# 09-23【陆　玫】腾讯会议 吴文俊数学重点实验室组合图论系列讲座之183

1.题目: Saturation Numbers for Disjoint Stars

https://meeting.tencent.com/dm/STNw6LQLNzFF

A graph \$G\$ is called an \$H\$-saturated if \$G\$ does not contain \$H\$ as a subgraph, but the addition of any edge between two nonadjacent vertices in \$G\$ results in a copy of \$H\$ in \$G\$. The saturation number \$sat(n,H)\$ is the minimum number of edges in \$G\$ for all \$H\$-saturated graphs \$G\$ of order \$n\$. For a graph \$F\$, let \$mF\$ denote the disjoint union of \$m\$ copies of \$F\$. In 2011, Faudree, Faudree and Schmitt proposed a problem that is to determine \$sat(n,mK_{1,k})\$ for all \$m\$ and \$k\$.  In this talk, I will give a result on \$ sat(n,mK_{1,k})\$ when \$m\ge 2\$, \$k\ge 4\$ and \$n\ge 3mk^2\$. This work is joint with Zequn Lv and Zhen He.

2.题目：The maximum number of triangles in Fk-free graphs

https://meeting.tencent.com/dm/STNw6LQLNzFF

The generalized Tur´an number ex(n, Ks, H) is the maximum number of com- plete graph Ks in an H-free graph on vertices. Let Fk be the friendship graph con- sisting of triangles. Erd˝os and S´os (1976) determined the value of ex(n, K3, F2). Alon and Shikhelman (2016) proved that ex(n, K3, Fk≤ (9− 15)(+ 1)n. In this talk, we will report our new result on the exact value of ex(n, K3, Fk) and the extremal graph for any Fk when ≥ 4k3. 