The minimum number of clique-saturating edges

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 \(f_p(n, e)\) the minimum number of \(K_p\)-saturating edges that an \(n\)-vertex \(K_p\)-free graph with \(e\) edges can have. Erdős 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 \(p\ge 3\). The main result of this paper is to confirm the above conjecture of Balogh and Liu.