Graphs and Combinatorics

, Volume 32, Issue 5, pp 1641–1650 | Cite as

Lower Bounds for Locally Highly Connected Graphs

  • Anna Adamaszek
  • Michal Adamaszek
  • Matthias Mnich
  • Jens M. Schmidt
Original Paper

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.

Keywords

Local graph properties k-Connectivity Lower bounds Rigidity 

Mathematics Subject Classification

05C40 

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Copyright information

© Springer Japan 2016

Authors and Affiliations

  • Anna Adamaszek
    • 1
  • Michal Adamaszek
    • 2
  • Matthias Mnich
    • 3
  • Jens M. Schmidt
    • 4
  1. 1.Department of Computer Science (DIKU)University of CopenhagenCopenhagenDenmark
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  3. 3.Institut für InformatikUniversität BonnBonnGermany
  4. 4.Institute of MathematicsTU IlmenauIlmenauGermany

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