## Abstract

In this paper, we propose a simple and natural randomized algorithm to embed a tree *T* in a given graph *G*. The algorithm can be viewed as a “self-avoiding tree-indexed random walk“. The order of the tree *T* can be as large as a constant fraction of the order of the graph *G*, and the maximum degree of *T* can be close to the minimum degree of *G*. We show that our algorithm works in a variety of interesting settings. For example, we prove that any graph of minimum degree *d* without 4-cycles contains every tree of order *εd*
^{2} and maximum degree at most *d*-2*εd*-2. As there exist *d*-regular graphs without 4-cycles and with *O*(*d*
^{2}) vertices, this result is optimal up to constant factors. We prove similar nearly tight results for graphs of given girth and graphs with no complete bipartite subgraph *K*
_{
s,t
}.

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## Additional information

Research supported in part by NSF CAREER award DMS-0812005 and by an USA-Israeli BSF grant.

This work was done while the author was at Princeton University.

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### Cite this article

Sudakov, B., Vondrák, J. A randomized embedding algorithm for trees.
*Combinatorica* **30, **445–470 (2010). https://doi.org/10.1007/s00493-010-2422-5

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### Mathematics Subject Classification (2000)

- 05D40
- 05C81
- 05C05
- 05C35