On the bi-Lipschitz geometry of lamplighter graphs

TL;DR

This paper systematically studies the bi-Lipschitz geometry of lamplighter graphs, showing embeddings into Hamming cubes and ℓ₁, and characterizing superreflexivity and clique presence through these graphs.

Contribution

It provides new bi-Lipschitz embedding results for lamplighter graphs over trees and characterizes superreflexivity and clique presence via lamplighter graph properties.

Findings

Lamplighter graphs over trees embed into Hamming cubes with distortion ≤6.

Lamplighter graphs over countable trees embed into ℓ₁.

Presence of a clique in a graph implies a Hamming cube in its lamplighter graph.

Abstract

In this article we start a systematic study of the bi-Lipschitz geometry of lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly embed into Hamming cubes with distortion at most~$6$. It follows that lamplighter graphs over countable trees bi-Lipschitzly embed into $\ell_1$. We study the metric behaviour of the operation of taking the lamplighter graph over the vertex-coalescence of two graphs. Based on this analysis, we provide metric characterizations of superreflexivity in terms of lamplighter graphs over star graphs or rose graphs. Finally, we show that the presence of a clique in a graph implies the presence of a Hamming cube in the lamplighter graph over it. An application is a characterization in terms of a sequence of graphs with uniformly bounded degree of the notion of trivial Bourgain-Milman-Wolfson type for arbitrary metric spaces, similar to Ostrovskii's characterization previously obtained in \cite{ostrovskii:11}.

Equations

$$\tfrac{1}{2}\cdot d_{\operatorname{La}(T)}(x,y)\leqslant d_{\mathrm{H}}\big{(}f(x),f(y)\big{)}\leqslant 3\cdot d_{\operatorname{La}(T)}(x,y)$$

$$s\cdot d_{M}(u,v)\leqslant d_{N}\big{(}f(u),f(v)\big{)}\leqslant D\cdot s\cdot d_{M}(u,v)\ .$$

$$\mathrm{dist}(f)=\sup_{u\neq v}\frac{d_{N}\big{(}f(u),f(v)\big{)}}{d_{M}(u,v)}\cdot\sup_{u\neq v}\frac{d_{M}(u,v)}{d_{N}\big{(}f(u),f(v)\big{)}}\ .$$

$$c_{N}(M)=\inf\big{\{}\mathrm{dist}(f):\,f\colon M\to N\text{ is a bi-Lipschitz embedding}\big{\}}$$

$$Y\textrm{ has trivial BMW-type if and only if }\displaystyle\sup_{k\in\mathbb{N}}c_{Y}\big{(}\operatorname{La}(\mathrm{B}_{k})\big{)}<\infty.$$

$$d_{\mathrm{H}}(\epsilon ,\delta )=\sum _{i\in I}|{\epsilon }_i-{\delta }_i|.$$

$$d_{\operatorname{La}(G)}\big{(}(A,x),(B,y)\big{)}=\operatorname{tsp}_{G}(x,A\bigtriangleup B,y)+\lvert A\bigtriangleup B\rvert\ .$$

$$d_{\square}\big{(}(x,y),(v,z)\big{)}=d_{G}(x,v)+d_{H}(y,z)\ .$$

$$\tfrac{2}{3}\cdot d_{\operatorname{La}(\mathrm{P}_{k})}(x,y)\leqslant d_{\square}\big{(}f(x),f(y)\big{)}\leqslant 2\cdot d_{\operatorname{La}(\mathrm{P}_{k})}(x,y)\ .$$

$${\epsilon }_i^A=\{\begin{array}{ll}1& \text{ if }{v}_{i-1}\in A,\\ 0& \text{ if }{v}_{i-1}\notin A.\end{array}$$

$$\displaystyle d_{\square}\big{(}f(A,v_{m}),$$

$$\displaystyle=d_{\mathrm{B}}\big{(}(\varepsilon^{A}_{i})_{i=1}^{m},(\varepsilon^{B}_{i})_{i=1}^{n})\big{)}+d_{\mathrm{B}}\big{(}(\varepsilon^{A}_{k+1-i})_{i=0}^{k-m},(\varepsilon^{B}_{k+1-i})_{i=0}^{k-n}\big{)}$$

$$=n-m+(k-m)-(k-n)$$

$$\displaystyle=2\cdot d_{\operatorname{La}(\mathrm{P}_{k})}\big{(}(A,v_{m}),(B,v_{n})\big{)}\ .$$

$$\displaystyle d_{\square}\big{(}f(A,v_{m}),$$

$$\displaystyle=d_{\mathrm{B}}\big{(}(\varepsilon^{A}_{i})_{i=1}^{m},(\varepsilon^{B}_{i})_{i=1}^{m}\big{)}+d_{\mathrm{B}}\big{(}(\varepsilon^{A}_{k+1-i})_{i=0}^{k-m},(\varepsilon^{B}_{k+1-i})_{i=0}^{k-m}\big{)}$$

$$=0+2=2.$$

$$\displaystyle d_{\mathrm{B}}\big{(}(\varepsilon^{A}_{i})_{i=1}^{m},(\varepsilon^{B}_{i})_{i=1}^{n}\big{)}$$

$$=\{\begin{array}{ll}n-m& \text{if }\ell >m,\\ m+n+2-2\ell & \text{if }\ell ⩽m.\end{array}$$

$$\displaystyle d_{\mathrm{B}}\big{(}(\varepsilon^{A}_{k+1-i})_{i=0}^{k-m},$$

$$=\{\begin{array}{ll}(n+1)-(m+1)& \text{if }r⩽n,\\ (r+1)-(m+1)+(r+1)-(n+1)& \text{if }r>n,\end{array}$$

$$=\{\begin{array}{ll}n-m& \text{if }r⩽n,\\ 2r-m-n& \text{if }r>n.\end{array}$$

$$d_{\operatorname{La}(\mathrm{P}_{k})}\big{(}(A,v_{m}),(B,v_{n})\big{)}\leqslant m+n+2-2\ell+r-\ell+1=r+m+n-3(\ell-1)\\ \leqslant 3(n-\ell+1)=\tfrac{3}{2}\cdot d_{\square}\big{(}f(A,v_{m}),f(B,v_{n})\big{)}\ .$$

$$d_{\operatorname{La}(\mathrm{P}_{k})}\big{(}(A,v_{m}),(B,v_{n})\big{)}\leqslant m-n+2r-2\ell+r-\ell+1=m-n-2+3(r-\ell+1)\\ \leqslant 3(r-\ell+1)=\tfrac{3}{2}\cdot d_{\square}\big{(}f(A,v_{m}),f(B,v_{n})\big{)}\ .$$

$$d_{\operatorname{La}(\mathrm{P}_{k})}\big{(}(A,v_{m}),(B,v_{n})\big{)}\leqslant n-m+r-\ell+1\leqslant n+r-2m\\ \leqslant 2(n-m)=d_{\square}\big{(}f(A,v_{m}),f(B,v_{n})\big{)}\ .$$

$$d_{\operatorname{La}(\mathrm{P}_{k})}\big{(}(A,v_{m}),(B,v_{n})\big{)}\leqslant 2r-2-m-n+r-\ell+1=3r-2-m-n-(\ell-1)\\ \leqslant 3r-3m=\tfrac{3}{2}\cdot d_{\square}\big{(}f(A,v_{m}),f(B,v_{n})\big{)}\ .$$

$$\big{\{}\varepsilon\colon\mathbb{Z}_{\leqslant n}\to\{0,1\}:\,n\in\mathbb{Z},\ \varepsilon\text{ has finite support}\big{\}}$$

$$(A,x)\cdot (B,y)=(A△xB,xy)$$

$$S^{\prime}=\big{\{}(\emptyset,s):\,s\in S\big{\}}\cup\big{\{}\big{(}\{e\},e\big{)}\big{\}}$$

$$V_{H}^{(V_{G})}\times V_{G}=\big{\{}(f,v):\,f\colon V_{G}\to V_{H}\text{ finitely supported, }v\in V_{G}\}\ ,$$

$$\operatorname{tsp}_{T}(x,A,y)=2\big{\lvert}[x,A]\setminus[x,y]\big{\rvert}+\big{\lvert}[x,y]\big{\rvert}\ .$$

$$w=(y_n,y_{n-1},\dots ,y_{i+1},y_i=x_i,x_{i+1},x_{i+2},\dots ,x_m)$$

$$p=(x_i,x_{i+1},\dots ,x_k=y_{\ell },y_{\ell -1},\dots ,y_{i+1},y_i)$$

$$p(w_j,y)=(w_j,x_{i-1},x_i,\dots ,x_m),$$

Full text

The first named author was supported by the National Science Foundation under Grant Number DMS-1800322. The second named author was supported by National Science Foundation under Grant Numbers DMS-1600600 and DMS-1912897. The third named author was supported by the National Science Foundation under Grant Numbers DMS-1464713 and DMS-1711076. The fourth author was supported by the 2018 Workshop in Analysis and Probability at Texas A&M University.

1. Introduction

Wreath products of groups provide a wealth of fundamental examples with various algebraic, spectral and geometric properties. Given two groups $\Gamma_{1}$ and $\Gamma_{2}$, we denote by $\Gamma_{2}^{(\Gamma_{1})}$ the set of all functions $f\colon\Gamma_{1}\to\Gamma_{2}$ with finite support, i.e., with $\{x\in\Gamma_{1}:\,f(x)\neq e_{\Gamma_{2}}\}$ finite, where $e_{\Gamma_{2}}$ is the identity element of $\Gamma_{2}$. This is a group with pointwise multiplication. We let \lambda\colon\Gamma_{1}\to\operatorname{Aut}\big{(}\Gamma_{2}^{(\Gamma_{1})}\big{)} denote the left-regular representation given by $\lambda(x)(f)=f^{x}$, where $f^{x}(y)=f(x^{-1}y)$. The *(restricted) wreath product $\Gamma_{2}\wr\Gamma_{1}$ of $\Gamma_{2}$ with $\Gamma_{1}$ *is then defined as the semi-direct product $\Gamma_{2}^{(\Gamma_{1})}\rtimes_{\lambda}\Gamma_{1}$. It is the group of all pairs $(f,x)$, where $f\in\Gamma_{2}^{(\Gamma_{1})}$ and $x\in\Gamma_{1}$, equipped with the product $(f,x)\cdot(g,y)=(fg^{x},xy)$. When $\Gamma_{2}=\mathbb{Z}_{2}$ (the cyclic group of order $2$), the wreath product $\mathbb{Z}_{2}\wr\Gamma_{1}$ is commonly referred to as the lamplighter group of $\Gamma_{1}$. We shall often identify $\mathbb{Z}_{2}^{(\Gamma_{1})}$ with the set of all finite subsets of $\Gamma_{1}$. Under this identification, pointwise product becomes symmetric difference, and hence the group operation of $\mathbb{Z}_{2}\wr\Gamma_{1}$ is given by $(A,x)\cdot(B,y)=(A\bigtriangleup xB,xy)$, where $xB=\{xb:\,b\in B\}$.

The group $\mathbb{Z}_{2}\wr\mathbb{Z}$ is an example of an amenable group with exponential growth. Random walks on wreath product groups have been extensively studied and are well known to exhibit interesting behaviours. In an influential article [kv:83], Kaĭmanovich and Vershik showed that $\mathbb{Z}_{2}\wr\mathbb{Z}$ is an example of a group of exponential growth for which the simple random walk on the Cayley graph has zero speed. The variety of geometric features of wreath products of groups has also come to play an important role, sometimes quite unexpectedly, in metric geometry. For instance, the geometry of $\mathbb{Z}\wr\mathbb{Z}$ is closely connected to the extension of Lipschitz maps [naor-peres:11], and is also used in distinguishing bi-Lipschitz invariants, namely Enflo type and edge Markov type [naor-peres:08].

