IBS/KAIST Joint Discrete Math Seminar

Department of Mathematics, Rutgers University, USA

Seminar series on discrete mathematics @ Dept. of Mathematical Sciences, KAIST.

IBS/KAIST Joint Discrete Math Seminar

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.

IBS/KAIST Joint Discrete Math Seminar

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.

IBS/KAIST Joint Discrete Math Seminar

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.

IBS/KAIST Joint Discrete Math Seminar

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.

IBS/KAIST Joint Discrete Math Seminar

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.