A course in graph embedding

Jaehoon Kim (김재훈)

School of Mathematics, Birmingham University, Birmingham, UK

2018/6/25-29 10:30AM-12PM, 2:30PM-4PM (Room 3433, Bldg. E6-1)

In this lecture, we aim to learn several techniques to find sufficient conditions on a dense graph G to contain a sparse graph H as a subgraph.

Tentative plan for the course (June 25, 26, 27, [...]

The post (Intensive Lecture Series) Jaehoon Kim, A course in graph embedding appeared first on KAIST Discrete Math Seminar.

]]>A course in graph embedding

Jaehoon Kim (김재훈)

School of Mathematics, Birmingham University, Birmingham, UK

School of Mathematics, Birmingham University, Birmingham, UK

2018/6/25-29 10:30AM-12PM, 2:30PM-4PM (Room 3433, Bldg. E6-1)

In this lecture, we aim to learn several techniques to find sufficient conditions on a dense graph G to contain a sparse graph H as a subgraph.

**Tentative plan for the course (June 25, 26, 27, 28, 29)**

Lecture 1 : Basic probabilistic methods

Lecture 2 : Turan numbers

Lecture 3 : Bipartite Turan numbers and dependent random choice

Lecture 4 : The regularity lemma and its applications

Lecture 5 : Embedding large graphs

Lecture 6 : The blow-up lemma and its applications I

Lecture 7 : The blow-up lemma and its applications II

Lecture 8 : The absorbing method

Lecture 9 : Graph packing problems and iterative absorption I

Lecture 10 : Graph packing problems and iterative absorption II

The post (Intensive Lecture Series) Jaehoon Kim, A course in graph embedding appeared first on KAIST Discrete Math Seminar.

]]>Grassmanians and Pseudosphere Arrangements

Michael Dobbins

Department of Mathematics, Binghamton University, Binghamton, NY, USA

2018/5/30 Wednesday 5PM

In this talk I will present a metric space of pseudosphere arrangements, as in the topological representation theorem of oriented matroids, where each pseudosphere is assigned a weight. This gives an extension of the space of [...]

The post Michael Dobbins, Grassmanians and Pseudosphere Arrangements appeared first on KAIST Discrete Math Seminar.

]]>Grassmanians and Pseudosphere Arrangements

Michael Dobbins

Department of Mathematics, Binghamton University, Binghamton, NY, USA

Department of Mathematics, Binghamton University, Binghamton, NY, USA

2018/5/**30** Wednesday 5PM

In this talk I will present a metric space of pseudosphere arrangements, as in the topological representation theorem of oriented matroids, where each pseudosphere is assigned a weight. This gives an extension of the space of full rank vector configurations of fixed size in a fixed dimension that has nicer combinatorial and topological properties. In rank 3 these spaces, modulo SO(3), are homotopy equivalent to Grassmanians, and the subspaces representing a fixed oriented matroid are contractible. Work on these spaces was partly motivated by combinatorial tools for working with vector bundles.

The post Michael Dobbins, Grassmanians and Pseudosphere Arrangements appeared first on KAIST Discrete Math Seminar.

]]>Jinyoung Park (박진영)

Department of Mathematics, Rutgers, Piscataway, NJ, USA

2018/06/26 Tuesday 5PM

We discuss the number of proper colorings of the hypercube (the covering graph of the Boolean algebra) given q colors. When q=2, it is easy to see that there are only 2 possible colorings. However, it is already highly nontrivial to figure out the [...]

The post Jinyoung Park (박진영), Coloring hypercubes appeared first on KAIST Discrete Math Seminar.

]]>Jinyoung Park (박진영)

Department of Mathematics, Rutgers, Piscataway, NJ, USA

Department of Mathematics, Rutgers, Piscataway, NJ, USA

2018/06/26 Tuesday 5PM

We discuss the number of proper colorings of the hypercube (the covering graph of the Boolean algebra) given q colors. When q=2, it is easy to see that there are only 2 possible colorings. However, it is already highly nontrivial to figure out the number of colorings when q=3. Since Galvin (2002) proved the asymptotics for the number of 3-colorings, the rest cases remained open so far. In this talk, I will introduce a recent work about the number of 4-colorings, mainly focusing on 1. how entropy can be used in counting and 2. isoperimetric properties of hypercube. This is joint work with Jeff Kahn.

The post Jinyoung Park (박진영), Coloring hypercubes appeared first on KAIST Discrete Math Seminar.

]]>Mark Siggers

Department of Mathematics, Kyungpook National University, Daegu

2018/04/03 Tue 5PM

For problems with a discrete set of solutions, a reconfiguration problem defines solutions to be adjacent if they meet some condition of closeness, and then asks for two given solutions it there is a path between them in the set of [...]