In geometric group theory, the theory of compression exponents has undergone a detailed study, in particular the behaviour of compression exponents under taking wreath products. Compression exponents were introduced by Guentner and Kaminker in order to measure how well an infinite, finitely generated group that does not admit a bi-Lipschitz embedding into a certain metric space, can be faithfully represented in it. A deep result of Naor and Peres states that the $\ell_{1}$-compression of a lamplighter group over a group with at least quadratic growth is $1$. This result includes the case of the planar lamplighter group $\mathbb{Z}_{2}\wr\mathbb{Z}^{2}$. However, it is not known whether $\mathbb{Z}_{2}\wr\mathbb{Z}^{2}$ bi-Lipschitzly embeds into $\ell_{1}$. This challenging problem was raised by Naor and Peres in [naor-peres:11]. Understanding the $\ell_{1}$-embeddability of graphs is motivated by its profound connections with the design of efficient algorithms for some NP-hard problems (see [deza-laurent:97]*Chapter 10, [got:18]*Chapter 8, Chapter 43, and [naor:10]). Very little is known about the bi-Lipschitz embeddability of lamplighter groups into Banach spaces. The Euclidean distortion of $\mathbb{Z}_{2}\wr\mathbb{Z}_{k}$ is of the order $\sqrt{\log k}$. The lower bound was proved in [lnp:09] and the upper bound in [anv:10]. It was shown in [naor-peres:08] that $\mathbb{Z}_{2}\wr\mathbb{Z}_{k}$ bi-Lipschitzly embeds into $\ell_{1}$ with some distortion independent of $k$ (and thus so does $\mathbb{Z}_{2}\wr\mathbb{Z}$). In [RO2018], it was proved that a Banach space is superreflexive if and only if it does not contain bi-Lipschitz copies of $\mathbb{Z}_{2}\wr\mathbb{Z}_{k}$ (for every $k\in\mathbb{N}$ and with uniformly bounded distortions). In [csv:12], Cornulier, Stadler and Valette proved that for a finitely generated group $\Gamma$ and for a finitely generated free group $\mathrm{F}$, the equivariant $L_{1}$-compression of $\Gamma\wr\mathrm{F}$ is equal to that of $\Gamma$. It follows from this that $\mathbb{Z}_{2}\wr\mathrm{F}$ bi-Lipschitzly embeds into $\ell_{1}$.

Working with groups might be restrictive because relatively few graphs can be realized as Cayley graphs of groups. In this paper we consider the most general graph-theoretic setting and we will be concerned with the metric geometry of lamplighter graphs. We anticipate that working in this more flexible framework will be fruitful to construct new graphs with subtle geometric properties. Moreover, lamplighter graphs are generalizations of the wreath product construction in group theory and our results apply to lamplighter groups as well. Indeed, in the context of graph theory it is possible to define a notion of the wreath product of two graphs that is compatible with the wreath product construction in group theory in the sense that the wreath product of two Cayley graphs of groups is the Cayley graph of the wreath product of the two groups for a well-chosen set of generators (cf.  [donno:15]). For practical purposes which will be explained in the next section, we chose to work with the *walk/switch model *of the lamplighter graph over a graph $G$, simply denoted $\operatorname{La}(G)$. Specifically, $\operatorname{La}(G)$ is the graph whose vertex set consists of all pairs $(A,x)$ where $A$ is a finite subset of the vertex set of $G$, and $x$ is a vertex of $G$. Vertices $(A,x)$ and $(B,y)$ of $\operatorname{La}(G)$ are joined by an edge if and only if *either *$A=B$ and $xy$ is an edge in $G$ *or *$x=y$ and $A\bigtriangleup B=\{x\}$. A well known description of this graph is as follows. Assume there is a lamp attached to each vertex of $G$ and a lamplighter is able to walk along edges of $G$ and switch lights on and off. A vertex $(A,x)$ corresponds to the lamplighter standing at vertex $x$ of $G$ with $A$ being the set of lamps that are currently lit. The lamplighter can make one of two types of moves: he can either move to a neighbouring vertex of $G$ without changing the configuration of lamps that are lit, or he can change the state of the lamp at vertex $x$ and stay at vertex $x$. We will refer to these as horizontal and *vertical *moves, respectively. (See Figure 1.)

Other models with different available moves can also be considered, such as the move-and-switch/move model or the like. Note that just as different finite generating sets of a group lead to bi-Lipschitzly equivalent Cayley graphs, it is also easy to verify whether two models of lamplighter graphs are bi-Lipschitzly equivalent. Here we are talking about graphs as metric spaces with the geodesic distance. We will recall this and other standard graph-theoretic notions in Section 2.

Our first main result is about lamplighter graphs over arbitrary trees.

Theorem A**.**

Let $T$ be a (non-empty) tree. Then there is a set $I$ such that $\operatorname{La}(T)$ bi-Lipschitzly embeds into the Hamming cube $\mathrm{H}_{I}$. More precisely, there exists a map $f\colon\operatorname{La}(T)\to\mathrm{H}_{I}$ such that

[TABLE]

for all $x,y\in\operatorname{La}(T)$. Moreover, if $T$ is finite or countable, then $I$ can also be chosen to be finite or countable, respectively.

It follows from Theorem A that the lamplighter graph over a countable tree bi-Lipschitzly embeds into $\ell_{1}$. In particular, this applies to the lamplighter group of a finitely generated free group; as we mentioned in the Introduction, this result also follows from more general results by Cornulier, Stadler and Valette  [csv:12]. Unlike  [csv:12] that relies on geometric group-theoretic arguments, our approach is based on elementary metric techniques.

Our second main result is a technical structural result (Theorem 12) that relates the geometry of the lamplighter graph over the vertex-coalescence of two graphs with the geometry of the coalesced components. By combining this structural result together with several embedding results that are discussed in Section 6, we extend the metric characterizations of superreflexivity in terms of lamplighter groups of [RO2018] to characterizations in terms of lamplighter graphs over graphs that are built by coalescing several copies of elementary graphs such as cycles or paths. In order to state our next result, we recall some basic definitions from metric geometry. Let $(M,d_{M})$ and $(N,d_{N})$ be two metric spaces. A map $f\colon M\to N$ is called a *bi-Lipschitz embedding *if there exist $s>0$ and $D\geqslant 1$ such that for all $u,v\in M$, we have

[TABLE]

The *distortion *$\mathrm{dist}(f)$ of a bi-Lipschitz embedding $f$ is given by

[TABLE]

As usual,

[TABLE]

denotes the $N$-distortion of $M$. If there is no bi-Lipschitz embedding from $M$ into $N$, then we set $c_{N}(M)=\infty$. A sequence $(M_{k})_{k\in\mathbb{N}}$ of metric spaces is said to *equi-bi-Lipschitzly embed *into a metric space $N$ if $\sup_{k\in\mathbb{N}}c_{N}(M_{k})<\infty$.

Denote by $\mathrm{St}_{n,k}$ the *star graph *with $n$ branches of length $k$, and by $\mathrm{Ro}_{n,k}$ the *rose graph *whose $n$ leaves are $k$-cycles (see Figure 2; definitions will be given in Section 4).

While these graphs can be easily embedded into every finite-dimensional Banach space of a sufficiently large dimension, it is far from being the case for lamplighter graphs over them.

Theorem B**.**

*Let $Y$ be a Banach space and $n\in\mathbb{N}$. The following assertions are equivalent.

• (i)

$Y$* is superreflexive;*

• (ii)

\displaystyle\sup_{k\in\mathbb{N}}c_{Y}\big{(}\operatorname{La}(\mathrm{St}_{n,k})\big{)}=\infty;

• (iii)

\displaystyle\sup_{k\in\mathbb{N}}c_{Y}\big{(}\operatorname{La}(\mathrm{Ro}_{n,k})\big{)}=\infty.

If $(M_{k})_{k\in\mathbb{N}}$ and $(N_{k})_{k\in\mathbb{N}}$ are sequences of metric spaces, we say that $(M_{k})_{k\in\mathbb{N}}$ equi-bi-Lipschitzly embeds into $(N_{k})_{k\in\mathbb{N}}$, or $(N_{k})_{k\in\mathbb{N}}$ equi-bi-Lipschitzly contains $(M_{k})_{k\in\mathbb{N}}$, if $\sup_{k}\inf_{\ell}c_{N_{\ell}}(M_{k})<\infty$, or equivalently, if there is a $C>0$ such that for all $k\in\mathbb{N}$ there exists $\ell\in\mathbb{N}$ such that $M_{k}$ bi-Lipschitzly embeds into $N_{\ell}$ with distortion at most $C$. We say that $(M_{k})_{k\in\mathbb{N}}$ and $(N_{k})_{k\in\mathbb{N}}$ are Lipschitz-comparable if $(M_{k})_{k\in\mathbb{N}}$ equi-bi-Lipschitzly embeds into $(N_{k})_{k\in\mathbb{N}}$ and $(N_{k})_{k\in\mathbb{N}}$ equi-bi-Lipschitzly embeds into $(M_{k})_{k\in\mathbb{N}}$. In Section 6 we prove that the lamplighter graph over $\mathrm{K}_{k}$, the complete graph with $k$ vertices, contains a bi-Lipschitz copy of the $k$-dimensional Hamming cube $\mathrm{H}_{k}$ with distortion independent of $k$. Together with Theorem A, it follows that the geometry of lamplighter graphs over complete graphs or over binary trees is essentially the same as the geometry of the Hamming cubes.

Theorem C**.**

The sequences \big{(}\operatorname{La}(\mathrm{K}_{k})\big{)}_{k\in\mathbb{N}}, \big{(}\operatorname{La}(\mathrm{B}_{k})\big{)}_{k\in\mathbb{N}} and \big{(}\mathrm{H}_{k}\big{)}_{k\in\mathbb{N}} are pairwise Lipschitz-comparable.

Theorem C has an important consequence regarding characterizations of the notion of trivial Bourgain-Milman-Wolfson type [bmw:86] (BMW-type in short). In 1986, Bourgain, Milman, and Wolfson showed that a metric space $Y$ has trivial BMW-type if and only if \sup_{k\in\mathbb{N}}c_{Y}\big{(}\mathrm{H}_{k})<\infty. This result is a nonlinear analogue of the Maurey-Pisier theorem for trivial type. The Hamming cube $\mathrm{H}_{k}$ is a $k$-regular graph and thus \big{(}\mathrm{H}_{k}\big{)}_{k\in\mathbb{N}} is a sequence of graphs with unbounded degree. The notion of BMW-type comes from the local theory of Banach spaces and a natural question is whether trivial BMW-type can be characterized as above using a sequence of graphs $(G_{k})_{k\in\mathbb{N}}$ with uniformly bounded degree. For Banach spaces, Ostrovskii [ostrovskii:11] answered this question positively and it is not difficult to see that the sequence of graphs with maximum degree 3 from [ostrovskii:11] is actually Lipschitz-comparable to the sequence of Hamming cubes and thus also settles the question for arbitrary metric spaces. Since every graph in the sequence \big{(}\operatorname{La}(\mathrm{B}_{k})\big{)}_{k\in\mathbb{N}} has maximum degree $4$, Theorem C also provides a sought after sequence $(G_{k})_{k\in\mathbb{N}}$.

Corollary D**.**

Let $(Y,d_{Y})$ be a metric space. Then,

[TABLE]

Note that the similar question for the nonlinear analogue of the Maurey-Pisier theorem for trivial cotype, obtained by Mendel and Naor [mendel-naor:08], has a simple solution for arbitrary metric spaces (see [ostrovskii:11] for a proof and a discussion of these questions).

Acknowledgment

We thank the referees for their careful reading and the many suggestions that helped improve the exposition and clarity of the paper.

2. Preliminaries on lamplighter graphs

We shall use standard graph theory terminology as can be found in [bollobas:98]. In particular, a graph $G$ is a pair $(V,E)$ where $V=V(G)$ is an arbitrary set (the set of vertices) and $E=E(G)$ is the set of edges, i.e., a set consisting of some unordered pairs of distinct vertices. (So edges are not directed and there are no multiple edges or loops.) We shall often write $x\in G$ instead of $x\in V$ for a vertex $x$. The edge connecting distinct vertices $x$ and $y$ is simply denoted by $xy$ (which is the same as $yx$). A *walk in *$G$ is a finite sequence $w=(x_{0},x_{1},\dots,x_{n})$ of vertices of $G$ with $n\geqslant 0$ such that $x_{i-1}x_{i}$ is an edge of $G$ for all $1\leqslant i\leqslant n$. We call $w$ a *walk from $x=x_{0}$ to $y=x_{n}$ *and call $n$ the length of $w$. If $w$ has no repetition of vertices other than the first and last vertices, i.e., if $x_{i}\neq x_{j}$ whenever $1<j-i<n$, then $w$ is called a path (from $x$ to $y$). If $w$ is a walk and $x_{r}=x_{s}$ for some $r,s$ with $1<s-r<n$, then $(x_{0},\dots,x_{r-1},x_{s},x_{s+1},\dots,x_{n})$ is a strictly shorter walk from $x$ to $y$. It follows that if $w^{\prime}$ is a subsequence of $w$ of minimal length such that $w^{\prime}$ is a walk from $x$ to $y$, then $w^{\prime}$ is in fact a path. We say that the graph $G$ is *connected *if any two vertices are connected in $G$ by a walk (or, equivalently, by a path).

