Bounded-degree spanning trees in randomly perturbed graphs

Michael Krivelevich, Matthew Kwan, Benny Sudakov

Research output: Contribution to journalArticlepeer-review

36 Scopus citations

Abstract

We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T. This combines the viewpoints of the well-studied problems of embedding trees into fixed dense graphs and into random graphs, and extends a sizable body of existing research on randomly perturbed graphs. Specifically, we show that there is c = c(α, Δ) such that if G is an n-vertex graph with minimum degree at least αn, and T is an n-vertex tree with maximum degree at most Δ, then if we add cn uniformly random edges to G, the resulting graph will contain T asymptotically almost surely (as n ⇒ ∞). Our proof uses a lemma concerning the decomposition of a dense graph into superregular pairs of comparable sizes, which may be of independent interest.

Original languageEnglish
Pages (from-to)155-171
Number of pages17
JournalSIAM Journal on Discrete Mathematics
Volume31
Issue number1
DOIs
StatePublished - 2017

Keywords

  • Bounded-degree spanning trees
  • Random graphs
  • Smoothed analysis

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