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9084 results in Discrete Mathematics, Information Theory and Coding

Page 36 of 455

  • Triangles in randomly perturbed graphs

  • Part of
  • Julia Böttcher, Olaf Parczyk, Amedeo Sgueglia, Jozef Skokan
  • Journal:
    Combinatorics, Probability and Computing / Volume 32 / Issue 1 / January 2023
    Published online by Cambridge University Press:
    05 July 2022, pp. 91-121
    • Article
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  • We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$-vertex graph $G$ satisfying a given minimum degree condition and the binomial random graph $G(n,p)$. We prove that asymptotically almost surely $G \cup G(n,p)$ contains at least $\min \{\delta (G), \lfloor n/3 \rfloor \}$ pairwise vertex-disjoint triangles, provided $p \ge C \log n/n$, where $C$ is a large enough constant. This is a perturbed version of an old result of Dirac.

    Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516] in a strong form. We also prove a stability version of our result, which in the case of pairwise vertex-disjoint triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159–176], this fully resolves the existence of triangle factors in randomly perturbed graphs.

    We believe that the methods introduced in this paper are useful for a variety of related problems: we discuss possible generalisations to clique factors, cycle factors, and $2$-universality.

Page 36 of 455