A connected graph $G$ becomes a metric space in a natural way. For vertices $x$ and $y$ of $G$, we denote by $d_{G}(x,y)$ (or sometimes simply by $d(x,y)$) the length of a shortest path in $G$ (called a geodesic) from $x$ to $y$. It is easy to verify that $d_{G}$ is a metric. An important example for us are Hamming cubes. For an arbitrary set $I$, the Hamming cube $\mathrm{H}_{I}$ has vertex set $\{0,1\}^{(I)}$ consisting of all functions $\varepsilon\colon I\to\{0,1\}$ with finite support, i.e., the set $\{i\in I:\,\varepsilon_{i}=1\}$ is a finite subset of $I$. Two vertices $\varepsilon$ and $\delta$ are joined by an edge if and only if they differ in exactly one coordinate, i.e., there is a unique $i\in I$ with $\varepsilon_{i}\neq\delta_{i}$. The graph distance on $\mathrm{H}_{I}$, denoted $d_{\mathrm{H}}$ and referred to as the Hamming metric, is the $\ell_{1}$-metric given by

[TABLE]

We shall often identify $\{0,1\}^{(I)}$ with the set of all finite subsets of $I$. Under this identification, the Hamming metric becomes the symmetric difference metric given by $d_{\mathrm{H}}(A,B)=\lvert A\bigtriangleup B\rvert$ for finite subsets $A$ and $B$ of $I$.

2.1. A closed formula for the lamplighter graph metric

Let us recall the definition of the lamplighter graph $\operatorname{La}(G)$ of a graph $G$. The vertex set of $\operatorname{La}(G)$ consists of all pairs $(A,x)$ with $x\in G$ and $A$ a finite subset of $G$. Two vertices $(A,x)$ and $(B,y)$ are joined by an edge in $\operatorname{La}(G)$ if and only if *either *$A=B$ and $xy$ is an edge in $G$ *or *$x=y$ and $A\bigtriangleup B=\{x\}$, and these correspond, respectively, to horizontal and vertical moves by the lamplighter.

It is clear that if $\operatorname{La}(G)$ is connected, then so is $G$. Indeed, the horizontal moves in a path in $\operatorname{La}(G)$ from $(\emptyset,x)$ to $(\emptyset,y)$ correspond to a path from $x$ to $y$ in $G$. The converse also holds and its proof yields a formula for the graph metric of $\operatorname{La}(G)$ given in Proposition 1 below. Computing the distance in $\operatorname{La}(G)$ boils down to the problem of finding a shortest walk in $G$ from a vertex $x$ to another vertex $y$ that visits all vertices in a given subset $C$ of $G$. This is a well known and famous problem, the *travelling salesman problem *for $G$. We shall denote by $\operatorname{tsp}_{G}(x,C,y)$ the length of a solution to this problem, i.e., the least $n\geqslant 0$ for which there is a walk $(x_{0},x_{1},\dots,x_{n})$ from $x=x_{0}$ to $y=x_{n}$ such that $C\subset\{x_{0},x_{1},\dots,x_{n}\}$.

Proposition 1**.**

Let $G$ be a connected graph. Then the lamplighter graph $\operatorname{La}(G)$ is also connected with graph metric given by

[TABLE]

Proof.

Let us fix vertices $(A,x)$ and $(B,y)$. The lamplighter clearly needs at least $\lvert A\bigtriangleup B\rvert$ vertical moves in getting from $(A,x)$ to $(B,y)$ in order to switch all lamps in $A\setminus B$ off and to lit all lamps in $B\setminus A$. As the lamplighter can only alter the state of the lamp at the vertex he is currently at, his horizontal moves must visit all vertices in $A\bigtriangleup B$ while travelling from $x$ to $y$. Thus, the right-hand side in the expression above is a lower bound for the distance. It is easy to see that this lower bound is attained. Indeed, let $n=\operatorname{tsp}_{G}(x,A\bigtriangleup B,y)$ and let $w=(x_{0},x_{1},\dots,x_{n})$ be a walk in $G$ from $x=x_{0}$ to $y=x_{n}$ such that $A\bigtriangleup B\subset\{x_{0},x_{1},\dots,x_{n}\}$. Let $m=\lvert A\bigtriangleup B\rvert$ and let $0\leqslant i_{1}<i_{2}<\dots<i_{m}\leqslant n$ be such that $A\bigtriangleup B=\{x_{i_{1}},\dots,x_{i_{m}}\}$. Set $i_{m+1}=n$. Now consider the following path of length $m+n$ in $\operatorname{La}(G)$ from $(A,x)$ to $(B,y)$. Start with horizontal moves $(A,x_{i})$, $0\leqslant i\leqslant i_{1}$, from $(A,x)$ to $(A,x_{i_{1}})$. Having reached the vertex \big{(}A\bigtriangleup\{x_{i_{1}},\dots,x_{i_{j-1}}\},x_{i_{j}}\big{)} for some $1\leqslant j\leqslant m$, make the vertical move \big{(}A\bigtriangleup\{x_{i_{1}},\dots,x_{i_{j}}\},x_{i_{j}}\big{)} followed by horizontal moves \big{(}A\bigtriangleup\{x_{i_{1}},\dots,x_{i_{j}}\},x_{i}\big{)} for $i_{j}<i\leqslant i_{j+1}$. These moves end at the vertex \big{(}A\bigtriangleup\{x_{i_{1}},\dots,x_{i_{j}}\},x_{i_{j+1}}\big{)} which becomes $(B,y)$ when $j=m$. ∎

In general, the Travelling Salesman Problem is NP-hard. However, for some graphs it is possible to find explicit algorithms. We present one such algorithm for trees in Section 3.1. This is essentially a pre-order traversal algorithm that also accounts for backtracking.

2.2. A first example: the lamplighter graph over a path

It was shown, amongst other things, in [stein-taback:12] that the group $\mathbb{Z}_{q}\wr\mathbb{Z}$ (with a specific generating set) embeds bi-Lipschitzly with distortion at most $4$ in the Cartesian product of two infinite $(q+1)$-regular trees. Their proof is based on a lengthy and tricky computation of the word-length of certain group elements. We provide a simpler proof of the finite analogue for the case $q=2$ in the purely graph-theoretic context. We replace $\mathbb{Z}$ with $\mathrm{P}_{k}$, the path of length $k$, which has vertices $v_{0},v_{1},\dots,v_{k}$ and edges $v_{i-1}v_{i}$, $1\leqslant i\leqslant k$, and replace the lamplighter group $\mathbb{Z}_{2}\wr\mathbb{Z}$ with the lamplighter graph $\operatorname{La}(\mathrm{P}_{k})$. After the proof we explain how our argument extends to the infinite case, and hence shows the result for $\mathbb{Z}_{2}\wr\mathbb{Z}$ just mentioned.

We first describe the binary tree $\mathrm{B}_{k}$ of height $k$, and introduce some notation. The vertex set of $\mathrm{B}_{k}$ is $\bigcup_{i=0}^{k}\{0,1\}^{i}$. Let $\delta=(\delta_{1},\dots,\delta_{m})$ and $\varepsilon=(\varepsilon_{1},\dots,\varepsilon_{n})$ be vertices of $\mathrm{B}_{k}$. We write $\delta\prec\varepsilon$ if $m<n$ and $\delta_{i}=\varepsilon_{i}$ for $1\leqslant i\leqslant m$. Then $\delta\varepsilon$ is an edge of $\mathrm{B}_{k}$ if and only if $\lvert m-n\rvert=1$ and either $\delta\prec\varepsilon$ or $\varepsilon\prec\delta$. We will write $\delta\preccurlyeq\varepsilon$ if $\delta=\varepsilon$ or $\delta\prec\varepsilon$. We define the length of $\delta$ to be $\lvert\delta\rvert=m$. If $m\geqslant 1$, we let $\delta^{\prime}=(\delta_{1},\dots,\delta_{m-1})$. Note that if $\lvert\delta\rvert\leqslant\lvert\varepsilon\rvert$, then $\delta\varepsilon$ is an edge of $\mathrm{B}_{k}$ if and only if $\delta=\varepsilon^{\prime}$. The unique vertex of length zero will be denoted by $\emptyset$ and the graph-metric by $d_{\mathrm{B}}$ regardless of the value of $k$.

Given graphs $G$ and $H$, the *Cartesian product graph *$G\operatorname{\square}H$ of $G$ and $H$ has vertex set $V(G)\times V(H)$, and vertices $(x,y)$ and $(v,z)$ are joined by an edge if and only if *either *$x=v$ and $yz$ is an edge in $H$ *or *$y=z$ and $xv$ is an edge in $G$. Observe that the graph metric on the Cartesian product is given by

[TABLE]

Note that the Hamming cube $\mathrm{H}_{n}=\mathrm{H}_{\{1,\dots,n\}}$ is the $n$-fold Cartesian product graph $\mathrm{P}_{1}\operatorname{\square}\dots\operatorname{\square}\mathrm{P}_{1}$.

Proposition 2**.**

Let $k\in\mathbb{N}$. There exists a map $f\colon\operatorname{La}(\mathrm{P}_{k})\to\mathrm{B}_{k+1}\operatorname{\square}\mathrm{B}_{k+1}$ such that for all $x,y\in\operatorname{La}(\mathrm{P}_{k})$ we have

[TABLE]

Proof.

For $(A,v_{m})\in\operatorname{La}(\mathrm{P}_{k})$ let f(A,v_{m})=\big{(}(\varepsilon^{A}_{i})_{i=1}^{m},(\varepsilon^{A}_{k+1-i})_{i=0}^{k-m}\big{)} where

[TABLE]

Let $(A,v_{m}),(B,v_{n})\in\operatorname{La}(\mathrm{P}_{k})$ and assume without loss of generality that $m\leqslant n$. If $A\bigtriangleup B=\emptyset$, then $A=B$, $\varepsilon^{A}=\varepsilon^{B}$ and

[TABLE]

If $v_{m}=v_{n}$ and $A\bigtriangleup B=\{v_{m}\}$, then $\varepsilon^{A}_{i}=\varepsilon^{B}_{i}$ if and only if $i\neq m+1$, and hence

[TABLE]

Since $d_{\operatorname{La}(\mathrm{P}_{k})}$ is a graph metric, it is sufficient to estimate the Lipschitz constant on adjacent vertices, and it follows from (4) and (5) that $f$ is $2$-Lipschitz.

Assume now that $A\bigtriangleup B\neq\emptyset$. Set \ell=\min\{i:\,v_{i-1}\in A\bigtriangleup B\}=\min\big{\{}i:\,\varepsilon^{A}_{i}\neq\varepsilon^{B}_{i}\big{\}} and r=\max\{i:\,v_{i-1}\in A\bigtriangleup B\}=\max\big{\{}i:\,\varepsilon^{A}_{i}\neq\varepsilon^{B}_{i}\big{\}}. From the definition of $\ell$ and $r$ it follows that

[TABLE]

and

[TABLE]

Obtaining a lower bound on d_{\square}\big{(}f(A,v_{m}),f(B,v_{n})\big{)} using (6) and (7) naturally splits into four cases. In all cases we will use the estimate $\lvert A\bigtriangleup B\rvert\leqslant r-\ell+1$.

• Case 1:

$\ell\leqslant m$ and $r\leqslant n$. In this case $\operatorname{tsp}_{\mathrm{P}_{k}}(v_{m},A\bigtriangleup B,v_{n})=m+n+2-2\ell$ as the salesman moves from $v_{m}$ to $v_{\ell-1}$ and then to $v_{n}$. We then get

[TABLE]

• Case 2:

$\ell\leqslant m$ and $r>n$. In this case $\operatorname{tsp}_{\mathrm{P}_{k}}(v_{m},A\bigtriangleup B,v_{n})=m-n+2r-2\ell$ as the salesman moves from $v_{m}$ to $v_{\ell-1}$, then to $v_{r-1}$ and finally to $v_{n}$. Thus,

[TABLE]

• Case 3:

$\ell>m$ and $r\leqslant n$. Then $\operatorname{tsp}_{\mathrm{P}_{k}}(v_{m},A\bigtriangleup B,v_{n})=n-m$ as the optimal walk for the salesman is from $v_{m}$ to $v_{n}$. Therefore,

[TABLE]

• Case 4:

$\ell>m$ and $r>n$. In this range $\operatorname{tsp}_{\mathrm{P}_{k}}(v_{m},A\bigtriangleup B,v_{n})=2r-2-m-n$ as the salesman moves from $v_{m}$ to $v_{r-1}$ and to $v_{n}$. Using (6) and (7) for the last time, we get

[TABLE]

∎

Remark*.*

Let $\mathrm{Z}$ denote the double-infinite path. This graph has vertex set $\mathbb{Z}$ and edges between consecutive integers. Let $\mathrm{T}_{3}$ be the $3$-regular (infinite) tree. A description of $\mathrm{T}_{3}$ is as follows. For $n\in\mathbb{Z}$ denote by $\mathbb{Z}_{\leqslant n}$ the initial segment $\{m\in\mathbb{Z}:\,m\leqslant n\}$ of $\mathbb{Z}$. Then $\mathrm{T}_{3}$ has vertex set

[TABLE]

and vertices $(\delta_{i})_{i=-\infty}^{m}$ and $(\varepsilon_{i})_{i=-\infty}^{n}$ with $m\leqslant n$ are joined by an edge if and only if $n=m+1$ and $\delta_{i}=\varepsilon_{i}$ for all $i\leqslant m$. An almost identical argument as the one used in the proof above shows that the map $f\colon\operatorname{La}(\mathrm{Z})\to\mathrm{T}_{3}\operatorname{\square}\mathrm{T}_{3}$ defined by f(A,n)=\big{(}(\varepsilon^{A}_{i})_{i=-\infty}^{n},(\varepsilon^{A}_{-i})_{i=-\infty}^{-n-1}\big{)} has distortion at most 3, where $\varepsilon^{A}$ denotes the indicator function of the finite subset $A$ of $\mathbb{Z}$. It is clear that $\operatorname{La}(\mathrm{Z})$ is isometric to $\mathbb{Z}_{2}\wr\mathbb{Z}$ with respect to a suitable set of generators.

