TY - JOUR
T1 - Embedding nearly-spanning bounded degree trees
AU - Alon, Noga
AU - Krivelevich, Michael
AU - Sudakov, Benny
N1 - Funding Information:
* 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.
PY - 2007/11
Y1 - 2007/11
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=44649140055&partnerID=8YFLogxK
U2 - 10.1007/s00493-007-2182-z
DO - 10.1007/s00493-007-2182-z
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AN - SCOPUS:44649140055
SN - 0209-9683
VL - 27
SP - 629
EP - 644
JO - Combinatorica
JF - Combinatorica
IS - 6
ER -