This is a problem inspired by Hard planar graph problem.
Let $\nu$ be the average vertex degree of a graph $\Gamma$. Is it always possible to find an edge $\{u, v\}$ of $\Gamma$ such that $\deg(u) + \deg(v) \le 2\nu$ Intuitively, it seems like that this is the case, but I can't seem to find a nice way to prove it.
Edit/Update: Since the result seems to be trivially false for disconnected graphs, let us consider the case of connected graphs.
It seems intuitively true that this should be the case (atleast for connected graphs).