Nonexistence of embeddings with uniformly bounded distortions of Laakso graphs into diamond graphs

TL;DR

This paper proves that Laakso graphs cannot be embedded into diamond graphs with uniformly bounded distortions, highlighting fundamental differences in their metric structures.

Contribution

It establishes the nonexistence of uniformly bilipschitz embeddings of Laakso graphs into diamond graphs, filling a gap in the understanding of their metric embedding properties.

Findings

Laakso graphs are not embeddable into diamond graphs with uniform distortion.

Diamond graphs are not uniformly doubling, unlike Laakso graphs.

The result clarifies the limitations of embedding Laakso graphs into diamond graphs.

Abstract

Diamond graphs and Laakso graphs are important examples in the theory of metric embeddings. Many results for these families of graphs are similar to each other. In this connection, it is natural to ask whether one of these families admits uniformly bilipschitz embeddings into the other. The well-known fact that Laakso graphs are uniformly doubling but diamond graphs are not, immediately implies that diamond graphs do not admit uniformly bilipschitz embeddings into Laakso graphs. The main goal of this paper is to prove that Laakso graphs do not admit uniformly bilipschitz embeddings into diamond graphs.