KPPY 72

Jaehyun Kim, Jihye Park, Jongyoon Hyun, Seog-Jin Kim, Seung Hyun Shin

KPPY 72

Sept 05 2015. 11:00am-5:50pm at PNU

Schedule
11:00am - 11:50 Jaehyun Kim
POSTECH
Rational and Non-rational Varieties
12pm - 1:25 Lunch
1:30 - 2:20 Jihye Park
Yeungnam University
Colouring of a Digraph
2:30 - 3:20 Jongyoon Hyun
Ewha Woman's University
Bent functions, linear codes and strongly regular graphs
3:30 - 4:20 Seog-Jin Kim
Konkuk University
Coloring the square of graphs whose maximum average degrees are less than 4
4:30 - 5:20 Seung Hyun Shin
POSTECH
Spherical Designs and Codes

Abstracts



Jaehyun Kim
Rational and Non-rational Varieties
In algebraic geometry there is an equivalence relation coarser than isomorphic (biregular), called birational equivalence. As we can know from the name, it means that two varieties are related under not biregular but birational map. In this field (birational geometry), one topic is a classification of varieties of given dimension n under this equivalence relation. But more basically we want to determine whether given n dimensional variety is rational or not. That is, it is birational to n dimensional projective space. Although there are many beautiful theorems and examples in algebraic geometry, as a beginner, I will consider only few algebraic curves and algebraic surfaces. They are node, nonsingular cubic plane curve (elliptic curve), the surface of product of two projective lines, nonsingular cubic surface, pencil of conics and pencil of elliptic curves.
Jihye Park
Colouring of a Digraph
An acyclic $n$-colouring of a digraph $D$ is a function $f: V(D)\rightarrow I_n$ such that $f^{-1}(i)$ induces an acyclic subdigraph in $D$ for each $i\in I_n$. The dichromatic number $d_k (D)$ of $D$ is the minimum $n$ such that there exists an acyclic $n$-colouring of $D$. [V. Neumann-Lara, The Dichromatic Number of a Digraph, J. Combin. Theory Ser. B 33 (1982), 265-270] In this talk, we present several properties of the colouring of a digraph.
Jongyoon Hyun
Bent functions, linear codes and strongly regular graphs
A class of bent functions has been one of major subjects in combinatorial design theory, coding theory, association schemes and cryptography since it is introduced by Rothhaus in 1976. In this talk, we construct a three-weight linear code and a strongly regular graph from bent functions.
Seog-Jin Kim
Coloring the square of graphs whose maximum average degrees are less than 4
The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. The maximum average degree of $G$, $mad (G)$, is the maximum among the average degrees of the subgraphs of $G$.

It is known that there is no constant $C$ such that every graph $G$ with $mad(G) < 4$ has $\chi(G^2) \leq \Delta(G) + C$. Charpentier (2014) conjectured that there exists an integer $D$ such that every graph $G$ with $\Delta(G)\ge D$ and $mad(G)<4$ 2 has $\chi(g^2) \leq \delta(g)$.

In this paper, we show for $c\ge 2$, if $mad(G) < 4 - \frac{1}{c}$ and $\Delta(G) \geq 14c-7$, then $\chi_\ell(G^2) \leq 2 \Delta(G)$, which improves a result of Bonamy, L\ ́eveque, and Pinlou (2014). We also show that for every integer $D$, there is a graph $G$ with $\Delta(G)\ge D$ such that $mad(G)<4$, and $\chi(g^2) \geq 2\delta(g) +2$, which disproves charpentier's conjecture. in addition, we give counterexamples to another conjecture of charpentier's, stating that for every integer $k\ge 3$, there is an $d_k$ such graph $g$ with $mad(g)<2k$ $\delta(g)\ge d_k$ has \leq k\delta(g) -k$.

This is joint work with Boram Park.
Seung Hyun Shin
Spherical Designs and Codes
A finite non-empty subset $X$ of Euclidean sphere $S^{d} (\subset \mathbb{R}^{d+1})$ is called code or design if it satisfies some conditions. We are mainly interested in minimizing spherical designs and maximizing spherical codes. In this talk, I will introduce the concept of spherical designs and codes. And I will also introduce tight designs and linear programming bound which are main results about minimizing designs and maximizing codes.