# KPPY 72

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

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

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

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

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

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

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 $mad(G)<4$ has $\chi(G^2) \leq 2 \Delta(G)$.

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 integer $D_k$ such that every graph $G$ with $mad(G)<2k$ and $\Delta(G)\ge D_k$ has $\chi(G^2) \leq k\Delta(G) -k$.

This is joint work with Boram Park.

*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 $mad(G)<4$ has $\chi(G^2) \leq 2 \Delta(G)$.

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 integer $D_k$ such that every graph $G$ with $mad(G)<2k$ and $\Delta(G)\ge D_k$ has $\chi(G^2) \leq k\Delta(G) -k$.

This is joint work with Boram Park.

Seung Hyun Shin

Spherical Designs and Codes

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.