Lower Bounds for Locally Highly Connected Graphs

We propose a conjecture regarding the lower bound for the number of edges in locally k-connected graphs and we prove it for k = 2. In particular, we show that every connected locally 2-connected graph is M3-rigid. For the special case of surface triangulations, this fact was known before using topological methods. We generalize this result to all locally 2-connected graphs and give a purely combinatorial proof. Our motivation to study locally k-connected graphs comes from lower bound conjectures for flag triangulations of manifolds, and we discuss some more specific problems in this direction.