KPPY 72

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.

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.

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.

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 $mad(G)<4$ 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.

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.