Ilkyoo Choi (최일규)

Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si

2018/11/26 Mon 5PM-6PM (Bldg. E6-1, Room 1401)

For a graph G, let f2(G) denote the largest number of vertices in a 2-regular subgraph of G. We determine the minimum of f2(G) over 3-regular n-vertex simple graphs G.

To do this, [...]

The post Ilkyoo Choi (최일규), Largest 2-regular subgraphs in 3-regular graphs appeared first on KAIST Discrete Math Seminar.

]]>Ilkyoo Choi (최일규)

Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si

Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si

2018/11/26 Mon 5PM-6PM (Bldg. E6-1, Room 1401)

For a graph G, let f_{2}(G) denote the largest number of vertices in a 2-regular subgraph of G. We determine the minimum of f_{2}(G) over 3-regular n-vertex simple graphs G.

To do this, we prove that every 3-regular multigraph with exactly c cut-edges has a 2-regular subgraph that omits at most max{0,⎣(c-1)/2⎦} vertices.

More generally, every n-vertex multigraph with maximum degree 3 and m edges has a 2-regular subgraph that omits at most max{0, ⎣(3n-2m+c-1)/2⎦} vertices.

These bounds are sharp; we describe the extremal multigraphs.

This is joint work with Ringi Kim, Alexandr V. Kostochka, Boram Park, and Douglas B. West.

To do this, we prove that every 3-regular multigraph with exactly c cut-edges has a 2-regular subgraph that omits at most max{0,⎣(c-1)/2⎦} vertices.

More generally, every n-vertex multigraph with maximum degree 3 and m edges has a 2-regular subgraph that omits at most max{0, ⎣(3n-2m+c-1)/2⎦} vertices.

These bounds are sharp; we describe the extremal multigraphs.

This is joint work with Ringi Kim, Alexandr V. Kostochka, Boram Park, and Douglas B. West.

The post Ilkyoo Choi (최일규), Largest 2-regular subgraphs in 3-regular graphs appeared first on KAIST Discrete Math Seminar.

]]>Jaehoon Kim (김재훈)

Mathematics Institute, University of Warwick, UK

2018/10/15 5PM

Graphs are mathematical structures used to model pairwise relations between objects.

Graph decomposition problems ask to partition the edges of large/dense graphs into small/sparse graphs.

In this talk, we introduce several famous graph decomposition problems, related puzzles and known results on the problems.

The post Jaehoon Kim, Introduction to Graph Decomposition appeared first on KAIST Discrete Math Seminar.

]]>Jaehoon Kim (김재훈)

Mathematics Institute, University of Warwick, UK

Mathematics Institute, University of Warwick, UK

2018/10/15 5PM

Graphs are mathematical structures used to model pairwise relations between objects.

Graph decomposition problems ask to partition the edges of large/dense graphs into small/sparse graphs.

In this talk, we introduce several famous graph decomposition problems, related puzzles and known results on the problems.

Graph decomposition problems ask to partition the edges of large/dense graphs into small/sparse graphs.

In this talk, we introduce several famous graph decomposition problems, related puzzles and known results on the problems.

The post Jaehoon Kim, Introduction to Graph Decomposition appeared first on KAIST Discrete Math Seminar.

]]>Jaehoon Kim (김재훈)

Mathematics Institute, University of Warwick, UK

2018/10/15 2:30PM

We say a subgraph H of an edge-colored graph is rainbow if all edges in H has distinct colors. The concept of rainbow subgraphs generalizes the concept of transversals in latin squares.

In this talk, we discuss how these concepts are related and we [...]

The post Jaehoon Kim, Rainbow subgraphs in graphs appeared first on KAIST Discrete Math Seminar.

]]>Jaehoon Kim (김재훈)

Mathematics Institute, University of Warwick, UK

Mathematics Institute, University of Warwick, UK

2018/10/15 2:30PM

We say a subgraph H of an edge-colored graph is rainbow if all edges in H has distinct colors. The concept of rainbow subgraphs generalizes the concept of transversals in latin squares.

In this talk, we discuss how these concepts are related and we introduce a result regarding approximate decompositions of graphs into rainbow subgraphs. This has implications on transversals in latin square. It is based on a joint work with Kühn, Kupavskii and Osthus.

In this talk, we discuss how these concepts are related and we introduce a result regarding approximate decompositions of graphs into rainbow subgraphs. This has implications on transversals in latin square. It is based on a joint work with Kühn, Kupavskii and Osthus.

