Cycle lengths in randomly perturbed graphs

Research output: Contribution to journalArticlepeer-review

Abstract

Let (Figure presented.) be an (Figure presented.) -vertex graph, where (Figure presented.) for some (Figure presented.). A result of Bohman, Frieze and Martin from 2003 asserts that if (Figure presented.), then perturbing (Figure presented.) via the addition of (Figure presented.) random edges, a.a.s. yields a Hamiltonian graph. We prove several improvements and extensions of the aforementioned result. In particular, keeping the bound on (Figure presented.) as above and allowing for (Figure presented.), we determine the correct order of magnitude of the number of random edges whose addition to (Figure presented.) a.a.s. yields a pancyclic graph. Moreover, we prove similar results for sparser graphs, and assuming the correctness of Chvátal's toughness conjecture, we handle graphs having larger independent sets. Finally, under milder conditions, we determine the correct order of magnitude of the number of random edges whose addition to (Figure presented.) a.a.s. yields a graph containing an almost spanning cycle.

Original languageEnglish
Pages (from-to)867-884
Number of pages18
JournalRandom Structures and Algorithms
Volume63
Issue number4
DOIs
StatePublished - Dec 2023

Keywords

  • cycle lengths
  • hamiltonicity
  • independence number
  • pancyclicity
  • randomly perturbed graphs
  • toughness

All Science Journal Classification (ASJC) codes

  • Software
  • Applied Mathematics
  • General Mathematics
  • Computer Graphics and Computer-Aided Design

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