Low-distortion embeddings of graphs with large girth

TL;DR

This paper constructs a sequence of constant-degree graphs with increasing girth that can be embedded into  with bounded distortion, addressing a longstanding open problem in graph embeddings.

Contribution

It provides a construction of graphs with large girth and constant degree that embed into  with uniformly bounded distortion, solving an open problem from 2002.

Findings

Graphs with arbitrarily large girth and constant degree admit low-distortion  embeddings.

The construction answers an open problem in graph embedding theory.

The result has implications for metric embedding and graph theory.

Abstract

The main purpose of the paper is to construct a sequence of graphs of constant degree with indefinitely growing girths admitting embeddings into $\ell_1$ with uniformly bounded distortions. This result answers the problem posed by N. Linial, A. Magen, and A. Naor (2002).