The post Jaehoon Kim, Rainbow subgraphs in graphs appeared first on KAIST Discrete Math Seminar.

]]>Dong Yeap Kang (강동엽)

Department of Mathematical Sciences, KAIST

2018/11/5 Mon 5PM-6PM

The extremal number ex(n,F) of a graph F is the maximum number of edges in an n-vertex graph not containing F as a subgraph. A real number r∈[1,2] is realisable if there exists a graph F with ex(n , F) = [...]

The post Dong Yeap Kang (강동엽), On the rational Turán exponents conjecture appeared first on KAIST Discrete Math Seminar.

]]>Dong Yeap Kang (강동엽)

Department of Mathematical Sciences, KAIST

Department of Mathematical Sciences, KAIST

2018/11/5 Mon 5PM-6PM

The extremal number ex(n,F) of a graph F is the maximum number of edges in an n-vertex graph not containing F as a subgraph. A real number r∈[1,2] is *realisable* if there exists a graph F with ex(n , F) = Θ(n^{r}). Several decades ago, Erdős and Simonovits conjectured that every rational number in [1,2] is realisable. Despite decades of effort, the only known realisable numbers are 1,7/5,2, and the numbers of the form 1+(1/m), 2-(1/m), 2-(2/m) for integers m≥1. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than two numbers 1 and 2.

We discuss some recent progress on the conjecture of Erdős and Simonovits. First, we show that 2-(a/b) is realisable for any integers a,b≥1 with b>a and b≡±1 (mod a). This includes all previously known ones, and gives infinitely many limit points 2-(1/m) in the set of all realisable numbers as a consequence.

Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.

This is joint work with Jaehoon Kim and Hong Liu.

We discuss some recent progress on the conjecture of Erdős and Simonovits. First, we show that 2-(a/b) is realisable for any integers a,b≥1 with b>a and b≡±1 (mod a). This includes all previously known ones, and gives infinitely many limit points 2-(1/m) in the set of all realisable numbers as a consequence.

Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.

This is joint work with Jaehoon Kim and Hong Liu.

The post Dong Yeap Kang (강동엽), On the rational Turán exponents conjecture appeared first on KAIST Discrete Math Seminar.

]]>JinHoo Ahn (안진후)

Department of Mathematics, Yonsei University, Seoul

2018/10/1 Mon 5PM-6PM (E6-1, Room 1401)

Any structure whose language is finite has a model of graph theory which is bi-interpretable with it. From this idea, Mekler further developed a way of interpreting a model into a group. This Mekler's construction preserves various model-theoretic properties [...]

The post JinHoo Ahn (안진후), Mekler’s Construction on NTP1 Theory appeared first on KAIST Discrete Math Seminar.

]]>JinHoo Ahn (안진후)

Department of Mathematics, Yonsei University, Seoul

Department of Mathematics, Yonsei University, Seoul

2018/10/1 Mon 5PM-6PM (E6-1, Room 1401)

Any structure whose language is finite has a model of graph theory which is bi-interpretable with it. From this idea, Mekler further developed a way of interpreting a model into a group. This Mekler’s construction preserves various model-theoretic properties such as stability, simplicity, and NTP2, thus helps us find new group examples in model theory. In this talk, I will introduce to you what Mekler’s construction is and briefly show that this preserves NTP1.

The post JinHoo Ahn (안진후), Mekler’s Construction on NTP1 Theory appeared first on KAIST Discrete Math Seminar.

]]>Joonkyung Lee (이준경)

Universität Hamburg, Hamburg, Germany

2018/9/17 Monday 5PM

One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if H is a bipartite graph with maximum degree r on one side, then there is a constant [...]

The post Joonkyung Lee (이준경), The extremal number of subdivisions appeared first on KAIST Discrete Math Seminar.

]]>Joonkyung Lee (이준경)

Universität Hamburg, Hamburg, Germany

Universität Hamburg, Hamburg, Germany

2018/9/17 Monday 5PM

One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if H is a bipartite graph with maximum degree r on one side, then there is a constant C such that every graph with n vertices and C n^{2 – 1/r} edges contains a copy of H. This result is tight up to the constant when H contains a copy of K_{r,s} with s sufficiently large in terms of r. We conjecture that this is essentially the only situation in which Füredi’s result can be tight and prove this conjecture for r = 2. More precisely, we show that if H is a C_{4}-free bipartite graph with maximum degree 2 on one side, then there are positive constants C and δ such that every graph with n vertices and C n^{3/2 – δ} edges contains a copy of H. This answers a question by Erdős from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest. This is joint work with David Conlon.

