A model theoretical approach to sparsity

Patrice Ossona de Mendez

CNRS, France

2019/06/25 Tue 4:30PM-5:30PM

We discuss how the model theoretic notion of first-order transduction allows to define a notion of structural sparsity, and give some example of applications, like existence of low shrub-depth decompositions for tranductions of bounded expansion classes, characterization of transductions of [...]

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]]>A model theoretical approach to sparsity

Patrice Ossona de Mendez

CNRS, France

CNRS, France

2019/06/25 Tue 4:30PM-5:30PM

We discuss how the model theoretic notion of first-order transduction allows to define a notion of structural sparsity, and give some example of applications, like existence of low shrub-depth decompositions for tranductions of bounded expansion classes, characterization of transductions of classes with bounded pathwidth, decompositions of graphs with bounded rank-width into cographs.

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]]>An odd [1,b]-factor in regular graphs from eigenvalues

Suil O (오수일)

Department of Applied Mathematics and Statistics, SUNY-Korea

2019/06/19 Wed 4:30PM-5:30PM

An odd [1,b]-factor of a graph is a spanning subgraph H such that for every vertex v∈V(G), 1≤dH(v)≤b, and dH(v) is odd. For positive integers r≥3 and b≤r, Lu, Wu, and Yang gave an [...]

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]]>An odd [1,b]-factor in regular graphs from eigenvalues

Suil O (오수일)

Department of Applied Mathematics and Statistics, SUNY-Korea

Department of Applied Mathematics and Statistics, SUNY-Korea

2019/06/19 Wed 4:30PM-5:30PM

An odd [1,b]-factor of a graph is a spanning subgraph H such that for every vertex v∈V(G), 1≤d_{H}(v)≤b, and d_{H}(v) is odd. For positive integers r≥3 and b≤r, Lu, Wu, and Yang gave an upper bound for the third largest eigenvalue in an r-regular graph with even number of vertices to guarantee the existence of an odd [1,b]-factor. In this talk, we improve their bound.

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]]>The number of maximal independent sets in the Hamming cube

Jinyoung Park (박진영)

Department of Mathematics, Rutgers University, USA

2019/06/03 Monday 4:30PM-5:30PM (IBS, Room B232)

Let $Q_n$ be the $n$-dimensional Hamming cube (hypercube) and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically $2n2^{N/4}$, as was conjectured by Ilinca and [...]

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]]>The number of maximal independent sets in the Hamming cube

Jinyoung Park (박진영)

Department of Mathematics, Rutgers University, USA

Department of Mathematics, Rutgers University, USA

2019/06/03 Monday 4:30PM-5:30PM (IBS, Room B232)

Let $Q_n$ be the $n$-dimensional Hamming cube (hypercube) and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically $2n2^{N/4}$, as was conjectured by Ilinca and Kahn in connection with a question of Duffus, Frankl and Rödl. The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof of the upper bound draws on various tools, among them “stability” results for maximal independent set counts and old and new results on isoperimetric behavior in $Q_n$. This is joint work with Jeff Kahn.

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]]>A complexity dichotomy for critical values of the b-chromatic number of graphs

Lars Jaffke

University of Bergen

2019/05/20 Mon 4:30PM-5:30PM (IBS, B232)

A b-coloring of a graph G is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. [...]

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]]>A complexity dichotomy for critical values of the b-chromatic number of graphs

Lars Jaffke

University of Bergen

University of Bergen

2019/05/20 Mon 4:30PM-5:30PM (IBS, B232)

A b-coloring of a graph G is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The b-Coloring problem asks whether a graph G has a b-coloring with k colors.The b-chromatic number of a graph G, denoted by χ_{b}(G), is the maximum number k such that G admits a b-coloring with k colors. We consider the complexity of the b-Coloring problem, whenever the value of k is close to one of two upper bounds on χ_{b}(G): The maximum degree Δ(G) plus one, and the m-degree, denoted by m(G), which is defined as the maximum number i such that G has i vertices of degree at least i−1. We obtain a dichotomy result stating that for fixed k∈{Δ(G)+1−p,m(G)−p}, the problem is polynomial-time solvable whenever p∈{0,1} and, even when k=3, it is NP-complete whenever p≥2.

We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree Δ(G) of the input graph G and give two FPT-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FPT parameterized by Δ(G). Second, we show that b-Coloring is FPT parameterized by Δ(G)+ℓ_{k}(G), where ℓ_{k}(G) denotes the number of vertices of degree at least k.

This is joint work with Paloma T. Lima.

We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree Δ(G) of the input graph G and give two FPT-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FPT parameterized by Δ(G). Second, we show that b-Coloring is FPT parameterized by Δ(G)+ℓ

This is joint work with Paloma T. Lima.

