TY - JOUR

T1 - Bounded-degree spanning trees in randomly perturbed graphs

AU - Krivelevich, Michael

AU - Kwan, Matthew

AU - Sudakov, Benny

N1 - Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics.

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - Bounded-degree spanning trees

KW - Random graphs

KW - Smoothed analysis

UR - http://www.scopus.com/inward/record.url?scp=85018702644&partnerID=8YFLogxK

U2 - 10.1137/15M1032910

DO - 10.1137/15M1032910

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AN - SCOPUS:85018702644

SN - 0895-4801

VL - 31

SP - 155

EP - 171

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 1

ER -