Lower Bounds for Locally Highly Connected Graphs
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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 \(M_3\)-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.
KeywordsLocal graph properties k-Connectivity Lower bounds Rigidity
Mathematics Subject Classification05C40
We thank Jean-Marie Droz and Carsten Thomassen for related discussions, and the referee for detailed remarks which greatly improved the paper.