# 信息来源：发布日期: 2022-11-30　 浏览次数: _showDynClicks("wbnews", 1756464364, 1631)

Let $G$ be a $K_p$-free graph. We say $e$ is a $K_p$-saturating edge of $G$ if $e\notin E(G)$ and $G+e$ contains a copy of $K_p$.Denote by $f_p(n, m)$ the minimum number of $K_p$-saturating edges that an $n$-vertex $K_p$-free graph with $m$ edges can have.

Erd\H{o}s and Tuza conjectured that$f_4(n,\lfloor n^2/4\rfloor+1)=\left(1 +o(1)\right)\frac{n^2}{16}.$Balogh and Liu disproved this by showing$f_4(n,\lfloor n^2/4\rfloor+1)=(1+o(1))\frac{2n^2}{33}$.They believed that a natural generalization of their construction for $K_p$-free graph should also be optimal and made a conjecture that $f_{p+1}(n,\ex(n,K_p)+1)=\left(\frac{2(p-2)^2}{p(4p^2-11p+8)}+o(1)\right)n^2$ for all integers $p\ge 3$.

In this talk, we confirm the above conjecture of Balogh and Liu. The result is the joint work with Jie Ma, Fuhong Ma and Xinyang Ye.

何家林，本科与博士均毕业于中国科学技术大学。目前在香港科技大学数学系做博士后。主要研究方向为极值图论中的图兰类问题，目前在JCTB与CSIAM共发表论文4篇。