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]]>On equitable tree-colorings of graphs

Xin Zhang (张欣)

Xidian Univeristy, China

2019/05/16 4:30PM-5:30PM (IBS, B232)

An equitable tree-k-coloring of a graph is a vertex coloring using k distinct colors such that every color class (i.e, the set of vertices in a common color) induces a forest and the sizes of any two color classes differ [...]

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]]>On equitable tree-colorings of graphs

Xin Zhang (张欣)

Xidian Univeristy, China

Xidian Univeristy, China

2019/05/16 4:30PM-5:30PM (IBS, B232)

An equitable tree-k-coloring of a graph is a vertex coloring using k distinct colors such that every color class (i.e, the set of vertices in a common color) induces a forest and the sizes of any two color classes differ by at most one. The minimum integer k such that a graph G is equitably tree-k-colorable is the equitable vertex arboricity of G, denoted by va_{eq}(G). A graph that is equitably tree-k-colorable may admits no equitable tree-k′-coloring for some k′>k. For example, the complete bipartite graph K_{9,9} has an equitable tree-2-coloring but is not equitably tree-3-colorable. In view of this a new chromatic parameter so-called the equitable vertex arborable threshold is introduced. Precisely, it is the minimum integer k such that G has an equitable tree-k′-coloring for any integer k′≥k, and is denoted by va^{∗}_{eq}(G). The concepts of the equitable vertex arboricity and the equitable vertex arborable threshold were introduced by J.-L. Wu, X. Zhang and H. Li in 2013. In 2016, X. Zhang also introduced the list analogue of the equitable tree-k-coloring. There are many interesting conjectures on the equitable (list) tree-colorings, one of which, for example, conjectures that every graph with maximum degree at most Δ is equitably tree-k-colorable for any integer k≥(Δ+1)/2, i.e, va^{∗}_{eq}(G)≤⌈(Δ+1)/2⌉. In this talk, I review the recent progresses on the studies of the equitable tree-colorings from theoretical results to practical algorithms, and also share some interesting problems for further research.

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]]>On strong Sidon sets of integers

Sang June Lee

Duksung Women's University, Seoul

2019/05/08 Wed 4:30PM-5:30PM (IBS, Room B232)

Let N be the set of natural numbers. A set A⊂N is called a Sidon set if the sums a1+a2, with a1,a2∈S and a1≤a2, are distinct, or equivalently, if

|(x+w)−(y+z)|≥1

for every x,y,z,w∈S with x<y≤z<w. We define strong [...]

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]]>On strong Sidon sets of integers

Sang June Lee

Duksung Women’s University, Seoul

Duksung Women’s University, Seoul

2019/05/08 Wed 4:30PM-5:30PM (IBS, Room B232)

Let N be the set of natural numbers. A set A⊂N is called a Sidon set if the sums a_{1}+a_{2}, with a_{1},a_{2}∈S and a_{1}≤a_{2}, are distinct, or equivalently, if

|(x+w)−(y+z)|≥1

for every x,y,z,w∈S with x<y≤z<w. We define strong Sidon sets as follows:

|(x+w)−(y+z)|≥1

for every x,y,z,w∈S with x<y≤z<w. We define strong Sidon sets as follows:

For a constant α with 0≤α<1, a set S⊂N is called an α-strong Sidon set if

|(x+w)−(y+z)|≥w^{α}

for every x,y,z,w∈S with x<y≤z<w.

The motivation of strong Sidon sets is that a strong Sidon set generates many Sidon sets by altering each element a bit. This infers that a dense strong Sidon set will guarantee a dense Sidon set contained in a sparse random subset of N.

In this talk, we are interested in how dense a strong Sidon set can be. This is joint work with Yoshiharu Kohayakawa, Carlos Gustavo Moreira and Vojtěch Rödl.

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]]>Circle graphs are polynomially chi-bounded

Rose McCarty

University of Waterloo, Waterloo, Canada

2019/04/26 Fri 4PM-5PM (IBS, Room B232)

Circle graphs are the intersection graphs of chords on a circle; vertices correspond to chords, and two vertices are adjacent if their chords intersect. We prove that every circle graph with clique number k has chromatic [...]

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]]>Circle graphs are polynomially chi-bounded

Rose McCarty

University of Waterloo, Waterloo, Canada

University of Waterloo, Waterloo, Canada

2019/04/26 Fri 4PM-5PM (IBS, Room B232)

Circle graphs are the intersection graphs of chords on a circle; vertices correspond to chords, and two vertices are adjacent if their chords intersect. We prove that every circle graph with clique number k has chromatic number at most 4k^{2}. Joint with James Davies.

