On Generalized Random Railways

@article{Garmo2004OnGR,
  title={On Generalized Random Railways},
  author={Hans Garmo and Svante Janson and Michal Karonski},
  journal={Combinatorics, Probability and Computing},
  year={2004},
  volume={13},
  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. 
Rainbow Hamilton cycles in random regular graphs
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A rainbow subgraph of an edge-coloured graph has all edges of distinct colours and has a rainbow Hamilton cycle with probability tending to 1 as n tends to infinity, provided d is at least 8.

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