A randomized embedding algorithm for trees

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 d and maximum degree at most d − 2 d − 2. As there exist d-regular graphs without 4-cycles and with O(d) 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 Ks,t.