On the number of K4-saturating edges

Let G be a K4-free graph, an edge in its complement is a K4-saturating edge if the addition of this edge to G creates a copy of K4. Erdős and Tuza conjectured that for any n-vertex K4-free graph G with bn2/4c + 1 edges, one can find at least (1 + o(1)) 2 16 K4-saturating edges. We construct a graph with only 2n2 33 K4-saturating edges. Furthermore, we prove that it is best possible, i.e., one can always find at least (1 + o(1)) 2 33 K4-saturating edges in an n-vertex K4-free graph with bn2/4c+ 1 edges.