The genus of a random graph and the fragile genus property

Mihyun Kang (강미현)

TU Graz

2019/08/20 Tue 4:30PM-5:30PM

In this talk we shall discuss how quickly the genus of the Erdős-Rényi random graph grows as the number of edges increases and how dramatically a small number of random edges can increase the genus of [...]

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]]>The genus of a random graph and the fragile genus property

Mihyun Kang (강미현)

TU Graz

TU Graz

2019/08/20 Tue 4:30PM-5:30PM

In this talk we shall discuss how quickly the genus of the Erdős-Rényi random graph grows as the number of edges increases and how dramatically a small number of random edges can increase the genus of a randomly perturbed graph. (Joint work with Chris Dowden and Michael Krivelevich)

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]]>Integrality of set covering polyhedra and clutter minors

Dabeen Lee (이다빈)

IBS Discrete Mathematics Group

2019/07/16 Tue 4:30PM-5:30PM

Given a finite set of elements $V$ and a family $\mathcal{C}$ of subsets of $V$, the set covering problem is to find a minimum cardinality subset of $V$ intersecting every subset in the family $\mathcal{C}$. The set [...]

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]]>Integrality of set covering polyhedra and clutter minors

Dabeen Lee (이다빈)

IBS Discrete Mathematics Group

IBS Discrete Mathematics Group

2019/07/16 Tue 4:30PM-5:30PM

Given a finite set of elements $V$ and a family $\mathcal{C}$ of subsets of $V$, the set covering problem is to find a minimum cardinality subset of $V$ intersecting every subset in the family $\mathcal{C}$. The set covering problem, also known as the hitting set problem, admits a simple integer linear programming formulation. The constraint system of the integer linear programming formulation defines a polyhedron, and we call it the set covering polyhedron of $\mathcal{C}$. We say that a set covering polyhedron is integral if every extreme point is an integer lattice point. Although the set covering problem is NP-hard in general, conditions under which the problem becomes polynomially solvable have been studied. If the set covering polyhedron is integral, then it is straightforward that the problem can be solved using a polynomial-time algorithm for linear programming.

In this talk, we will focus on the question of when the set covering polyhedron is integral. We say that the family $\mathcal{C}$ is a clutter if every subset in $\mathcal{C}$ is inclusion-wise minimal. As taking out non-minimal subsets preserves integrality, we may assume that $\mathcal{C}$ is a clutter. We call $\mathcal{C}$ ideal if the set covering polyhedron of it is integral. To understand when a clutter is ideal, the notion of clutter minors is important in that $\mathcal{C}$ is ideal if and only if so is every minor of it. We will study two fundamental classes of non-ideal clutters, namely, deltas and the blockers of extended odd holes. We will characterize when a clutter contains either a delta or the blocker of an extended odd hole as a minor.

This talk is based on joint works with Ahmad Abdi and G\’erard Cornu\’ejols.

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]]>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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