Embedding nearly-spanning bounded degree trees

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.

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Correspondence to Noga Alon.

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Research supported in part by a USA-Israeli BSF grant, by NSF grant CCR-0324906, by a Wolfensohn fund and by the State of New Jersey.

Research supported in part by USA-Israel BSF Grant 2002-133, and by grants 64/01 and 526/05 from the Israel Science Foundation.

Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred P. Sloan fellowship.

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Alon, N., Krivelevich, M. & Sudakov, B. Embedding nearly-spanning bounded degree trees. Combinatorica 27, 629–644 (2007). https://doi.org/10.1007/s00493-007-2182-z

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Mathematics Subject Classification (2000)

  • 05C35
  • 05C80
  • 05C05