2.3. Lamplighter graphs vs lamplighter groups

We conclude this section by making precise the connection between lamplighter graphs and lamplighter groups. As previously mentioned, the lamplighter group of a group $\Gamma$ is the (restricted) wreath product $\mathbb{Z}_{2}\wr\Gamma$. This can be thought of as the set of all pairs $(A,x)$ with $A$ a finite subset of $\Gamma$ and $x\in\Gamma$ with multiplication defined by

[TABLE]

where $xB=\{xb:\,b\in B\}$. Now assume that $\Gamma$ is generated by $S\subset\Gamma$. We assume that the identity $e\notin S$ and that $x^{-1}\in S$ whenever $x\in S$. The *(right) Cayley graph *$\operatorname{Cay}(\Gamma,S)$ of $\Gamma$ with respect to $S$ has vertex set $\Gamma$, and $x,y\in\Gamma$ are joined by an edge if and only if $y^{-1}x\in S$. Since $S$ generates $\Gamma$, it follows that $\operatorname{Cay}(\Gamma,S)$ is connected. It is easy to verify that

[TABLE]

generates $\mathbb{Z}_{2}\wr\Gamma$. Moreover, the Cayley graph $\operatorname{Cay}(\mathbb{Z}_{2}\wr\Gamma,S^{\prime})$ is the lamplighter graph \operatorname{La}\big{(}\operatorname{Cay}(\Gamma,S)\big{)}.

Remark*.*

It is possible to define the wreath product of graphs which generalizes the notion of wreath product of groups. Let $G=(V_{G},E_{G})$ and $H=(V_{H},E_{H})$ be two graphs, and let $v_{0}$ be a distinguished point in $V_{H}$. A function $f\colon V_{G}\to V_{H}$ is called finitely supported if $f(v)=v_{0}$ for all but finitely many $v\in V_{G}$. The *wreath product $H\wr G$ of $H$ with $G$ *is the graph with vertex set

[TABLE]

and two vertices $(f,x)$ and $(g,y)$ are connected by an edge if and only if *either *$f=g$ and $xy$ is an edge in $G$ *or *$x=y$, $f(v)=g(v)$ for every $v\in G\setminus\{x\}$, and $f(x)g(x)$ is an edge in $H$. As in the special case above, it is easy to verify that if $G$ and $H$ are Cayley graphs of groups $\Gamma$ and $\Delta$, respectively, then $H\wr G$ is the Cayley graph of $\Delta\wr\Gamma$ with respect to a suitable generating set.

In this paper we are concerned with lamplighter graphs. It is not too hard to verify that some of our results extend fairly easily to more general wreath products. As wreath products with $\mathbb{Z}_{2}$ are of greatest interest, we prefer to state and prove our results only for such products. One justification for concentrating on lamplighter graphs instead of more general wreath products is as follows. Any finite graph $H$ is Lipschitz isomorphic to the complete graph $S$ on the vertex set of $H$ with distortion $D$ depending on $H$. This naturally induces a Lipschitz isomorphism between $H\wr G$ and $S\wr G$ with distortion at most $D$ for any graph $G$. An argument similar to [naor-peres:11, Lemma 2.1] shows that the bi-Lipschitz embeddability into $L_{p}$ of $S\wr G$ and $\operatorname{La}(G)$ are the same up to universal constants.

3. Embeddability of lamplighter graphs over trees into Hamming cubes

3.1. The Travelling Salesman Problem for trees

A *tree *is a connected acyclic graph, i.e., a connected graph in which there is no path $(x_{0},\dots,x_{n})$ with $n\geqslant 3$ and $x_{0}=x_{n}$. Equivalently, a tree is a graph such that for any two vertices $x$ and $y$ there is a unique path from $x$ to $y$. E.g., every binary tree is a tree.

We now fix a tree $T$ for the rest of this section. For vertices $x,y\in T$ we denote by $p(x,y)$ the unique path in $T$ from $x$ to $y$. If $p(x,y)=(x_{0},x_{1},\dots,x_{n})$, then we let $p_{i}(x,y)=x_{i}$ for $0\leqslant i\leqslant n$, and we also let $[x,y]=\{x_{i-1}x_{i}:\,1\leqslant i\leqslant n\}$ be the set of edges on the path $p(x,y)$. By definition of a path, every edge in $[x,y]$ occurs exactly once, and so \big{\lvert}[x,y]\big{\rvert}=d_{T}(x,y). It is also clear that if $p(x,y)=(x_{0},x_{1},\dots,x_{n})$, then $p(y,x)=(x_{n},x_{n-1},\dots,x_{1},x_{0})$, and hence $[x,y]=[y,x]$.

For $x\in T$ and for $A\subset T$ we let $[x,A]=\bigcup_{a\in A}[x,a]$. Note that if $A$ is finite, then so is $[x,A]$. We are now ready to provide a closed formula for the Travelling Salesman Problem on a tree.

Theorem 3**.**

For $x,y\in T$ and a finite $A\subset T$, we have

[TABLE]

We begin the proof with a couple of simple lemmas.

Lemma 4**.**

Let $x,y,a$ be vertices of $T$. Then $[x,a]\setminus[x,y]=[y,a]\setminus[x,y]$.

Proof.

We may assume that $x,y,a$ are pairwise distinct, otherwise the result is clear. Let $p(x,y)=(x_{0},x_{1},\dots,x_{m})$ and $p(x,a)=(y_{0},y_{1},\dots,y_{n})$. Then $x_{0}=y_{0}=x$, $x_{m}=y$ and $y_{n}=a$. Choose $i$ maximal with $0\leqslant i\leqslant\min(m,n)$ such that $x_{j}=y_{j}$ for $0\leqslant j\leqslant i$. Then

[TABLE]

is a walk from $a$ to $y$. We show that $w$ is in fact a path. If it is not, then we must have $i<\min(m,n)$ and $x_{k}=y_{\ell}$ for some $k,\ell$ with $i+1\leqslant k\leqslant m$ and $i+1\leqslant\ell\leqslant n$. Choosing $k$ minimal, we obtain a cycle

[TABLE]

in $T$. Indeed, $x_{i}=y_{i}$ and there is no other repetition of vertices by minimality of $k$. Thus, $p$ is a path from $x_{i}$ to $y_{i}$. Moreover, since $x_{i+1}\neq y_{i+1}$, either $k>i+1$ or $\ell>i+1$, and hence the length $(k-i)+(\ell-i)$ of $p$ is at least 3. This contradiction completes the proof that $w$ is a path, and so $p(a,y)=w$.

Now let $e\in[x,a]\setminus[x,y]$. Since $e\in[x,a]$, we have $e=y_{j-1}y_{j}$ for some $1\leqslant j\leqslant n$, and since $e\notin[x,y]$, we must have $i<j$. It follows that $e$ is also on the path $w=p(a,y)$, i.e., that $e\in[y,a]$. The inclusion $[x,a]\setminus[x,y]\subset[y,a]\setminus[x,y]$ follows, and the reverse inclusion holds by symmetry in $x,y$. ∎

The next lemma shows that any walk from $x$ to $y$ must travel through every edge in the unique path from $x$ to $y$.

Lemma 5**.**

Let $x,y\in T$ and $w=(w_{0},w_{1},\dots,w_{n})$ be a walk from $x$ to $y$. Then for every $e\in[x,y]$ there exists $1\leqslant j\leqslant n$ such that $e=w_{j-1}w_{j}$.

Proof.

Let $p(x,y)=(x_{0},x_{1},\dots,x_{m})$. Then $e=x_{i-1}x_{i}$ for some $1\leqslant i\leqslant m$. We observed at the start of the previous section that in any graph, every walk between vertices contains a subsequence which is a path between the same vertices. It follows that $p(x,y)$ is a subsequence of $w$. Hence there is a maximal $j$, $1\leqslant j\leqslant n$, such that $w_{j-1}=x_{i-1}$. If $w_{j}=x_{i}$, then we are done. So let us assume $w_{j}\neq x_{i}$. Then $w_{j}\notin\{x_{k}:\,i\leqslant k\leqslant m\}$ since otherwise we obtain a cycle in $T$. It follows that

[TABLE]

which therefore must be a subsequence of the walk $(w_{j},w_{j+1},\dots,w_{n})$. In particular, $x_{i-1}=w_{k-1}$ for some $j<k\leqslant n$, which contradicts the maximality of $j$. ∎

We are now ready to prove the lower bound for the Travelling Salesman Problem in $T$.

Proposition 6**.**

For $x,y\in T$ and a finite $A\subset T$, we have

[TABLE]

Proof.

Let $n=\operatorname{tsp}_{T}(x,A,y)$ and let $w=(w_{0},w_{1},\dots,w_{n})$ be a walk from $x$ to $y$ such that $A\subset\{w_{0},w_{1},\dots,w_{n}\}$. By Lemma 5, for every $e\in[x,y]$, there is at least one $j\in\{1,2,\dots,n\}$ such that $e=w_{j-1}w_{j}$.

Now assume $e\in[x,A]\setminus[x,y]$. Then $e\in[x,a]\setminus[x,y]$ for some $a\in A$. Choose $i$ with $0\leqslant i\leqslant n$ and $a=w_{i}$. Then $(w_{0},w_{1},\dots,w_{i})$ is a walk from $x$ to $a$, and hence by Lemma 5, $e=w_{j-1}w_{j}$ for some $1\leqslant j\leqslant i$. On the other hand, by Lemma 4 we also have $e\in[a,y]$. Since $(w_{i},w_{i+1},\dots,w_{n})$ is a walk from $a$ to $y$, it follows that $e=w_{k-1}w_{k}$ for some $i<k\leqslant n$. Since $j\neq k$, it follows that every edge in $[x,A]\setminus[x,y]$ appears at least twice in the walk $w$. The result follows. ∎

We next introduce some more notation. For $x\in T$ we denote by $N_{x}$ the set of *neighbours of $x$ *given by $N_{x}=\{y\in T:\,xy\in E(T)\}$. For $y\in N_{x}$ we let $T_{x,y}=\{z\in T:\,p_{1}(x,z)=y\}$, and for $y\in N_{x}$ and $A\subset T$ we let $A_{x,y}=A\cap T_{x,y}$. We now establish some simple properties.

Lemma 7**.**

*Fix $x\in T$ and $A\subset T$. We have then the following.

• (i)

$T=\{x\}\cup\bigcup_{y\in N_{x}}T_{x,y}$.

• (ii)

For $y\in N_{x}$ and $z\in T_{x,y}$, we have $[x,z]=\{xy\}\cup[y,z]$. Moreover, the endvertices of an edge in $[x,z]$ lie in $\{x\}\cup T_{x,y}$.

• (iii)

$A\setminus\{x\}=\bigcup_{y\in N_{x}}A_{x,y}$* and $[x,A]=\bigcup_{y\in N_{x}}[x,A_{x,y}]$.*

• (iv)

Given $y\in N_{x}$, if $A_{x,y}\neq\emptyset$, then $[x,A_{x,y}]=\{xy\}\cup[y,A_{x,y}]$.

Furthermore, all unions above are disjoint unions.

Proof.

Given a vertex $z\neq x$, let $p(x,z)=(x_{0},x_{1},\dots,x_{n})$. Then $n\geqslant 1$, $y=x_{1}$ is a neighbour of $x_{0}=x$, and $z\in T_{x,y}$. It is clear that $x\notin T_{x,y}$ for any $y\in N_{x}$. Moreover, it is immediate from definition that $T_{x,y}\cap T_{x,z}=\emptyset$ for distinct neighbours $y,z$ of $x$. Thus, (i) follows.