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]]>Introduction to Boolean functions with Artificial Neural Network

Jon-Lark Kim (김종락)

Department of Mathematics, Sogang University, Seoul

2019/04/18 Thu 11:00AM-12:00PM (IBS, Room B232)

A Boolean function is a function from the set Q of binary vectors of length n (i.e., the binary n-dimensional hypercube) to F2={0,1}. It has several applications to complexity theory, digital circuits, [...]

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]]>Introduction to Boolean functions with Artificial Neural Network

Jon-Lark Kim (김종락)

Department of Mathematics, Sogang University, Seoul

Department of Mathematics, Sogang University, Seoul

2019/04/18 Thu 11:00AM-12:00PM (IBS, Room B232)

A Boolean function is a function from the set Q of binary vectors of length n (i.e., the binary n-dimensional hypercube) to F_{2}={0,1}. It has several applications to complexity theory, digital circuits, coding theory, and cryptography.In this talk we give a connection between Boolean functions and Artificial Neural Network. We describe how to represent Boolean functions by Artificial Neural Network including linear and polynomial threshold units and sigmoid units. For example, even though a linear threshold function cannot realize XOR, a polynomial threshold function can do it. We also give currently open problems related to the number of (Boolean) linear threshold functions and polynomial threshold functions.

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]]>Large cliques in hypergraphs with forbidden substructures

Andreas Holmsen

Department of Mathematical Sciences, KAIST

2019/03/12 Tue 4:30PM-5:30PM (Room B232, IBS)

A result due to Gyárfás, Hubenko, and Solymosi, answering a question of Erdős, asserts that if a graph G does not contain K2,2 as an induced subgraph yet has at least c n(n-1)/2 edges, then [...]

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]]>Large cliques in hypergraphs with forbidden substructures

Andreas Holmsen

Department of Mathematical Sciences, KAIST

Department of Mathematical Sciences, KAIST

2019/03/12 Tue 4:30PM-5:30PM (Room B232, IBS)

A result due to Gyárfás, Hubenko, and Solymosi, answering a question of Erdős, asserts that if a graph G does not contain K_{2,2} as an induced subgraph yet has at least c n(n-1)/2 edges, then G has a complete subgraph on at least c^2 n /10 vertices. In this paper we suggest a “higher-dimensional” analogue of the notion of an induced K_{2,2}, which allows us to extend their result to k-uniform hypergraphs. Our result also has interesting consequences in topological combinatorics and abstract convexity, where it can be used to answer questions by Bukh, Kalai, and several others.

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]]>IBS/KAIST Joint Discrete Math Seminar

Signed colouring and list colouring of k-chromatic graphs

Seog-Jin Kim (김석진)Department of Mathematics Education, Konkuk University, Seoul

2019/1/28 Mon 4PM-5PM (Room B232, IBS)

A signed graph is a pair (G, σ), where G is a graph and σ: E(G) → {1,-1} is a signature of G. A set S of integers is symmetric if [...]

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]]>Signed colouring and list colouring of k-chromatic graphs

Seog-Jin Kim (김석진)

Department of Mathematics Education, Konkuk University, Seoul

Department of Mathematics Education, Konkuk University, Seoul

2019/1/28 Mon 4PM-5PM (Room B232, IBS)

A *signed graph* is a pair (G, σ), where G is a graph and σ: E(G) → {1,-1} is a signature of G. A set S of integers is *symmetric* if I∈S implies that -i∈S. A *k-colouring* of (G,σ) is a mapping f:V(G) → N_{k} such that for each edge e=uv, f(x)≠σ(e) f(y), where N_{k} is a symmetric integer set of size k. We define the *signed chromatic number* of a graph G to be the minimum integer k such that for any signature σ of G, (G, σ) has a k-colouring.
Let f(n,k) be the maximum signed chromatic number of an n-vertex k-chromatic graph. This paper determines the value of f(n,k) for all positive integers n ≥ k. Then we study list colouring of signed graphs. A list assignment L of G is called *symmetric* if L(v) is a symmetric integer set for each vertex v. The *weak signed choice number* ch_{±}^{w}(G) of a graph G is defined to be the minimum integer k such that for any symmetric k-list assignment L of G, for any signature σ on G, there is a proper L-colouring of (G, σ). We prove that the difference ch_{±}^{w}(G)-χ_{±}(G) can be arbitrarily large. On the other hand, ch_{±}^{w}(G) is bounded from above by twice the list vertex arboricity of G. Using this result, we prove that ch_{±}^{w}(K_{2⋆n})= χ_{±}(K_{2⋆n}) = ⌈2n/3⌉ + ⌊2n/3⌋. This is joint work with Ringi Kim and Xuding Zhu.

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