## 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.

This is a preview of subscription content, access via your institution.

## References

- [1]
M. Ajtai, J. Komlós and E. Szemerédi: The longest path in a random graph,

*Combinatorica***1(1)**(1981), 1–12. - [2]
N. Alon, M. Capalbo, Y. Kohayakawa, V. Rödl, A. Ruciński and E. Szemerédi: Near-optimal universal graphs for graphs with bounded degrees, in

*Proc.*5^{th}*Int. Workshop on Randomization and Approximation techniques in Computer Science (RANDOM-APPROX 2001)*, Berkeley 2001, 170–180. - [3]
N. Alon and J. H. Spencer:

*The probabilistic method*, 2^{nd}ed., Wiley, New York, 2000. - [4]
S. N. Bhatt, F. Chung, F. T. Leighton and A. Rosenberg: Universal graphs for bounded-degree trees and planar graphs,

*SIAM J. Disc. Math.***2**(1989), 145–155. - [5]
B. Bollobás: Long paths in sparse random graphs,

*Combinatorica***2(3)**(1982), 223–228. - [6]
B. Bollobás:

*Random graphs*, 2^{nd}ed., Cambridge Studies in Advanced Mathematics,**73**, Cambridge University Press, Cambridge, 2001. - [7]
F. R. K. Chung and R. L. Graham: On graphs which contain all small trees,

*J. Combin. Theory Ser. B***24**(1978), 14–23. - [8]
F. R. K. Chung and R. L. Graham: On universal graphs,

*Ann. New York Acad. Sci.***319**(1979), 136–140. - [9]
F. R. K. Chung and R. L. Graham: On universal graphs for spanning trees,

*Proc. London Math. Soc.***27**(1983), 203–211. - [10]
F. R. K. Chung, R. L. Graham and N. Pippenger: On graphs which contain all small trees II, in

*Proc. 1976 Hungarian Colloquium on Combinatorics*, 1978, 213–223. - [11]
W. Fernandez de la Vega: Long paths in random graphs,

*Studia Sci. Math. Hungar.***14**(1979), 335–340. - [12]
W. Fernandez de la Vega: Trees in sparse random graphs,

*J. Combin. Theory Ser. B***45**(1988), 77–85. - [13]
J. Friedman: On the second eigenvalue and random walks in random

*d*-regular graphs,*Combinatorica***11(4)**(1991), 331–362. - [14]
J. Friedman, J. Kahn and E. Szemerédi: On the second eigenvalue in random regular graphs, in

*Proc. of*21^{th}*ACM STOC*(1989), 587–598. - [15]
J. Friedman and N. Pippenger: Expanding graphs contain all small trees,

*Combinatorica***7(1)**(1987), 71–76. - [16]
A. Frieze: On large matchings and cycles in sparse random graphs,

*Discrete Math.***59**(1986), 243–256. - [17]
M. Krivelevich and B. Sudakov: Sparse pseudo-random graphs are Hamiltonian,

*J. Graph Theory***42**(2003), 17–33. - [18]
L. Pósa: Hamiltonian circuits in random graphs,

*Discrete Math.***14**(1976), 359–364.

## Author information

### Affiliations

### Corresponding author

## Additional 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.

## Rights and permissions

## About this article

### Cite this article

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

Received:

Published:

Issue Date:

### Mathematics Subject Classification (2000)

- 05C35
- 05C80
- 05C05