The post Joonkyung Lee (이준경), The extremal number of subdivisions appeared first on KAIST Discrete Math Seminar.

]]>Euiwoong Lee (이의웅)

Courant Institute of Mathematical Sciences, NYU, New York, USA

2018/11/13 Tue 5PM (Bldg. E6-1, Room 1401)

In the k-cut problem, we are given an edge-weighted graph G and an integer k, and have to remove a set of edges with minimum total weight so that G has at [...]

The post Euiwoong Lee (이의웅), Faster Exact and Approximate Algorithms for k-Cut appeared first on KAIST Discrete Math Seminar.

]]>Euiwoong Lee (이의웅)

Courant Institute of Mathematical Sciences, NYU, New York, USA

Courant Institute of Mathematical Sciences, NYU, New York, USA

2018/11/13 Tue 5PM (Bldg. E6-1, Room 1401)

In the k-cut problem, we are given an edge-weighted graph G and an integer k, and have to remove a set of edges with minimum total weight so that G has at least k connected components. This problem has been studied various algorithmic perspectives including randomized algorithms, fixed-parameter tractable algorithms, and approximation algorithms. Their proofs of performance guarantees often reveal elegant structures for cuts in graphs.

It has still remained an open problem to (a) improve the runtime of exact algorithms, and (b) to get better approximation algorithms. In this talk, I will give an overview on recent progresses on both exact and approximation algorithms. Our algorithms are inspired by structural similarities between k-cut and the k-clique problem.

It has still remained an open problem to (a) improve the runtime of exact algorithms, and (b) to get better approximation algorithms. In this talk, I will give an overview on recent progresses on both exact and approximation algorithms. Our algorithms are inspired by structural similarities between k-cut and the k-clique problem.

The post Euiwoong Lee (이의웅), Faster Exact and Approximate Algorithms for k-Cut appeared first on KAIST Discrete Math Seminar.

]]>Bridging Continuous and Discrete Optimization through the Lens of Approximation

Euiwoong Lee (이의웅)

Courant Institute of Mathematical Sciences, NYU, New York, USA

2018/11/12 Mon 4PM (Bldg. E3-1, Room 1501)

Mathematical optimization is a field that studies algorithms, given an objective function and a feasible set, to find the element that maximizes or minimizes the objective [...]

The post Euiwoong Lee (이의웅), Bridging Continuous and Discrete Optimization through the Lens of Approximation appeared first on KAIST Discrete Math Seminar.

]]>Bridging Continuous and Discrete Optimization through the Lens of Approximation

Euiwoong Lee (이의웅)

Courant Institute of Mathematical Sciences, NYU, New York, USA

Courant Institute of Mathematical Sciences, NYU, New York, USA

2018/11/12 Mon 4PM (Bldg. E3-1, Room 1501)

Mathematical optimization is a field that studies algorithms, given an objective function and a feasible set, to find the element that maximizes or minimizes the objective function from the feasible set. While most interesting optimization problems are NP-hard and unlikely to admit efficient algorithms that find the exact optimal solution, many of them admit efficient “approximation algorithms” that find an approximate optimal solution.

Continuous and discrete optimization are the two main branches of mathematical optimization that have been primarily studied in different contexts. In this talk, I will introduce my recent results that bridge some of important continuous and discrete optimization problems, such as matrix low-rank approximation and Unique Games. At the heart of the connection is the problem of approximately computing the operator norm of a matrix with respect to various norms, which will allow us to borrow mathematical tools from functional analysis.

Continuous and discrete optimization are the two main branches of mathematical optimization that have been primarily studied in different contexts. In this talk, I will introduce my recent results that bridge some of important continuous and discrete optimization problems, such as matrix low-rank approximation and Unique Games. At the heart of the connection is the problem of approximately computing the operator norm of a matrix with respect to various norms, which will allow us to borrow mathematical tools from functional analysis.

The post Euiwoong Lee (이의웅), Bridging Continuous and Discrete Optimization through the Lens of Approximation appeared first on KAIST Discrete Math Seminar.

]]>Jineon Baek (백진언)

NIMS, Daejeon

2018/09/10 Mon 5PM-6PM (Room 1401 of Bldg. E6-1)

The infamous Erdős-Szekeres conjecture, posed in 1935, states that the minimum number ES(n) of points on a plane in general position (that is, no three colinear points) that guarantees a subset of n points in convex position is equal [...]