The post Mark Siggers, The reconfiguration problem for graph homomorpisms appeared first on KAIST Discrete Math Seminar.

]]>Mark Siggers

Department of Mathematics, Kyungpook National University, Daegu

Department of Mathematics, Kyungpook National University, Daegu

2018/04/03 Tue 5PM

For problems with a discrete set of solutions, a reconfiguration problem defines solutions to be adjacent if they meet some condition of closeness, and then asks for two given solutions it there is a path between them in the set of all solutions.

The reconfiguration problem for graph homomorphisms has seen fair activity in recent years. Fixing a template, the problem Recol(H) for a graph H asks if one can get from one H-colouring of a graph G to another by changing one vertex at a time, always remaining an H-colouring. If the changed vertex has a loop, it must move to an adjecent vertex

Depending on H, the problem seems to be either polynomial time solvable or PSPACE-complete. We discuss many recent results in the area and work towards conjecturing for which H the problem is PSPACE-complete.

This is joint work with Rick Brewster, Jae-baek Lee, Ben Moore and Jon Noel.

The reconfiguration problem for graph homomorphisms has seen fair activity in recent years. Fixing a template, the problem Recol(H) for a graph H asks if one can get from one H-colouring of a graph G to another by changing one vertex at a time, always remaining an H-colouring. If the changed vertex has a loop, it must move to an adjecent vertex

Depending on H, the problem seems to be either polynomial time solvable or PSPACE-complete. We discuss many recent results in the area and work towards conjecturing for which H the problem is PSPACE-complete.

This is joint work with Rick Brewster, Jae-baek Lee, Ben Moore and Jon Noel.

The post Mark Siggers, The reconfiguration problem for graph homomorpisms appeared first on KAIST Discrete Math Seminar.

]]>Hong Liu

Mathematics Institute, University of Warwick, Warwick, UK

2018/4/10 Tue 5PM

Given graphs H1,..., Hk, a graph G is (H1,..., Hk)-free if there is a k-edge-colouring of G with no Hi in colour-i for all i in {1,2,...,k}. Fix a function f(n), the Ramsey-Turán function rt(n,H1,...,Hk,f(n)) is the maximum size of an n-vertex [...]

The post Hong Liu, Two conjectures in Ramsey-Turán theory appeared first on KAIST Discrete Math Seminar.

]]>Hong Liu

Mathematics Institute, University of Warwick, Warwick, UK

Mathematics Institute, University of Warwick, Warwick, UK

2018/4/10 Tue 5PM

Given graphs H_{1},…, H_{k}, a graph G is (H_{1},…, H_{k})-free if there is a k-edge-colouring of G with no H_{i} in colour-i for all i in {1,2,…,k}. Fix a function f(n), the Ramsey-Turán function rt(n,H_{1},…,H_{k},f(n)) is the maximum size of an n-vertex (H_{1},…, H_{k})-free graph with independence number at most f(n). We determine rt(n,K_{3},K_{s},δn) for s in {3,4,5} and sufficiently small δ, confirming a conjecture of Erdős and Sós from 1979. It is known that rt(n,K_{8},f(n)) has a phase transition at f(n)=Θ(√(n\log n)). We prove that rt(n,K_{8},o(√(n\log n)))=n^{2}/4+o(n^{2}), answering a question of Balogh, Hu and Simonovits. The proofs utilise, among others, dependent random choice and results from graph packings. Joint work with Jaehoon Kim and Younjin Kim.

The post Hong Liu, Two conjectures in Ramsey-Turán theory appeared first on KAIST Discrete Math Seminar.

]]>Antoine Vigneron

School of Electrical and Computer Engineering, UNIST, Ulsan, South Korea

2018/3/27 Tue 5PM

Given a planar subdivision with n vertices, each face having a cone of possible directions of travel, our goal is to decide which vertices of the subdivision can be reached from a given starting point s. [...]

The post Antoine Vigneron, Reachability in a Planar Subdivision with Direction Constraints appeared first on KAIST Discrete Math Seminar.

]]>Antoine Vigneron

School of Electrical and Computer Engineering, UNIST, Ulsan, South Korea

School of Electrical and Computer Engineering, UNIST, Ulsan, South Korea

2018/3/27 Tue 5PM

Given a planar subdivision with n vertices, each face having a cone of possible directions of travel, our goal is to decide which vertices of the subdivision can be reached from a given starting point s. We give an O(n log n)-time algorithm for this problem, as well as an Ω(n log n) lower bound in the algebraic computation tree model. We prove that the generalization where two cones of directions per face are allowed is NP-hard.

The post Antoine Vigneron, Reachability in a Planar Subdivision with Direction Constraints appeared first on KAIST Discrete Math Seminar.

