Abstract
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.
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Notes
According to our definition the complete graph \(K_k\) is k-connected, which is not the case if one uses the alternative definition via k vertex-disjoint paths between every pair of vertices.
It is \({\texttt {O}}^{\sim \sim }{\texttt {em]uj[vmsZTUrfFwN}}^{\sim }\) in the notation of the graph package nauty [7].
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Acknowledgments
We thank Jean-Marie Droz and Carsten Thomassen for related discussions, and the referee for detailed remarks which greatly improved the paper.
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A. Adamaszek partially supported by the Danish Council for Independent Research DFF-MOBILEX mobility grant. Research of M. Adamaszek done at the University of Bremen and supported by DFG Grant FE 1058/1-1. Research of J. M. Schmidt done partly at the Max Planck Institute for Informatics in Saarbrücken.
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Adamaszek, A., Adamaszek, M., Mnich, M. et al. Lower Bounds for Locally Highly Connected Graphs. Graphs and Combinatorics 32, 1641–1650 (2016). https://doi.org/10.1007/s00373-016-1686-y
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DOI: https://doi.org/10.1007/s00373-016-1686-y