To see (ii), let $p(x,z)=(x_{0},x_{1},\dots,x_{n})$. Then $n\geqslant 1$, $x_{0}=x$, $x_{1}=y$ and $p(y,z)=(x_{1},\dots,x_{n})$. Hence $[x,z]=\{xy\}\cup[y,z]$, and $xy\notin[y,z]$ since the vertices $x_{0},x_{1},\dots,x_{n}$ are pairwise distinct. For $1\leqslant i\leqslant n$, we have $p(x,x_{i})=(x_{0},x_{1},\dots,x_{i})$, and hence $p_{1}(x,x_{i})=y$. This implies that $\{x_{0},x_{1},\dots,x_{n}\}\subset\{x\}\cup T_{x,y}$ and the second part of (ii) follows.

It follows from (i) that $A\setminus\{x\}=\bigcup_{y\in N_{x}}A_{x,y}$ and that this is a disjoint union. The second part of (iii) now follows:

[TABLE]

and moreover, since the endvertices of an edge in $[x,A_{x,y}]$ lie in $\{x\}\cup T_{x,y}$, it follows that the sets $[x,A_{x,y}]$, $y\in N_{x}$, are pairwise disjoint.

Finally, we establish (iv). If $A_{x,y}\neq\emptyset$, then from (ii) it follows that

[TABLE]

and the union is a disjoint union. ∎

We next prove Theorem 3 in a special case.

Theorem 8**.**

For $x\in T$ and a finite $A\subset T$, we have \operatorname{tsp}_{T}(x,A,x)=2\big{\lvert}[x,A]\big{\rvert}.

Proof.

We may assume $A\neq\emptyset$ otherwise the result is clear. Let us define $h=h(x,A)=\max_{a\in A}d(x,a)$. We construct by recursion on $h$ a walk $w$ from $x$ to $x$ of length $n=2\lvert[x,A]\rvert$ visiting all vertices in $A$. Together with Proposition 6, this will complete the proof.

If $h=0$, then $A=\{x\}$ and the result is clear. Let us now assume that $h\geqslant 1$. Set $N=N_{x}$ and $A_{y}=A_{x,y}$ for each $y\in N$. Let $M=\{y\in N:\,A_{y}\neq\emptyset\}$.

Fix $y\in M$. For every $a\in A_{y}$, we have $p(x,a)=(x,p(y,a))$, and thus, $h(y,A_{y})\leqslant h-1$. By recursion, there is a walk $w^{(y)}$ from $y$ to $y$ of length 2\big{\lvert}[y,A_{y}]\big{\rvert} visiting all vertices of $A_{y}$. Now since $A$ is finite, so is $M$, which we can then enumerate as $y_{1},y_{2},\dots,y_{k}$. Then

[TABLE]

is a walk from $x$ to $x$ visiting all vertices in $\bigcup_{y\in M}A_{y}$. It follows from Lemma 7 that $w$ visits all vertices in $A$ and has length

[TABLE]

∎

We are finally ready to complete the proof of our main result.

Proof of Theorem 3.

We proceed by induction on $d_{T}(x,y)$ and construct a walk from $x$ to $y$ of length 2\big{\lvert}[x,A]\setminus[x,y]\big{\rvert}+\big{\lvert}[x,y]\big{\rvert} visiting all vertices of $A$. Together with Proposition 6, this will complete the proof.

When $d_{T}(x,y)=0$, the result follows from Theorem 8. Now assume $d_{T}(x,y)\geqslant 1$ and set $x_{1}=p_{1}(x,y)$. Let $N=N_{x}$ and $A_{y}=A_{x,y}$ for all $y\in N$. Set $A_{0}=\bigcup_{z\in N,z\neq x_{1}}A_{z}$ and $A_{1}=A_{x_{1}}$. From Lemma 7 we have the following.

[TABLE]

and moreover, all unions are disjoint unions. From this we obtain

[TABLE]

By Theorem 8, there is a walk $w^{(0)}$ from $x$ to $x$ of length \ell_{0}=2\big{\lvert}[x,A_{0}]\big{\rvert} visiting all vertices in $A_{0}$. By induction hypothesis, there is a walk $w^{(1)}$ from $x_{1}$ to $y$ of length \ell_{1}=2\big{\lvert}[x_{1},A_{1}]\setminus[x_{1},y]\big{\rvert}+\big{\lvert}[x_{1},y]\big{\rvert} visiting all vertices in $A_{1}$. It follows that $w=(w^{(0)},w^{(1)})$ is a walk from $x$ to $y$ visiting all vertices in $\{x\}\cup A_{0}\cup A_{1}=\{x\}\cup A$. Let $\ell$ be the length of $w$. Then from (8) we obtain

[TABLE]

as required. ∎

Remark*.*

It is not hard to see that the proof of Theorem 3 yields an efficient algorithm for finding optimal walks in $T$ for the Travelling Salesman Problem.

3.2. Embeddability into Hamming cubes

We will show that the lamplighter graph over a tree bi-Lipschitzly embeds into a Hamming cube, and thus prove Theorem A. Let us fix a tree $T$. By Proposition 1 and Theorem 3, the graph metric in the lamplighter graph $\operatorname{La}(T)$ is given by

[TABLE]

For $C\subset T$ let $[C]=\bigcup_{x,y\in C}[x,y]$ be the minimal set of edges needed to travel between different vertices of $C$. Define

[TABLE]

For $A\subset T$, $x\in T$ and $e\in E(T)$, let A_{x,e}=\big{\{}a\in A:\,e\in[x,a]\big{\}}. We now define a map into the Hamming cube $\mathrm{H}_{I}$ whose role is to capture the first of the three summands in the right-hand side of (9).

Lemma 9**.**

Define $f\colon\operatorname{La}(T)\to\mathrm{H}_{I}$ as follows. For $(A,x)\in\operatorname{La}(T)$ and $i\in I$ we let

[TABLE]

Then for vertices $(A,x)$ and $(B,y)$ of $\operatorname{La}(T)$ we have

[TABLE]

Proof.

We first check that $f$ is well-defined, i.e., that $f(A,x)$ has finite support. Given $i=(e,C)\in I$, if $f(A,x)_{i}=1$, then $C=A_{x,e}\neq\emptyset$, and hence $e\in[x,A]$. It follows that the support of $f(A,x)$ has at most (in fact, exactly) \big{\lvert}[x,A]\big{\rvert} elements. Since $A$ is finite, so is $[x,A]$, and hence $f(A,x)$ is finitely supported.

We now turn to the inequalities. Given $i=(e,C)\in I$, we have

[TABLE]

Thus, setting $E=\{e\in E(T):\,A_{x,e}\neq B_{y,e}\text{ and \emph{either }}A_{x,e}\neq\emptyset\text{\ \emph{or }}B_{y,e}\neq\emptyset\}$, we have

[TABLE]

To estimate $\lvert E\rvert$, let us first consider an edge $e\in E\setminus[x,y]$. By definition of $E$, there is a vertex $c\in A_{x,e}\bigtriangleup B_{y,e}$. Hence, using Lemma 4, we have $e\in[x,c]\setminus[x,y]=[y,c]\setminus[x,y]$. It follows that $c\in A\bigtriangleup B$, and thus $e\in[x,A\bigtriangleup B]\setminus[x,y]$. This shows the upper bound

[TABLE]

Next consider $e\in[x,A\bigtriangleup B]\setminus[x,y]$. Then, using Lemma 4 again, we have some $c\in A\bigtriangleup B$ such that $e\in[x,c]\setminus[x,y]=[y,c]\setminus[x,y]$. It follows that $c\in A_{x,e}\bigtriangleup B_{y,e}$, and hence $e\in E$. This yields the lower bound

[TABLE]

Combining the inequalities (10), (11) and (12) completes the proof of the lemma. ∎

After some definitions, we will state and prove the main result of this section, which then immediately yields Theorem A. Note that for disjoint sets $J$ and $K$, the product $\mathrm{H}_{J}\operatorname{\square}\mathrm{H}_{K}$ is the Hamming cube $\mathrm{H}_{J\cup K}$. In the next result we identify the vertices of a Hamming cube $\mathrm{H}_{J}$ with finite subsets of $J$.

Theorem 10**.**

Let $T$ be a (non-empty) tree. Let $f\colon\operatorname{La}(T)\to\mathrm{H}_{I}$ be the map from Lemma 9. Fix $x_{0}\in T$ and define $F\colon\operatorname{La}(T)\to\mathrm{H}_{I}\operatorname{\square}\mathrm{H}_{E(T)}\operatorname{\square}\mathrm{H}_{T}$ by

[TABLE]

Then $F$ is a bi-Lipschitz embedding with distortion at most 6.

Proof.

Fix two vertices $(A,x)$ and $(B,y)$ in $\operatorname{La}(T)$. Then

[TABLE]

We first estimate the middle term. Let $p(x_{0},x)=(x_{0},x_{1},\dots,x_{m})$ and $p(x_{0},y)=(y_{0},y_{1},\dots,y_{n})$. As in the proof of Lemma 4, if $i\leqslant\min(m,n)$ is maximal such that $x_{j}=y_{j}$ for $0\leqslant j\leqslant i$, then

[TABLE]

It follows at once that

[TABLE]

Hence, using (9) and Lemma 9, we deduce that

[TABLE]

and that

[TABLE]

∎

4. Lamplighter graph over the vertex-coalescence of two graphs

The procedure that consists of gluing two graphs at a common vertex is known as *vertex-coalescence *or, simply, *coalescence *of two graphs. Consider two *pointed graphs *$G_{1}=(V_{1},E_{1},v_{1})$ and $G_{2}=(V_{2},E_{2},v_{2})$, i.e., graphs $G_{i}$, $i=1,2$, with vertex set $V_{i}$, edge set $E_{i}$ and a specified vertex $v_{i}\in V_{i}$. We define *the vertex-coalescence $G_{1}\ast G_{2}$ of $G_{1}$ and $G_{2}$ *by first taking the disjoint union of $G_{1}$ and $G_{2}$ followed by identifying the vertices $v_{1}$ and $v_{2}$. Formally, $G_{1}\ast G_{2}$ has vertex set

[TABLE]

where $v_{0}\notin V_{1}\times\{1\}\cup V_{2}\times\{2\}$, and edge set

[TABLE]

This formal definition is rather cumbersome. In practice, we shall either assume after relabeling that $V_{1}\cap V_{2}=\{v_{0}\}$ and $v_{0}=v_{1}=v_{2}$, in which case we can simply take $V=V_{1}\cup V_{2}$ and $E=E_{1}\cup E_{2}$, or, particularly in the case of gluing several copies of the same pointed graph together, we shall refer to vertices of the original graph as being in the $k^{\text{th}}$ copy in the coalesced graph for $k=1,2,\dots$. Note that if $G_{1}$ and $G_{2}$ are connected, then so is $G_{1}\ast G_{2}$ with graph metric given by

[TABLE]

In the next lemma we record the relationship between the Travelling Salesman Problem on the coalescence graph with the ones on its components. The proof is elementary and left to the reader.

Lemma 11**.**

Let $G_{1}\ast G_{2}$ be the coalescence of two connected pointed graphs $G_{1}$ and $G_{2}$ at a common vertex $v_{0}$. Let $x,y\in G_{1}\ast G_{2}$ and $C\subset G_{1}\ast G_{2}$ with $C$ finite.

If there exists $i\in\{1,2\}$ such that $x,y\in G_{i}$ and $C\subset G_{i}$, then

[TABLE]

If $x\in G_{1}$, $y\in G_{2}$ and $C=C_{1}\cup C_{2}$ with $C_{i}\subset G_{i}$ for $i=1,2$, then

[TABLE]

If $x\in G_{1}$, $y\in G_{1}$ and $C\cap G_{2}\neq\emptyset$, then

[TABLE]

where the minimum is taken over all sets $C^{\prime},C^{\prime\prime}\subset V_{1}$ with $C^{\prime}\cup C^{\prime\prime}=C\cap V_{1}$.

The purpose of the next theorem is to establish a metric connection between the lamplighter graph over the coalescence of two graphs with the lamplighter graphs over its components. To do this, we need to make use of clover graphs. Given $n\in\mathbb{N}$ and a pointed graph $G=(V,E,v_{0})$, *the clover graph *$\operatorname{Clo}(G,n)$ is obtained by coalescing $n$ copies of $G$ at $v_{0}$ in an obvious inductive fashion.

Theorem 12**.**

Let $G_{1}\ast G_{2}$ be the coalescence of two finite, connected pointed graphs $G_{1}$ and $G_{2}$ at a common vertex $v_{0}$. Then there exists a map

[TABLE]

such that

[TABLE]

for all $u,v\in\operatorname{La}(G_{1}\ast G_{2})$.