The post Jineon Baek, On the off-diagonal Erdős-Szekeres convex polygon problem appeared first on KAIST Discrete Math Seminar.

]]>Jineon Baek (백진언)

NIMS, Daejeon

NIMS, Daejeon

2018/09/10 Mon 5PM-6PM (Room 1401 of Bldg. E6-1)

The infamous Erdős-Szekeres conjecture, posed in 1935, states that the minimum number ES(n) of points on a plane in general position (that is, no three colinear points) that guarantees a subset of n points in convex position is equal to 2^{(n-2)} + 1. Despite many years of effort, the upper bound of ES(n) had not been better than O(4^{n – o(n)}) until Suk proved the groundbreaking result ES(n)≤2^{n+o(n)} in 2016.

In this talk, we focus on a variant of this problem by Erdős, Tuza and Valtr regarding the number ETV(a, b, n) of points needed to force either an a-cup, b-cap or a convex n-gon for varying a, b and n. They showed ETV(a, b, n) > \sum_{i=n-b}^{a-2} \binom{n}{i-2} by supplying a set of points with no a-cup, b-cap nor a n-gon of that many number, and conjectured that the inequality cannot be improved. Due to their construction, the conjecture is in fact equivalent to the Erdős-Szekeres conjecture. However, even the simplest equality ETV(4, n, n) = \binom{n+1}{2} + 1, which must be true if the Erdős-Szekeres conjecture is, has not been verified yet. To the best of our knowledge, the bound ETV(4, n, n) ≤ \binom{n + 2}{2} – 1, mentioned by Balko and Valtr in 2015, is currently the best bound known in literature.

The talk is divided into two parts. First, we introduce the mentioned works on the Erdős-Szekeres conjecture and observe that the argument of Suk can be directly adapted to prove an improved bound of ETV(a, n, n). Then we show the bound ETV(4, n, n) ≤ \binom{n+2}{2} – C \sqrt{n} for a fixed constant C>0, improving the previously known best bound of Balko and Valtr.

In this talk, we focus on a variant of this problem by Erdős, Tuza and Valtr regarding the number ETV(a, b, n) of points needed to force either an a-cup, b-cap or a convex n-gon for varying a, b and n. They showed ETV(a, b, n) > \sum_{i=n-b}^{a-2} \binom{n}{i-2} by supplying a set of points with no a-cup, b-cap nor a n-gon of that many number, and conjectured that the inequality cannot be improved. Due to their construction, the conjecture is in fact equivalent to the Erdős-Szekeres conjecture. However, even the simplest equality ETV(4, n, n) = \binom{n+1}{2} + 1, which must be true if the Erdős-Szekeres conjecture is, has not been verified yet. To the best of our knowledge, the bound ETV(4, n, n) ≤ \binom{n + 2}{2} – 1, mentioned by Balko and Valtr in 2015, is currently the best bound known in literature.

The talk is divided into two parts. First, we introduce the mentioned works on the Erdős-Szekeres conjecture and observe that the argument of Suk can be directly adapted to prove an improved bound of ETV(a, n, n). Then we show the bound ETV(4, n, n) ≤ \binom{n+2}{2} – C \sqrt{n} for a fixed constant C>0, improving the previously known best bound of Balko and Valtr.

The post Jineon Baek, On the off-diagonal Erdős-Szekeres convex polygon problem appeared first on KAIST Discrete Math Seminar.

]]>Hong Liu

Mathematics Institute, University of Warwick, Warwick, UK

2018/9/3 Mon 5PM

Counting problems on sets of integers with additive constraints have been extensively studied. In contrast, the counting problems for sets with multiplicative constraints remain largely unexplored. In this talk, we will discuss two such recent results, one on primitive sets [...]

The post Hong Liu, Enumerating sets of integers with multiplicative constraints appeared first on KAIST Discrete Math Seminar.

]]>Hong Liu

Mathematics Institute, University of Warwick, Warwick, UK

Mathematics Institute, University of Warwick, Warwick, UK

2018/9/3 Mon 5PM

Counting problems on sets of integers with additive constraints have been extensively studied. In contrast, the counting problems for sets with multiplicative constraints remain largely unexplored. In this talk, we will discuss two such recent results, one on primitive sets and the other on multiplicative Sidon sets. Based on joint work with Peter Pach, and with Peter Pach and Richard Palincza.

The post Hong Liu, Enumerating sets of integers with multiplicative constraints appeared first on KAIST Discrete Math Seminar.

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