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Bi-Lipschitz embedding properties of lamplighter graphs on weighted and unweighted trees

Charlotte Melby, Beata Randrianantoanina

TL;DR

This work provides counterexamples to the converse of a prior embedding result for lamplighter graphs by constructing trees W_{\theta,n} and WB_{\theta,n} in which complete graphs K_k cannot embed equi-bi-Lipschitzly, while Hamming cubes embed into the corresponding lamplighter graphs with distortion arbitrarily close to 1 (for certain parameters). It introduces an explicit near-optimal embedding of H_{2^k} into La(W_{2,n}) and La(WB_{2,n}) via carefully structured lamp configurations, and shows that, for large growth parameters, diamonds do not embed uniformly into La(W_{\theta,n}). The results yield nontrivial implications for Ribe-style characterizations of Banach space geometry and demonstrate delicate interactions between underlining tree geometry and lamplighter metrics, offering new counterexamples and sharpening the landscape of metric embeddings.

Abstract

In 2021 Baudier, Motakis, Schlumprecht, and Zsák proved that if a sequence of graphs $(G_k)_{k\in{\mathbb{N}}}$ contains the sequence of complete graphs with uniformly bounded distortion, then the sequence of lamplighter graphs on $G_k$'s contains Hamming cubes with uniformly bounded distortion and asked whether the converse holds. They suggested that a sequence of trees with edges replaced by paths of ``moderately growing'' lengths may be a counterexample. We prove that indeed this is the case, and that a sequence of ``moderately'' weighted trees is another counterexample. Further, we prove that diamond graphs do not embed with uniformly bounded distortion into lamplighter graphs on trees with edges replaced by paths with sufficiently fast growing lengths.

Bi-Lipschitz embedding properties of lamplighter graphs on weighted and unweighted trees

TL;DR

This work provides counterexamples to the converse of a prior embedding result for lamplighter graphs by constructing trees W_{\theta,n} and WB_{\theta,n} in which complete graphs K_k cannot embed equi-bi-Lipschitzly, while Hamming cubes embed into the corresponding lamplighter graphs with distortion arbitrarily close to 1 (for certain parameters). It introduces an explicit near-optimal embedding of H_{2^k} into La(W_{2,n}) and La(WB_{2,n}) via carefully structured lamp configurations, and shows that, for large growth parameters, diamonds do not embed uniformly into La(W_{\theta,n}). The results yield nontrivial implications for Ribe-style characterizations of Banach space geometry and demonstrate delicate interactions between underlining tree geometry and lamplighter metrics, offering new counterexamples and sharpening the landscape of metric embeddings.

Abstract

In 2021 Baudier, Motakis, Schlumprecht, and Zsák proved that if a sequence of graphs contains the sequence of complete graphs with uniformly bounded distortion, then the sequence of lamplighter graphs on 's contains Hamming cubes with uniformly bounded distortion and asked whether the converse holds. They suggested that a sequence of trees with edges replaced by paths of ``moderately growing'' lengths may be a counterexample. We prove that indeed this is the case, and that a sequence of ``moderately'' weighted trees is another counterexample. Further, we prove that diamond graphs do not embed with uniformly bounded distortion into lamplighter graphs on trees with edges replaced by paths with sufficiently fast growing lengths.

Paper Structure

This paper contains 6 sections, 14 theorems, 38 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions of the trees $W_{\theta,n}$ and $\mathsf{WB}_{\theta,n}$
  4. Proof of Theorem \ref{['main1']}
  5. Proof of Theorem \ref{['main2']}
  6. Proof of Theorem \ref{['nodiamonds']}

Key Result

Theorem 1.1

BMSZ21 The sequences $(\mathop{\mathrm{\mathsf{La}}}\nolimits(B_k))_{k\in \mathbb{N}}$, $(H_k)_{k\in \mathbb{N}}$, and $(K_k)_{k\in \mathbb{N}}$ pairwise equi-bi-Lipschitzly embed into each other, where, for $k\in\mathbb{N}$, $K_k$ is a complete graph on $k$ vertices and $H_k$ is the Hamming cube on

Figures (6)

  • Figure 3.1: The tree $W_{2,4}$
  • Figure 4.1: A possible configuration of an embedding of $K_4$ into $W_{2,n}$, where $s_1$ and $s_2$ are branching points.
  • Figure 5.1: The set $F^4_{1,1}\subset W_{2,4}$
  • Figure 5.2: Images of the four basis elements of $H_4$ in $\mathop{\mathrm{\mathsf{La}}}\nolimits(W_{2,4})$ - in each of them the lampligher is at the root of $W_{2,4}$
  • Figure 6.1: The set $F_n\subset W_{\theta,n}$ - the fork of depth 2
  • ...and 1 more figures

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Definition 2.1
  • Lemma 2.2
  • Lemma 4.1
  • ...and 13 more