Proof.

Observe that $2^{\lvert G_{2}\rvert}$ is the number of subsets of $G_{2}$, and thus we can index the $2^{\lvert G_{2}\rvert}$ copies of $G_{1}$ in \operatorname{Clo}\big{(}G_{1},2^{\lvert G_{2}\rvert}\big{)} by the collection of all subsets of $G_{2}$. For $x\in G_{1}$ and $S\subset G_{2}$ we denote by $\iota_{S}(x)$ the element $x$ considered in the copy of $G_{1}$ in \operatorname{Clo}\big{(}G_{1},2^{\lvert G_{2}\rvert}\big{)} that is indexed by $S$. We proceed in a similar way for $\operatorname{Clo}(G_{2},2^{\lvert G_{1}\rvert})$. We define the function

[TABLE]

as follows. Given a vertex $(A,x)$ of $\operatorname{La}(G_{1}\ast G_{2})$, we let $A_{i}=A\cap G_{i}$ for $i=1,2$, and set

[TABLE]

To establish (17), we fix vertices $(A,x)$ and $(B,y)$ in $\operatorname{La}(G_{1}\ast G_{2})$, we let $A_{i}=A\cap G_{i}$ and $B_{i}=B\cap G_{i}$ for $i=1,2$, and consider several cases. To make the notation less crowded, we will at times drop subscripts in the graph distance.

• Case 1:

$x,y\in G_{1}$ and $A\bigtriangleup B\subset G_{1}$. In this case we have $A\bigtriangleup B=A_{1}\bigtriangleup B_{1}$ and $A_{2}=B_{2}$. It follows that

[TABLE]

which implies (17) since

[TABLE]

• Case 2:

$x\in G_{1}$ and $y\in G_{2}$. Then,

[TABLE]

and we are again done as in the previous case.

• Case 3:

$x,y\in G_{1}$ and $A\bigtriangleup B\cap G_{2}\neq\emptyset$. Then $A_{2}\bigtriangleup B_{2}\neq\emptyset$, and thus $A_{2}\neq B_{2}$. Therefore,

[TABLE]

For any decomposition $C\cup D$ of $A_{1}\bigtriangleup B_{1}$ one has

[TABLE]

and hence

[TABLE]

For the lower bound, assume without loss of generality that $d_{G_{1}}(x,v_{0})\geqslant d_{G_{1}}(v_{0},y)$. Since $\operatorname{tsp}_{G_{1}}(x,A_{1}\bigtriangleup B_{1},y)+d_{G_{1}}(v_{0},y)\geqslant\operatorname{tsp}_{G_{1}}(x,A_{1}\bigtriangleup B_{1},v_{0})$, it follows that

[TABLE]

∎

We will now illustrate the utility of Theorem 12 with two applications. The first result (Proposition 15) is concerned with embeddings of lamplighter graphs over star graphs into non-superreflexive Banach spaces. Given $k,n\in\mathbb{N}$, we define the *star graph $\mathrm{St}_{n,k}$ *to be the clover graph $\operatorname{Clo}(\mathrm{P}_{k},n)$ obtained by coalescing $n$ copies of a path of length $k$ at an endvertex (see Figure 2).

Lemma 13**.**

Let $k,n\in\mathbb{N}$ and let $(E,\lVert\cdot\rVert)$ be an $n$-dimensional Banach space. Then $\mathrm{St}_{n,k}$ bi-Lipschitzly embeds into $E$ with distortion at most $2$.

Proof.

Let $(e_{i})_{i=1}^{n}$ be an Auerbach basis for $E$. By this we mean that $\lVert e_{i}\rVert=1$ for $i=1,\ldots,n$ and that there are functionals $(e^{*}_{i})_{i=1}^{n}$ in the dual of $E$ which are also normalized and such that $e^{*}_{i}(e_{j})=\delta_{ij}$ for all $i,j=1,\ldots,n$. Define $f\colon\mathrm{St}_{n,k}\to E$ by $f(x)=d_{\mathrm{P}_{k}}(v_{0},x)e_{i}$ if $x$ belongs to the $i^{\text{th}}$ copy of $\mathrm{P}_{k}$ in $\mathrm{St}_{n,k}$. Here $v_{0}$ is the endvertex of $\mathrm{P}_{k}$ at which the $n$ copies of $\mathrm{P}_{k}$ are coalesced. If, for some $i$, both $x$ and $y$ belong to the $i^{\text{th}}$ copy of $\mathrm{P}_{k}$ in $\mathrm{St}_{n,k}$, then

[TABLE]

If, for some $i\neq j$, we have that $x$ belongs to the $i^{\text{th}}$ copy and $y$ to the $j^{\text{th}}$ copy of $\mathrm{P}_{k}$ in $\mathrm{St}_{n,k}$, then

[TABLE]

On the other hand, $f$ is clearly $1$-Lipschitz by the triangle inequality. ∎

The following lemma says that under certain conditions one can embed a finite product of metric spaces into a Banach space if the metric spaces are themselves embeddable in a particular fashion. Similar arguments have already been used in metric geometry (cf.  [ostrovskii:14]*Theorem 1.7) and their proofs simply rely on basic functional analytic principles. We provide a proof for the convenience of the reader unfamiliar with those.

Lemma 14**.**

Let $M_{1},\dots,M_{n}$ be metric spaces, and let $Y$ be an infinite-dimensional Banach space. Assume that there exist positive real numbers $D_{1},\dots,D_{n}$ such that for every $i=1,\dots,n$ and for every finite-codimensional subspace $Z$ of $Y$, there is a bi-Lipschitz embedding $\varphi_{i,Z}$ of $M_{i}$ of distortion at most $D_{i}$ into a finite-dimensional subspace of $Z$. Then for every $\varepsilon>0$, the product $M=\prod_{i=1}^{n}M_{i}$ equipped with the $\ell_{1}$-metric bi-Lipschitzly embeds into $Y$ with distortion at most $(2+\varepsilon)n\max_{1\leqslant i\leqslant n}D_{i}$.

Proof.

We begin with a basic result from the geometry of Banach spaces. Given $\delta>0$ and a finite-dimensional subspace $E$ of $Y$, there is a finite-codimensional subspace $Z$ of $Y$ such that $\lVert x+z\rVert\geqslant(1-\delta)\lVert x\rVert$ for all $x\in E$ and $z\in Z$. Indeed, choose a $\delta$-net $x_{1},\dots,x_{K}$ in the unit sphere of $E$ together with norming functionals $x_{1}^{*},\dots,x_{K}^{*}$ in $Y^{*}$. Set $Z=\bigcap_{i=1}^{K}\ker x^{*}_{i}$. Given $x\in E$ and $z\in Z$, assuming as we may that $\lVert x\rVert=1$, choose $i\in\{1,\dots,K\}$ such that $\lVert x-x_{i}\rVert\leqslant\delta$. Then we have

[TABLE]

as required.

Let us now turn to the statement of the lemma. Firstly, after scaling, we may assume that for every $i=1,\dots,n$ and for every finite-codimensional subspace $Z$ of $Y$ we have

[TABLE]

Fix $\varepsilon>0$ and choose $\delta>0$ satisfying $2(1-\delta)^{-n}<2+\varepsilon$. We will recursively construct finite-codimensional subspaces $Z_{1},\dots,Z_{n}$ of $Y$ together with finite-dimensional subspaces $E_{i}$ of $Z_{i}$ as follows. At the $j^{\text{th}}$ step, having chosen $E_{i}\subset Z_{i}$ for $1\leqslant i<j$, we choose a finite-codimensional subspace $Z_{j}$ of $Y$ such that $\lVert x+z\rVert\geqslant(1-\delta)\lVert x\rVert$ for all $x\in E_{1}+\dots+E_{j-1}$ and for all $z\in Z_{j}$. We then choose a finite-dimensinal subspace $E_{j}$ of $Z_{j}$ containing $\varphi_{j,Z_{j}}(M_{j})$. This completes the recursive construction, which has the following consequence. Given $x_{i}\in E_{i}$ for $i=1,\dots,n$, we have

[TABLE]

for each $m=1,\dots,n$. It follows by the triangle-inequality and by the choice of $\delta$ that

[TABLE]

We now define $\varphi\colon M\to Y$ by $\varphi(\boldsymbol{\mathrm{u}})=\sum_{i=1}^{n}\varphi_{i,Z_{i}}(u_{i})$ for $\boldsymbol{\mathrm{u}}=(u_{1},\dots,u_{n})$ in the product space $M=\prod_{i=1}^{n}M_{i}$. We claim that $\varphi$ is bi-Lipschitz with distortion at most $(2+\varepsilon)Dn$ where $D=\max_{1\leqslant i\leqslant n}D_{i}$. Let us fix $\boldsymbol{\mathrm{u}}=(u_{i})_{i=1}^{n}$ and $\boldsymbol{\mathrm{v}}=(v_{i})_{i=1}^{n}$ in $M$. On the one hand, the triangle-inequality yields

[TABLE]

On the other hand, using (19) we obtain the following lower bound.

[TABLE]

Thus, $\varphi$ has distortion at most $(2+\varepsilon)Dn$, as claimed. ∎

Using Theorem 12 we can show that for fixed $n\in\mathbb{N}$, the sequence $(\operatorname{La}(\mathrm{St}_{n,k}))_{k\in\mathbb{N}}$ of lamplighter graphs equi-bi-Lipschiztly embeds into any non-superreflexive Banach space.

Proposition 15**.**

Let $Y$ be a non-superreflexive Banach space. For all $n\in\mathbb{N}$, there exist $C(n)\in(0,\infty)$ and maps $f_{n,k}\colon\operatorname{La}(\mathrm{St}_{n,k})\to Y$ such that

[TABLE]

for all $k\in\mathbb{N}$ and for all $x,y\in\operatorname{La}(\mathrm{St}_{n,k})$.

Proof.

It is sufficient to prove the proposition for each $n\in\{2^{i}:\,i\in\mathbb{N}\}$. Observe that $\mathrm{St}_{2^{i},k}=\mathrm{St}_{2^{i-1},k}\ast\mathrm{St}_{2^{i-1},k}$ and $\lvert\mathrm{St}_{2^{i-1},k}\rvert=k\cdot 2^{i-1}+1$. Set $\alpha_{i,k}=2^{k\cdot 2^{i-1}+1}$. Applying Theorem 12, $\operatorname{La}(\mathrm{St}_{2^{i},k})$ bi-Lipschitzly embeds with distortion at most $2$ into

[TABLE]

Now observe that $\operatorname{Clo}(\mathrm{St}_{r,s},t)=\mathrm{St}_{rt,s}$ for any $r,s,t\in\mathbb{N}$. If we apply Theorem 12 another $i-1$ times, we obtain that $\operatorname{La}(\mathrm{St}_{2^{i},k})$ bi-Lipschitzly embeds with distortion at most $2^{i}$ into the Cartesian product of $4\cdot 2^{i-1}+2\cdot 2^{i-2}+\dots+2\cdot 2+2=3\cdot 2^{i}-2$ graphs each of which is either $\operatorname{La}(\mathrm{P}_{k})$ or a graph of the form $\mathrm{St}_{r,k}$ for some $r\in\mathbb{N}$. Note that all these graphs admit bi-Lipschitz embeddings into every finite-codimensional subspace of any non-superreflexive Banach space. Indeed, by Bourgain’s metric characterization of superreflexivity [bourgain:86], for every $\varepsilon>0$, every binary tree of finite height admits a bi-Lipschitz embedding into every finite-codimensional subspace of any non-superreflexive Banach space with distortion at most $1+\varepsilon$. Therefore the conclusion follows by combining this result with Lemma 13, Proposition 2, and Lemma 14. ∎

Let $\mathrm{C}_{k}$ denote the $k$-cycle, i.e., the cycle of length $k$ with vertices $v_{1},\dots,v_{k}$ and edges $v_{i-1}v_{i}$ for $i=1,\dots,k$, where we set $v_{0}=v_{k}$. Given $k,n\in\mathbb{N}$, we define the *rose graph $\mathrm{Ro}_{n,k}$ *to be the clover graph $\operatorname{Clo}(\mathrm{C}_{k},n)$ obtained by coalescing $n$ copies of $\mathrm{C}_{k}$ at $v_{0}$. Using Theorem 12 together with the main result from [RO2018], we can show that for fixed $n\in\mathbb{N}$ the sequence \big{(}\operatorname{La}(\mathrm{Ro}_{n,k})\big{)}_{k\in\mathbb{N}} of lamplighter graphs equi-bi-Lipschiztly embeds into any non-superreflexive Banach space. First we need to prove that $\mathrm{Ro}_{n,k}$ can be well embedded into Euclidean spaces.

