A Note on Sparse Random Graphs and Cover Graphs

  • Tom Bohman
  • Alan Frieze
  • Miklós Ruszinkó
  • Lubos Thoma


It is shown in this note that with high probability it is enough to destroy all triangles in order to get a cover graph from a random graph $G_{n,p}$ with $p\le \kappa \log n/n$ for any constant $\kappa < 2/3$. On the other hand, this is not true for somewhat higher densities: If $p\ge \lambda (\log n)^3 / (n\log\log n)$ with $\lambda > 1/8$ then with high probability we need to delete more edges than one from every triangle. Our result has a natural algorithmic interpretation.

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