On the number of $K_4$-saturating edges

TL;DR

This paper investigates the minimum number of $K_4$-saturating edges in dense $K_4$-free graphs, disproving a conjecture by constructing a counterexample and establishing a tight lower bound.

Contribution

It provides the first construction of a $K_4$-free graph with fewer $K_4$-saturating edges than conjectured, and proves this bound is optimal.

Findings

Constructed a $K_4$-free graph with only $rac{2n^2}{33}$ $K_4$-saturating edges.

Proved that at least $(1+o(1))rac{2n^2}{33}$ $K_4$-saturating edges always exist.

Disproved Erd ext{"o}s and Tuza's conjecture on the minimum number of $K_4$-saturating edges.

Abstract

Let $G$ be a $K_4$-free graph, an edge in its complement is a $K_4$-\emph{saturating} edge if the addition of this edge to $G$ creates a copy of $K_4$. Erd\H{o}s and Tuza conjectured that for any $n$-vertex $K_4$-free graph $G$ with $\lfloor n^2/4\rfloor+1$ edges, one can find at least $(1+o(1))\frac{n^2}{16}$ $K_4$-saturating edges. We construct a graph with only $\frac{2n^2}{33}$ $K_4$-saturating edges. Furthermore, we prove that it is best possible, i.e., one can always find at least $(1+o(1))\frac{2n^2}{33}$ $K_4$-saturating edges in an $n$-vertex $K_4$-free graph with $\lfloor n^2/4\rfloor+1$ edges.