A conjecture I am thinking about:
Any graph with $n$ vertices in which at least 2 out of any 3 vertices are adjacent, where no such graph with fewer edges is possible, consists of 2 connected components which are cliques of size $\lfloor \frac{n}{2}\rfloor$ and $\lceil\frac{n}{2}\rceil$.
This seems like something that would be easy to prove if it were true, but I tried using induction on n and had some trouble working back to the IH. On the other hand, I can't think of any counterexamples.
Any thoughts?
If we think about the complement $H=\bar{G}$ then the condition is that $H$ has no triangle, and we want to maximize the number of edges of $H$. This is Turán's theorem. $H$ is (uniquely) a complete bipartite graph with parts as equal as possible. Hence $G$ is the disjoint union of two cliques with sizes as equal as possible, as conjectured.