Lemma 16**.**

Let $n\in\mathbb{N}$. There exist maps $g_{n,k}\colon\mathrm{Ro}_{n,k}\to\ell_{2}^{2n}$ such that

[TABLE]

for all $k\in\mathbb{N}$ and for all $x,y\in\mathrm{Ro}_{n,k}$.

Proof.

It was proven in [linial-magen:00] that the natural embedding of the $k$-cycle onto the vertices of the regular $k$-gon in $\mathbb{R}^{2}$ is optimal and has distortion exactly \frac{k}{2}\sin\big{(}\frac{\pi}{k}\big{)}\leqslant\frac{\pi}{2}. Therefore, there exist maps $\varphi_{k}\colon\mathrm{C}_{k}\to\ell_{2}^{2}$ with $\varphi_{k}(v_{0})=0$ and such that

[TABLE]

Let $E_{i}=\ell_{2}^{2}$ for all $i\in\mathbb{N}$ and define

[TABLE]

Observe that if $x$ and $y$ belong to the same copy of $\mathrm{C}_{k}$ in $\mathrm{Ro}_{n,k}$, then one has

[TABLE]

Otherwise,

[TABLE]

and

[TABLE]

∎

Proposition 17**.**

Let $Y$ be a non-superreflexive Banach space. For all $n\in\mathbb{N}$ there exist $D(n)\in(0,\infty)$ and maps $f_{n,k}\colon\operatorname{La}(\mathrm{Ro}_{n,k})\to Y$ such that

[TABLE]

for all $k\in\mathbb{N}$ and for all $x,y\in\operatorname{La}(\mathrm{Ro}_{n,k})$.

Proof.

It is sufficient to prove the proposition for each $n\in\{2^{i}:\,i\in\mathbb{N}\}$. Observe that $\mathrm{Ro}_{2^{i},k}=\mathrm{Ro}_{2^{i-1},k}\ast\mathrm{Ro}_{2^{i-1},k}$ and $\lvert\mathrm{Ro}_{2^{i-1},k}\rvert=(k-1)2^{i-1}+1$. Set $\beta_{i,k}=2^{(k-1)2^{i-1}+1}$. Applying Theorem 12, $\operatorname{La}(\mathrm{Ro}_{2^{i},k})$ bi-Lipschitzly embeds with distortion at most $2$ into

[TABLE]

Now observe that $\operatorname{Clo}(\mathrm{Ro}_{r,s},t)=\mathrm{Ro}_{rt,s}$ for any $r,s,t\in\mathbb{N}$. If we apply Theorem 12 another $i-1$ times, we obtain that $\operatorname{La}(\mathrm{Ro}_{2^{i},k})$ bi-Lipschitzly embeds with distortion at most $2^{i}$ into the Cartesian product of $4\cdot 2^{i-1}+2\cdot 2^{i-2}+\dots+2\cdot 2+2=3\cdot 2^{i}-2$ graphs each of which is either $\operatorname{La}(\mathrm{C}_{k})$ or a graph of the form $\mathrm{Ro}_{r,k}$ for some $r\in\mathbb{N}$. Note that all these graphs admit bi-Lipschitz embeddings into every finite-codimensional subspace of any non-superreflexive Banach space. Indeed, it was proved in [RO2018] that $\operatorname{La}(\mathrm{C}_{k})$ can be embedded into a product of 8 trees, and hence one can again use Bourgain’s metric characterization of superreflexivity [bourgain:86]. The conclusion follows by appealing to Lemma 16, Dvoretzky’s theorem, and Lemma 14. ∎

Remark*.*

By carefully keeping track of the distortions of embeddings in the proofs of Propositions 15 and 17, one obtains order $n^{2}$ upper bounds on $C(n)$ and $D(n)$.

Remark*.*

At this point we have established the implications “(ii)$\implies$(i)” and “(iii)$\implies$(i)” in Theorem B. The remaining implications will be shown in Section 6.

5. Induced maps between lamplighter graphs

A map $f\colon G\to H$ between two graphs induces a map $\accentset{\rule{2.79996pt}{0.6pt}}{f}\colon\operatorname{La}(G)\to\operatorname{La}(H)$ defined by \accentset{\rule{2.79996pt}{0.6pt}}{f}(A,x)=\big{(}f(A),f(x)\big{)}, where $f(A)=\{f(y):\,y\in A\}$. Moreover, if $G$ and $H$ are connected and for some $a,b\in[0,\infty]$ we have

[TABLE]

for all $x,y\in G$, then it is easy to see (cf. Remark following Lemma 18 below) that

[TABLE]

for all $u,v\in\operatorname{La}(G)$, where $a^{\prime}=\min\{1,a\}$ and $b^{\prime}=\max\{1,b\}$. Of course, this result is only interesting if $a>0$, $b<\infty$ or both, i.e., if $f$ is co-Lipschitz, Lipschitz or bi-Lipschitz, respectively. In particular, if $(G,d_{G})$ bi-Lipschitzly embeds into $(H,d_{H})$, then $\operatorname{La}(G)$ bi-Lipschitzly embeds into $\operatorname{La}(H)$. Observe that if $f$ is injective, then $b\geqslant 1$, and if in addition $0<a\leqslant 1$, then $\mathrm{dist}(\accentset{\rule{2.79996pt}{0.6pt}}{f})\leqslant\mathrm{dist}(f)$. However, there are bi-Lipschitz embeddings of interest where $a\to\infty$ with $b/a$ bounded. In this case, $b^{\prime}/a^{\prime}$ gets arbitrarily large. For this reason, we will consider more complicated induced maps in Lemmas 18 and 19 below.

Lemma 18**.**

Let $f\colon G\to H$ be a map between connected graphs $G$ and $H$, and let $a,b\in[0,\infty]$ be given so that

[TABLE]

Then for every $m\in\{0\}\cup\{1,\dots,\lceil a/2\rceil-1\}$, there is a map $\accentset{\rule{2.79996pt}{0.6pt}}{f}_{m}\colon\operatorname{La}(G)\to\operatorname{La}(H)$ induced by $f$ and $m$ such that $\accentset{\rule{2.79996pt}{0.6pt}}{f}_{0}=\accentset{\rule{2.79996pt}{0.6pt}}{f}$ and

[TABLE]

for all $u,v\in\operatorname{La}(G)$, where $a^{\prime}=\min\{a,m+1\}$ and $b^{\prime}=\max\{b,3m+1\}$.

Remark*.*

Assume that $f\colon G\to H$ is a bi-Lipschitz embedding, and thus that $b\geqslant 1$ and $a>0$. The observations made before the statement of the lemma follow immediately by taking $m=0$. On the other hand, choosing $m=\lceil a/2\rceil-1$, and by considering the fractions $\frac{b}{a}$, $\frac{b}{m+1}$, $\frac{3m+1}{a}$ and $\frac{3m+1}{m+1}$, it is easy to see that $\mathrm{dist}(\accentset{\rule{2.79996pt}{0.6pt}}{f}_{m})\leqslant b^{\prime}/a^{\prime}\leqslant 3b/a$. Hence we obtain the universal bound

[TABLE]

Proof.

If $m=0$ we define $\accentset{\rule{2.79996pt}{0.6pt}}{f}_{0}$ to be the natural induced map $\accentset{\rule{2.79996pt}{0.6pt}}{f}$, and the conclusion holds with $a^{\prime}=\min\{a,1\}$ and $b^{\prime}=\max\{b,1\}$. Indeed, let $x_{0},x_{1},\dots,x_{n}$ be vertices of $G$, and let $y_{i}=f(x_{i})$ for $0\leqslant i\leqslant n$. Consider a walk $w$ of length $\ell$ in $G$ from $x_{0}$ to $x_{n}$ visiting $x_{0},x_{1},\dots,x_{n}$ in this order. For each $i=1,\dots,n$, let $w_{i}$ be the section of $w$ from $x_{i-1}$ to $x_{i}$, and let $\ell_{i}$ be the length of $w_{i}$. Then $d_{H}(f(x_{i-1}),f(x_{i}))\leqslant b\cdot d_{G}(x_{i-1},x_{i})\leqslant b\cdot\ell_{i}$ for each $i=1,\dots,n$. Hence there is a walk in $H$ of length at most $b\cdot\ell$ from $y_{0}$ to $y_{n}$ visiting $y_{0},y_{1},\dots,y_{n}$ in this order. An essentially identical argument shows that if there is a walk in $H$ of length $\ell$ from $y_{0}$ to $y_{n}$ visiting $y_{0},y_{1},\dots,y_{n}$ in this order, then there is a walk in $G$ of length at most $\ell/a$ from $x_{0}$ to $x_{n}$ visiting $x_{0},x_{1},\dots,x_{n}$ in this order. It follows that

[TABLE]

for all $x,y\in G$ and for all finite $C\subset G$. Now fix vertices $(A,x)$ and $(B,y)$ of $\operatorname{La}(G)$. Observe that $f(A)\bigtriangleup f(B)\subset f(A\bigtriangleup B)$ and $\lvert f(A\bigtriangleup B)\rvert\leqslant\lvert A\bigtriangleup B\rvert$, and moreover equality holds when $a>0$. Hence, using Proposition 1 and (20), we have

[TABLE]

and if $a>0$, then

[TABLE]

Assume now that $1\leqslant m\leqslant\lceil a/2\rceil-1$ and in particular that $a>0$. Set $a^{\prime}=\min(a,m+1)$ and $b^{\prime}=\max(b,3m+1)$. Without loss of generality we will assume that $G$ has at least two vertices. For every vertex $y$ in the image of $f$, choose a path $(u_{0},u_{1},u_{2},\ldots,u_{m})$ in $H$ starting at $u_{0}=y$, and set $W_{y}=\{u_{0},u_{1},\dots,u_{m}\}$. This can always be done by picking another vertex $z\in f(G)$ and using $d_{H}(y,z)\geqslant a$ which in turn follows from the assumptions on $f$. It is easy to see that $\operatorname{tsp}_{H}(y,W_{y},y)=2m$ since the unique optimal walk for the salesman is the path from $u_{0}$ to $u_{m}$ and back. Note also that the sets $W_{y}$, $y\in f(G)$, are pairwise disjoint, since the vertices in $f(G)$ are $a$-separated and $2m<a$. For a finite set $C\subset f(G)$ we put $W_{C}=\bigcup_{y\in C}W_{y}$. Finally, we define $\accentset{\rule{2.79996pt}{0.6pt}}{f}_{m}\colon\operatorname{La}(G)\to\operatorname{La}(H)$ by letting \accentset{\rule{2.79996pt}{0.6pt}}{f}_{m}(A,x)=\big{(}W_{f(A)},f(x)\big{)}.

Given vertices $y,z$ and a finite subset $C$ in the image of $f$, we now obtain estimates on $\operatorname{tsp}_{H}(y,W_{C},z)$. Since $C\subset W_{C}$, we immediately obtain $\operatorname{tsp}_{H}(y,W_{C},z)\geqslant\operatorname{tsp}_{H}(y,C,z)$. On the other hand, consider the following walk. Start with a walk $w$ in $H$ of length $\operatorname{tsp}_{H}(y,C,z)$ from $y$ to $z$ visiting all vertices of $C$, and each time $w$ visits a vertex $u\in C$, insert a walk of length $2m=\operatorname{tsp}_{H}(u,W_{u},u)$ starting and ending at $u$ and visiting all vertices in $W_{u}$. The resulting walk from $y$ to $z$ visits all the vertices in $W_{C}$ and has length $\operatorname{tsp}_{H}(y,C,z)+2m\lvert C\rvert$. Therefore, we have

[TABLE]

Let us now fix vertices $(A,x)$ and $(B,y)$ in $\operatorname{La}(G)$. Observe that $W_{f(A)}\bigtriangleup W_{f(B)}=W_{f(A)\bigtriangleup f(B)}$ and $f(A)\bigtriangleup f(B)=f(A\bigtriangleup B)$. Combining Proposition 1, (21) and (20), we obtain

[TABLE]

and

[TABLE]

∎

In the last lemma of this section we consider a more sophisticated construction in order to improve the bound on the distortion. This construction is of a slightly different nature since it provides an embedding with a higher degree of faithfulness at the expense that we need to consider the lamplighter graph over a slightly bigger graph that contains the original graph $H$ under scrutiny. In some specific situations (see Proposition 23), this turns out not to be an issue and Lemma 19 can be efficiently used to significantly improve the distortion.

