Nearly optimal embeddings of trees

Abstract

In this paper we show how to find nearly optimal embeddings of large trees in several natural classes of graphs. The size of the tree T can be as large as a constant fraction of the size of the graph G, and the maximum degree of T can be close to the minimum degree of G. For example, we prove that any graph of minimum degree d without 4-cycles contains every tree of size \epsilon d^2 and maximum degree at most (1-2\epsilon)d - 2. As there exist d-regular graphs without 4-cycles of size O(d^2), this result is optimal up to constant factors. We prove similar nearly tight results for graphs of given girth, graphs with no complete bipartite subgraph K_{s,t}, random and certain pseudorandom graphs. These results are obtained using a simple and very natural randomized embedding algorithm, which can be viewed as a "self-avoiding tree-indexed random walk".