]]>Otfried Cheong

School of Computing, KAIST

2017/3/20 Tue 5PM

We prove a generalization of Pal's 1921 conjecture that if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360 degrees inside Q. We also prove a lower bound [...]

The post Otfried Cheong, The reverse Kakeya problem appeared first on KAIST Discrete Math Seminar.

]]>Otfried Cheong

School of Computing, KAIST

School of Computing, KAIST

2017/3/20 Tue 5PM

We prove a generalization of Pal’s 1921 conjecture that if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360 degrees inside Q. We also prove a lower bound of Ω(m n^{2}) on the number of combinatorially distinct maximal placements of a convex m-gon P in a convex n-gon Q. This matches the upper bound proven by Agarwal et al.

The post Otfried Cheong, The reverse Kakeya problem appeared first on KAIST Discrete Math Seminar.

]]>Ringi Kim (김린기)

Department of Mathematical Sciences, KAIST

2018/3/6 Tue 5PM

The chromatic number of a graph is the minimum k such that the graph has a proper k-coloring. It is known that if T is a tree, then every graph with large chromatic number contains T as a subgraph. In [...]

The post Ringi Kim (김린기), Characterization of forbidden subgraphs for bounded star-chromatic number appeared first on KAIST Discrete Math Seminar.

]]>Ringi Kim (김린기)

Department of Mathematical Sciences, KAIST

Department of Mathematical Sciences, KAIST

2018/3/6 Tue 5PM

The chromatic number of a graph is the minimum k such that the graph has a proper k-coloring. It is known that if T is a tree, then every graph with large chromatic number contains T as a subgraph. In this talk, we discuss this phenomena for star-coloring (a proper coloring forbidding a bicolored path on four vertices) and acyclic-coloring (a proper coloring forbidding bicolored cycles). Specifically, we will characterize all graphs T such that every graph with sufficiently large star-chromatic number (acyclic-chromatic number) contains T as a subgraph.

The post Ringi Kim (김린기), Characterization of forbidden subgraphs for bounded star-chromatic number appeared first on KAIST Discrete Math Seminar.

]]>O-joung Kwon (권오정)

Technische Universität Berlin, Berlin, Germany

2018/1/12 Fri 4PM-5PM

A chordless cycle in a graph G is an induced subgraph of G which is a cycle of length at least four. We prove that the Erdős-Pósa property holds for chordless cycles, which resolves the major open question concerning the [...]

The post O-joung Kwon (권오정), Erdős-Pósa property of chordless cycles and its applications appeared first on KAIST Discrete Math Seminar.

]]>O-joung Kwon (권오정)

Technische Universität Berlin, Berlin, Germany

Technische Universität Berlin, Berlin, Germany

2018/1/12 Fri 4PM-5PM

A chordless cycle in a graph G is an induced subgraph of G which is a cycle of length at least four. We prove that the Erdős-Pósa property holds for chordless cycles, which resolves the major open question concerning the Erdős-Pósa property. Our proof for chordless cycles is constructive: in polynomial time, one can find either k+1 vertex-disjoint chordless cycles, or ck^{2} log k vertices hitting every chordless cycle for some constant c. It immediately implies an approximation algorithm of factor O(OPT log OPT) for Chordal Vertex Deletion. We complement our main result by showing that chordless cycles of length at least ℓ for any fixed ℓ≥ 5 do not have the Erdős-Pósa property.

The post O-joung Kwon (권오정), Erdős-Pósa property of chordless cycles and its applications appeared first on KAIST Discrete Math Seminar.

]]>Joonkyung Lee (이준경)

Mathematical Institute, University of Oxford, Oxford, UK

2018/1/8 Mon 4PM-5PM

We prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies a conjecture of Kohayakawa, Nagle, Rödl, and Schacht on random-like counts for small graphs in locally dense graphs. This implies an [...]

The post Joonkyung Lee (이준경), Counting tree-like graphs in locally dense graphs appeared first on KAIST Discrete Math Seminar.

]]>Joonkyung Lee (이준경)

Mathematical Institute, University of Oxford, Oxford, UK

Mathematical Institute, University of Oxford, Oxford, UK

2018/1/8 Mon 4PM-5PM

We prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies a conjecture of Kohayakawa, Nagle, Rödl, and Schacht on random-like counts for small graphs in locally dense graphs. This implies an approximate version of the conjecture for graphs with bounded tree-width. We also prove an analogous result for odd cycles instead of complete multipartite graphs.

The proof uses a general information theoretic method to prove graph homomorphism inequalities for tree-like structured graphs, which may be of independent interest.

The proof uses a general information theoretic method to prove graph homomorphism inequalities for tree-like structured graphs, which may be of independent interest.

The post Joonkyung Lee (이준경), Counting tree-like graphs in locally dense graphs appeared first on KAIST Discrete Math Seminar.

]]>