Large holes in sparse random graphs

Abstract

We show that there is a function α(r) such that for each constantr≧3, almost everyr-regular graph onn vertices has a hole (vertex induced cycle) of size at least α(r)n asn→∞. We also show that there is a function β(c) such that forc>0 large enough,G n, p ,p=c/n almost surely has a hole of size at least β(c)n asn→∞.

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Frieze, A.M., Jackson, B. Large holes in sparse random graphs. Combinatorica 7, 265–274 (1987). https://doi.org/10.1007/BF02579303

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Keywords

  • Random Graph
  • Regular Graph
  • Large Hole
  • Sparse Graph
  • Queen Mary College