【数学与统计及交叉学科前沿论坛------高端学术讲座第159场】
报告题目：Maximum number of cliques in hypergraphs forbidding long Berge cycle and loose cycle
报 告 人：彭岳建 湖南大学
报告时间：6月23日（周一）16：40—17：20
报告地点：金沙6165总站线路检测311会议室
报告摘要：
Abstract: For an $r$-uniform hypergraph $\mathcal{H}$, an $r$-uniform hypergraph family $\mathcal{F}$ and an integer $n$, let $ex_{r}(n,\mathcal{H},\mathcal{F})$ denote the maximum possible number of copies of $\mathcal{H}$ in an $\mathcal{F}$-free graph on $n$ vertices.If $r=2$, then we simply write $ex(n,\mathcal{H},\mathcal{F})$.
Let $\mathcal{BC}_{k}$ be a set of Berge cycles of length $k$ and let $LC_{k}$ denote the loose cycle of length $k$.
Let $\{k_{i}\}_{i=1}^{\infty}$ be a sequence of positive integers such that $k_{1}
\begin{align*}
f(k,t)&=\begin{cases}
\frac{k-2}{2}t+1 & k\ \text{is even},\\
\frac{k-3}{2}t+t & k\ \text{is odd},
\end{cases}
\end{align*}
$g(k,t)=f(k,t)+2t-k-2$ and $s(k,t)=f(k,t)+\lfloor\frac{2t-2}{r-1}\rfloor-k$.
We prove the following results.
\begin{enumerate}
\item For $\frac{k_{1}}{2}\leq t
\leq k_{1}-1$ and $k_{i+1}-k_{i}\leq g(k_{1}-1,t)$,
we determine the exact value of
$ex_{r}(n,K^{r}_{t},\{\mathcal{BC}_{k_{1}},\mathcal{BC}_{k_{2}},\mathcal{BC}_{k_{3}},\cdots\})$, and characterized all extremal hypergraphs. \item For $\frac{k}{2}\leq t\leq k-1$,
we determine the exact value of
$ex_{r}(n,K^{r}_{t},\mathcal{BP}_{k})$, and characterized all extremal hypergraphs.
\item For $\frac{k_{1}}{2}(r-1)\leq t
\leq k_{1}(r-1)-1$ and $k_{i+1}-k_{i}\leq s(k_{1}-1,t)$,
we determine the exact value of
$ex_{r}(n,K^{r}_{t},\{LC_{k_{1}},LC_{k_{2}},LC_{k_{3}},\cdots\})$, and characterized all extremal hypergraphs.
\item For $\frac{k(r-1)+1}{2}\leq t\leq k(r-1)$,
we determine the exact value of
$ex_{r}(n,K^{r}_{t},LC_{k})$, and characterized all extremal hypergraphs.
\end{enumerate}
Our results generalize several known results. This is a joint wok with Zhao Xiaojun.
报告人简介：彭岳建，湖南大学数学学院教授、博士生导师。2001年于美国Emory大学获理学博士学位。主要研究领域为极值组合和结构图论，在J. Combin. Theory Ser. B、J. Combin. Theory Ser. A、Combin. Probab. Comput.、Siam Dis Math、J. Graph Theory等发表论文90余篇。主持国家自然科学基金面上项目和重点项目。
报告题目：The inversion number of oriented graphs
报 告 人：陆玫 清华大学
报告时间：6月23日（周一）17：20—18：00
报告地点：金沙6165总站线路检测311会议室
报告摘要：Abstract: For an oriented graph D, the inversion of X ⊆ V(D) in D is the digraph obtained from D by reversing the direction of all arcs with both ends in X. The inversion number of D is the minimum number of inversions needed to transform D into an acyclic digraph. In this talk, I will give some problems and results on the inversion number. This work is joint with Haozhe Wang and Yuxuan Yang.
报告人简介：陆玫，清华大学数学科学系教授，博士生导师。于1993年7月在中国科学院数学与系统科学研究院获博士学位，主要从事运筹学、图论与组合优化方面的研究。现任清华大学数学科学系计算数学与运筹学研究所所长。
