z
UID:5971-1661272200-1661275800@dimag.ibs.re.kr
SUMMARY:Raul Lopes\, Temporal Menger and related problems
DESCRIPTION:A temporal graph is a graph whose edges are available only at specific times. In this scenario\, the only valid walks are the ones traversing adjacent edges respecting their availability\, i.e. sequence of adjacent edges whose appearing times are non-decreasing. \nGiven a graph G and vertices s and t of G\, Menger’s Theorem states that the maximum number of (internally) vertex disjoint s\,t-paths is equal to the minimum size of a subset X for which G-X contains no s\,t-path. This is a classical result in Graph Theory\, taught in most basic Graph Theory courses\, and it holds also when G is directed and when edge disjoint paths and edge cuts are considered instead. A direct translation of Menger’s Theorem to the temporal context has been known not to hold since an example was shown in the seminal paper by Kempe\, Kleinberg and Kumar (STOC’00). In this talk\, an overview of possible temporal versions of Menger’s Theorem will be discussed\, as well as the complexity of the related problems.
URL:https://dimag.ibs.re.kr/event/2022-08-23/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220830T163000
DTEND;TZID=Asia/Seoul:20220830T173000
DTSTAMP:20221209T152900
CREATED:20220830T073000Z
LAST-MODIFIED:20220805T023825Z
UID:6018-1661877000-1661880600@dimag.ibs.re.kr
SUMMARY:Jun Gao\, Number of (k-1)-cliques in k-critical graph
DESCRIPTION:We prove that for $n>k\geq 3$\, if $G$ is an $n$-vertex graph with chromatic number $k$ but any its proper subgraph has smaller chromatic number\, then $G$ contains at most $n-k+3$ copies of cliques of size $k-1$. This answers a problem of Abbott and Zhou and provides a tight bound on a conjecture of Gallai. \nThis is joint work with Jie Ma.
URL:https://dimag.ibs.re.kr/event/2022-08-30/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220906T163000
DTEND;TZID=Asia/Seoul:20220906T173000
DTSTAMP:20221209T152900
CREATED:20220719T105944Z
LAST-MODIFIED:20220906T113808Z
UID:5974-1662481800-1662485400@dimag.ibs.re.kr
SUMMARY:Bjarne Schülke\, A local version of Katona's intersection theorem
DESCRIPTION:Katona’s intersection theorem states that every intersecting family $\mathcal F\subseteq[n]^{(k)}$ satisfies $\vert\partial\mathcal F\vert\geq\vert\mathcal F\vert$\, where $\partial\mathcal F=\{F\setminus x:x\in F\in\mathcal F\}$ is the shadow of $\mathcal F$.\nFrankl conjectured that for $n>2k$ and every intersecting family $\mathcal F\subseteq [n]^{(k)}$\, there is some $i\in[n]$ such that $\vert \partial \mathcal F(i)\vert\geq \vert\mathcal F(i)\vert$\, where $\mathcal F(i)=\{F\setminus i:i\in F\in\mathcal F\}$ is the link of $\mathcal F$ at $i$. \nHere\, we prove this conjecture in a very strong form for $n> \binom{k+1}{2}$. \nIn particular\, our result implies that for any $j\in[k]$\, there is a $j$-set $\{a_1\,\dots\,a_j\}\in[n]^{(j)}$ such that \[ \vert \partial \mathcal F(a_1\,\dots\,a_j)\vert\geq \vert\mathcal F(a_1\,\dots\,a_j)\vert.\]A similar statement is also obtained for cross-intersecting families.
URL:https://dimag.ibs.re.kr/event/2022-09-06/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220913T163000
DTEND;TZID=Asia/Seoul:20220913T173000
DTSTAMP:20221209T152900
CREATED:20220720T105001Z
LAST-MODIFIED:20220720T105001Z
UID:5990-1663086600-1663090200@dimag.ibs.re.kr
SUMMARY:Sebastian Wiederrecht\, Killing a vortex
DESCRIPTION:The Structural Theorem of the Graph Minors series of Robertson and Seymour asserts that\, for every $t\in\mathbb{N}\,$ there exists some constant $c_{t}$ such that every $K_{t}$-minor-free graph admits a tree decomposition whose torsos can be transformed\, by the removal of at most $c_{t}$ vertices\, to graphs that can be seen as the union of some graph that is embeddable to some surface of Euler genus at most $c_{t}$ and “at most $c_{t}$ vortices of depth $c_{t}$”. Our main combinatorial result is a “vortex-free” refinement of the above structural theorem as follows: we identify a (parameterized) graph $H_{t}$\, called shallow vortex grid\, and we prove that if in the above structural theorem we replace $K_{t}$ by $H_{t}\,$ then the resulting decomposition becomes “vortex-free”. Up to now\, the most general classes of graphs admitting such a result were either bounded Euler genus graphs or the so called single-crossing minor-free graphs. Our result is tight in the sense that\, whenever we minor-exclude a graph that is not a minor of some $H_{t}\,$ the appearance of vortices is unavoidable. Using the above decomposition theorem\, we design an algorithm that\, given an $H_{t}$-minor-free graph $G$\, computes the generating function of all perfect matchings of $G$ in polynomial time. This algorithm yields\, on $H_{t}$-minor-free graphs\, polynomial algorithms for computational problems such as the {dimer problem\, the exact matching problem}\, and the computation of the permanent. Our results\, combined with known complexity results\, imply a complete characterization of minor-closed graphs classes where the number of perfect matchings is polynomially computable: They are exactly those graph classes that do not contain every $H_{t}$ as a minor. This provides a sharp complexity dichotomy for the problem of counting perfect matchings in minor-closed classes. \nThis is joint work with Dimitrios M. Thilikos.
URL:https://dimag.ibs.re.kr/event/2022-09-13/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20220927T163000
DTEND;TZID=Asia/Seoul:20220927T173000
DTSTAMP:20221209T152900
CREATED:20220825T021718Z
LAST-MODIFIED:20220902T090425Z
UID:6074-1664296200-1664299800@dimag.ibs.re.kr
SUMMARY:Alexander Clifton\, Ramsey Theory for Diffsequences
DESCRIPTION:Van der Waerden’s theorem states that any coloring of $\mathbb{N}$ with a finite number of colors will contain arbitrarily long monochromatic arithmetic progressions. This motivates the definition of the van der Waerden number $W(r\,k)$ which is the smallest $n$ such that any $r$-coloring of $\{1\,2\,\cdots\,n\}$ guarantees the presence of a monochromatic arithmetic progression of length $k$. \nIt is natural to ask what other arithmetic structures exhibit van der Waerden-type results. One notion\, introduced by Landman and Robertson\, is that of a $D$-diffsequence\, which is an increasing sequence $a_1