Embedding nearly-spanning bounded degree trees

Noga Alon*, Michael Krivelevich, Benny Sudakov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1 - ε)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d ≥ 2 and 0 < ε < 1, there exists a constant c = c(d, ε) such that a random graph G(n, c/n) contains almost surely a copy of every tree T on (1 - ε)n vertices with maximum degree at most d. We also prove that if an (n, D, λ)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most λ in their absolute values) has large enough spectral gap D/λ as a function of d and ε, then G has a copy of every tree T as above.

Original languageEnglish
Pages (from-to)629-644
Number of pages16
JournalCombinatorica
Volume27
Issue number6
DOIs
StatePublished - Nov 2007

Funding

FundersFunder number
State of New Jersey
USA-Israel BSF2002-133, 526/05, 64/01
USA-Israeli BSF
National Science FoundationCCR-0324906
Directorate for Mathematical and Physical Sciences0546523
Alfred P. Sloan Foundation
Israel Science FoundationDMS-0546523, DMS-0355497

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