Nearly optimal embeddings of trees

TL;DR

This paper presents a natural randomized algorithm for embedding large trees into various classes of graphs, achieving nearly optimal results in terms of size and degree constraints, with broad applicability.

Contribution

It introduces a simple randomized embedding method that attains nearly optimal tree embeddings in graphs with specific structural properties, improving upon prior bounds.

Findings

Embedding trees of size proportional to d^2 in graphs without 4-cycles

Optimality of results up to constant factors in certain graph classes

Applicability to graphs with given girth, bipartite restrictions, and pseudorandom properties

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".