## 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 language | English |
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Pages (from-to) | 155-171 |

Number of pages | 17 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 31 |

Issue number | 1 |

DOIs | |

State | Published - 2017 |

## Keywords

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

## All Science Journal Classification (ASJC) codes

- General Mathematics