The minimum number of clique-saturating edges

TL;DR

This paper confirms a conjecture about the minimum number of edges that can be added to a large $K_p$-free graph to create a $K_p$, generalizing previous results and disproving earlier conjectures for specific cases.

Contribution

The paper proves the conjecture by Balogh and Liu regarding the asymptotic minimum number of $K_{p+1}$-saturating edges in large $K_p$-free graphs, extending previous work.

Findings

Confirmed the conjecture for all integers $p \\ge 3$.

Generalized previous results for specific cases.

Disproved earlier conjectures for certain parameters.

Abstract

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. 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 $p\ge 3$. The main result of this paper is to confirm the above conjecture of Balogh and Liu.