### 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 language | English (US) |
---|---|

Pages (from-to) | 629-644 |

Number of pages | 16 |

Journal | Combinatorica |

Volume | 27 |

Issue number | 6 |

DOIs | |

State | Published - Nov 1 2007 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics

## Fingerprint Dive into the research topics of 'Embedding nearly-spanning bounded degree trees'. Together they form a unique fingerprint.

## Cite this

*Combinatorica*,

*27*(6), 629-644. https://doi.org/10.1007/s00493-007-2182-z