On Generalized Random Railways

  title={On Generalized Random Railways},
  author={Hans Garmo and Svante Janson and Michal Karonski},
  journal={Combinatorics, Probability and Computing},
  pages={31 - 35}
We consider a random generalized railway defined as a random 3-regular multigraph where some vertices are regarded as switches that only allow traffic between certain pairs of attached edges. It is shown that the probability that the generalized railway is functioning is linear in the proportion of switches. Thus there is no threshold phenomenon for this property. 
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Asymptotic properties of the connectivity number of random railways
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  • 1999
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It is shown that every non-trivial monotone increasing property of subsets of a set has a threshold function. This generalises a number of classical results in the theory of random graphs.
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Some of the major results in random graphs and some of the more challenging open problems are reviewed, including those related to the WWW.
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Random railways as 3-regular graphs
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  • 1996