### 58th KPPY Combinatorics Workshop -- August 21 2013

 Date: August 21 2013 Time: 11am-5:00pm Location: Science Building 1, Room 319Department of Mathematics, Yeungnam University

 Schedule 11:00 - 11:50 Saieed Akbari IPM, Tehran, Iran Generalization of $c$-Sum Flows in Graphs and Hypergraphs 12:00 Lunch 1:30 -2:20 Suyoung Choi Ajou University Combinatorial description of Betti numbers of toric varieties 2:30 -3:20 Edgardo Roldán-Pensado KAIST A non-dual version of the Erdős-Szekeres Theorem 3:40 - 4:30 Meesue Yoo KIAS The combinatorics of HMZ operators applied to Schur functions 4:40 - 5:30 Dongseok Kim Kyonggi University Westbury diagram by combinatorial webs of invariants vectors of Lie alebras 6:00 - 8:00 Banquet
Program

## Abstracts

Saieed Akbari
Generalization of $c$-Sum Flows in Graphs and Hypergraphs
Let $G$ be a graph. For a real number $c$, a $c$-sum flow of $G$ is an assignment of non-zero real numbers to the edges of $G$ such that the sum of values of all edges incident with each vertex is $c$. Let $k$ be a natural number. A $c$-sum $k$-flow is a $c$-sum flow with values from the set \mbox{$\{\pm 1,\ldots ,\pm(k-1)\}$}. In this talk, we present known results on $c$-sum $k$-flows of graphs and propose several conjectures. For a hypergraph $H$, a $0$-sum flow , means a nowhere-zero real vector in the null space of the incidence matrix of $H$. Here, we state some results on $0$-sum flows of hypergraphs.
In this talk we use some linear algebraic tools and graph factorization methods to obtain some results in zero-sum flows and their generalizations. Let $A$ be an abelian group and $A^*=A\setminus \{0\}$. For a subset $S \subseteq A$, a map $\phi : E(G) \rightarrow S$ is called an $S$-flow . For a given $S$-flow of $G$, and every $v\in V(G)$, define $s(v) =\sum_{uv \in E(G)} \phi(uv)$. It is shown that if $G$ is a $2$-edge connected bipartite graph with two parts $X=\{x_1,\ldots,x_r\}$ and $Y=\{y_1,\ldots,y_s\}$ and $c_1,\ldots,c_r,d_1,\ldots,d_s$ are arbitrary integers, then there exists a $\mathbb{Z}^*$-flow of $G$ such that $s(x_i)=c_i$ and $s(y_j)=d_j$, for $1 \leq i \leq r$, $1 \leq j \leq s$ if and only if $\sum_{i=1}^{r}c_i=\sum_{j=1}^{s}d_j$.
Suyoung Choi
Combinatorial description of Betti numbers of toric varieties
Recently, I and Dr. Hanchul Park have computed the $i$-th (rational) Betti number of the real toric variety associated to a graph $G$. It can be calculated by a purely combinatorial method and is called the $i$-th $a$-number of $G$. In the talk, I am willing to introduce recent works on the topic from the viewpoint of combinatorics and poset theory.
For Lie algebra $\rm{sl}(2)$, the invariants vectors are known as Temperley Lieb algebras. Since the dimension of these invariants spaces is Catalan number, it endows with some combinatorial descriptions. We will explain how the Westbury diagram can be constructed for other Lie algebras including $\rm{sl}(3)$, $\rm{sl}(4)$ and more.