Perfect Graphs of Fixed Density: Counting and Homogeneous Sets

@article{Bttcher2012PerfectGO,
  title={Perfect Graphs of Fixed Density: Counting and Homogeneous Sets},
  author={Julia B{\"o}ttcher and Anusch Taraz and Andreas W{\"u}rfl},
  journal={Comb. Probab. Comput.},
  year={2012},
  volume={21},
  pages={661-682}
}
  • Julia Böttcher, Anusch Taraz, Andreas Würfl
  • Published 2012
  • Mathematics, Computer Science
  • Comb. Probab. Comput.
  • For c ∈ (0,1) let n(c) denote the set of n-vertex perfect graphs with density c and let n(c) denote the set of n-vertex graphs without induced C5 and with density c. We show that \[\lim_{n\to\infty}\log_2 |\cP_n(c)|/\binom{n}{2}=\lim_{n\to\infty}\log_2 |\cC_n(c)|/\binom{n}{2}=h(c)\] with otherwise, where H is the binary entropy function. Further, we use this result to deduce that almost all graphs in n(c) have homogeneous sets of linear size. This answers a question raised by Loebl and co… CONTINUE READING
    Random perfect graphs
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