A randomized embedding algorithm for trees

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-2εd-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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Correspondence to Benny Sudakov.

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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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