Let us fix a map $f\colon G\to H$ between two graphs. Let $Q=(V,E,v_{0})$ be a pointed graph and $W$ be a finite subset of $V$ with $v_{0}\in W$. Let $\widetilde{H}$ be the graph obtained by coalescing $H$ with $\lvert f(G)\rvert$ copies of $Q$ as follows. For each vertex $y$ in the image of $f$, we coalesce to $H$ at the vertex $y$ the copy of $Q$ that corresponds to $y$. This leads to a map $\tilde{f}\colon\operatorname{La}(G)\to\operatorname{La}(\widetilde{H})$ induced by $f$, $Q$ and $W$ and defined as follows. For $y\in f(G)$ we let $W_{y}$ denote the set $W$ considered in the copy of $Q$ that corresponds to $y$, and for a finite subset $C$ of $f(G)$ we let $W_{C}=\bigcup_{y\in C}W_{y}$. Finally, for a vertex $(A,x)$ of $\operatorname{La}(G)$ define \tilde{f}(A,x)=\big{(}W_{f(A)},f(x)\big{)}.

Lemma 19**.**

Let $f\colon G\to H$ be a map between connected graphs. Let $Q=(V,E,v_{0})$ be a connected pointed graph and $W$ be a finite subset of $V$ with $v_{0}\in V$. Let $\widetilde{H}$ and $\tilde{f}\colon\operatorname{La}(G)\to\operatorname{La}(\widetilde{H})$ be the map defined above. Assume that there exist $a,b\in[0,\infty]$ such that

[TABLE]

for all $x,y\in G$. Then it follows for all $u,v\in\operatorname{La}(G)$ that

[TABLE]

where $a^{\prime}=\min\{a,c\}$, $b^{\prime}=\max\{b,c\}$ and $c=\operatorname{tsp}_{Q}(v_{0},W,v_{0})+\lvert W\rvert$.

Remark*.*

This result can be very versatile. Assume that $a,b\in\mathbb{N}$ and $b-a\geqslant 2$. Assume further that $\lvert V\rvert\geqslant b$. Then $W$ can be chosen so that $a\leqslant c\leqslant b$, and hence $\mathrm{dist}(\tilde{f})\leqslant b/a$. Indeed, given a finite $W\subset V$, replacing $W$ by $W\cup\{q\}$ for some $q\in V\setminus W$ that is joined to a vertex in $W$, the value of $c$ increases by at most $3$. Hence, starting with $W=\{v_{0}\}$ and adding one vertex at a time, we eventually arrive at a set $W$ for which $a\leqslant c\leqslant b$ holds.

Proof.

Given vertices $y,z$ and a finite subset $C$ in the image of $f$, an optimal solution for computing \operatorname{tsp}_{\widetilde{H}}\big{(}y,W_{C},z) can be obtained as follows. Start with a walk $w$ in $H$ of length $\operatorname{tsp}_{H}(y,C,z)$ from $y$ to $z$ visiting all vertices of $C$, and each time $w$ visits a vertex $u\in C$ insert a walk of length $\operatorname{tsp}_{Q}(v_{0},W,v_{0})$ in the copy of $Q$ corresponding to $u$ that starts and ends at $v_{0}$ and visits all vertices in $W$. The resulting walk is easily seen to be optimal, and hence yields the formula

[TABLE]

Let us now fix vertices $(A,x)$ and $(B,y)$ in $\operatorname{La}(G)$. Observing that $W_{f(A)}\bigtriangleup W_{f(B)}=W_{f(A)\bigtriangleup f(B)}$, and combining Proposition 1 with (22), we obtain

[TABLE]

As before, we have $f(A)\bigtriangleup f(B)\subset f(A\bigtriangleup B)$ and $\lvert f(A\bigtriangleup B)\rvert\leqslant\lvert A\bigtriangleup B\rvert$, and moreover equality holds when $a>0$. Hence, by (20), which is still valid in this context, we obtain

[TABLE]

and if $a>0$, then

[TABLE]

∎

6. Binary trees and Hamming cubes in lamplighter graphs

It was observed in [lpp:96] that the lamplighter group $\mathbb{Z}_{2}\wr\mathbb{Z}$ contains a bi-Lipschitz copy of the infinite binary tree. We provide a simple proof of the finite version of this fact.

Lemma 20**.**

Let $k\in\mathbb{N}$. Then $\mathrm{B}_{k}$ bi-Lipschitzly embeds with distortion at most $2$ into $\operatorname{La}(\mathrm{P}_{k})$.

Proof.

Let $v_{0},\dots,v_{k}$ be the vertices of $\mathrm{P}_{k}$ with edges $v_{i-1}v_{i}$ for $1\leqslant i\leqslant k$. For any $\varepsilon=(\varepsilon_{1},\dots,\varepsilon_{n})\in\mathrm{B}_{k}$, let $A_{\varepsilon}=\{v_{s-1}:\,\varepsilon_{s}=1\}$, and define $f\colon\mathrm{B}_{k}\to\operatorname{La}(\mathrm{P}_{k})$ by setting $f(\varepsilon)=(A_{\varepsilon},v_{\lvert\varepsilon\rvert})$. We show that $f$ is a bi-Lipschitz embedding with distortion at most 2.

Let us fix $\delta,\varepsilon\in\mathrm{B}_{k}$ and assume without loss of generality that $\lvert\delta\rvert\leqslant\lvert\varepsilon\rvert$. Then by Proposition 1 we have

[TABLE]

Assume that $\delta$ and $\varepsilon$ are adjacent, and thus $\varepsilon=(\delta,\delta_{m+1})$, where $m=\lvert\delta\rvert$. If $\delta_{m+1}=0$, then $A_{\delta}\bigtriangleup A_{\varepsilon}=\emptyset$, otherwise $A_{\delta}\bigtriangleup A_{\varepsilon}=\{v_{m}\}$. Therefore,

[TABLE]

and thus $f$ is $2$-Lipschitz.

We now derive the lower bound. Let $\delta\wedge\varepsilon$ denote the last common ancestor of $\delta$ and $\varepsilon$. Thus, $\delta\wedge\varepsilon=(\delta_{1},\dots,\delta_{r})$, where $r=\max\{i\geqslant 0:\,\delta_{j}=\varepsilon_{j}\text{ for }1\leqslant j\leqslant i\}$. Since by definition we have $A_{\delta}\bigtriangleup A_{\varepsilon}\subset\{v_{\lvert\delta\wedge\varepsilon\rvert},\dots,v_{\lvert\varepsilon\rvert-1}\}$, an optimal solution for computing $\operatorname{tsp}_{\mathrm{P}_{k}}(v_{\lvert\delta\rvert},A_{\delta}\bigtriangleup A_{\varepsilon},v_{\lvert\varepsilon\rvert})$ starts at $v_{\lvert\delta\rvert}$, then travels to $v_{\lvert\delta\wedge\varepsilon\rvert}$, and finally travels to $v_{\lvert\varepsilon\rvert}$. Thus,

[TABLE]

It follows that

[TABLE]

∎

It is clear that a similar argument as in the proof above shows that the lamplighter graph $\operatorname{La}(\mathrm{P}_{\infty})$ over the infinite path $\mathrm{P}_{\infty}$ contains a bi-Lipschitz copy of the infinite binary tree. Since $\operatorname{La}(\mathrm{P}_{\infty})$ and $\mathbb{Z}_{2}\wr\mathbb{Z}$ are isometric (with suitable choice of generators for $\mathbb{Z}_{2}\wr\mathbb{Z}$), the observation from [lpp:96] can be recovered. Lemma 20 also provides the final result we need to complete the proof of Theorem B.

Proof of Theorem B.

The implications “(ii)$\implies$(i)” and “(iii)$\implies$(i)” follow from Propositions 15 and 17, respectively. To establish the reverse implications, fix $n\in\mathbb{N}$. Observe that for each $k\in\mathbb{N}$ the graphs $\mathrm{St}_{n,k}$ and $\mathrm{Ro}_{n,2k}$ contain isometric copies of $\mathrm{P}_{k}$, and hence by combining Lemmas 18 and 20, the binary tree $\mathrm{B}_{k}$ bi-Lipschitzly embeds with distortion at most 2 into the lamplighter graphs $\operatorname{La}(\mathrm{St}_{n,k})$ and $\operatorname{La}(\mathrm{Ro}_{n,2k})$. The implications “(i)$\implies$(ii)” and “(i)$\implies$(iii)” now follow from Bourgain’s metric characterization of superreflexivity [bourgain:86]. ∎

We now turn to the embeddability of Hamming cubes into lamplighter graphs. Here $\mathrm{K}_{n}$, for $n\in\mathbb{N}$, denotes the complete graph on $n$ vertices.

Lemma 21**.**

Let $k,m\in\mathbb{N}$. Then $\mathrm{H}_{k}$ bi-Lipschitzly embeds into $\operatorname{La}(\mathrm{K}_{km})$ with distortion at most $1+\frac{1}{2m}$.

Proof.

Recall that $\mathrm{H}_{k}$ can be thought of as the set of all subsets of $\{1,\dots,k\}$ and that under this identification the Hamming metric becomes the symmetric difference metric. Let us now partition the vertex set of $\mathrm{K}_{km}$ into $k$ sets $V_{1},\dots,V_{k}$ each of size $m$, and let us also fix a vertex $v_{0}$ of $\mathrm{K}_{km}$. Define $f\colon\mathrm{H}_{k}\to\operatorname{La}(\mathrm{K}_{km})$ by setting $f(I)=(V_{I},v_{0})$, where $V_{I}=\bigcup_{i\in I}V_{i}$.

To estimate the distortion of $f$, let us fix distinct elements $I,J\in\mathrm{H}_{k}$. Note that $V_{I}\bigtriangleup V_{J}=V_{I\bigtriangleup J}$, and hence

[TABLE]

It follows that

[TABLE]

Combining the above with Proposition 1 yields

[TABLE]

∎

Remark*.*

Lemma 21 shows in particular that for every $k\in\mathbb{N}$, there is a bi-Lipschitz embedding of $\mathrm{H}_{k}$ into $\operatorname{La}(\mathrm{K}_{k})$ of distortion at most $\frac{3}{2}$, and that for every $k\in\mathbb{N}$ and $\varepsilon>0$, there exists $n\in\mathbb{N}$ such that $\mathrm{H}_{k}$ bi-Lipschitzly embeds into $\operatorname{La}(\mathrm{K}_{n})$ with distortion at most $1+\varepsilon$, and moreover $n$ can be chosen to be $\frac{k}{2\varepsilon}$.

At this point, we need one more ingredient to prove Theorem C, which is the following well known fact.

Lemma 22**.**

Let $k\in\mathbb{N}$ and $\varepsilon>0$. Then $\mathrm{K}_{k}$ embeds with distortion at most $1+\varepsilon$ into $\mathrm{B}_{n}$ whenever $n\geqslant\log_{2}(k)\cdot\frac{1+\varepsilon}{\varepsilon}$.

Proof.

Choose $s,t\in\mathbb{N}$ such that $2^{s}\geqslant k$ and $\frac{s+t}{t+1}<1+\varepsilon$. We show that $n=s+t$ works.

By a *leaf *of the binary tree $\mathrm{B}_{s}$ of height $s$, we mean a vertex $\varepsilon$ with $\lvert\varepsilon\rvert=s$. The binary tree $\mathrm{B}_{n}$ of height $n=s+t$ can be considered as being constructed by coalescing $2^{s}$ copies of the binary tree $\mathrm{B}_{t}$ to the leaves of the binary tree $\mathrm{B}_{s}$ as follows. For each leaf $\varepsilon$ of $\mathrm{B}_{s}$, we coalesce a copy of $\mathrm{B}_{t}$ at $\emptyset$, its root, to $\mathrm{B}_{s}$ at $\varepsilon$.

Pick $k$ leaves $\ell_{1},\dots,\ell_{k}$ of $\mathrm{B}_{n}$, one from each of $k$ different copies of $\mathrm{B}_{t}$. Let $v_{1},\dots,v_{k}$ be the vertices of $\mathrm{K}_{k}$, and define $f\colon\mathrm{K}_{k}\to\mathrm{B}_{n}$ by $f(v_{i})=\ell_{i}$, $i=1,\dots,k$. We then have

[TABLE]

for all $i\neq j$. Thus, $f$ has distortion at most $\frac{s+t}{t+1}$, which in turn is at most $1+\varepsilon$ by the choice of $s$ and $t$. ∎

Proof of Theorem C.

It follows from Theorem A that $\operatorname{La}(\mathrm{B}_{k})$ embeds with distortion at most $6$ into a finite Hamming cube. In turn, by Lemma 21, the Hamming cube $\mathrm{H}_{k}$ embeds into $\operatorname{La}(\mathrm{K}_{k})$ with distortion at most $\frac{3}{2}$. It remains to show that \big{(}\operatorname{La}(\mathrm{K}_{k})\big{)}_{k\in\mathbb{N}} equi-bi-Lipschitzly embeds into \big{(}\operatorname{La}(\mathrm{B}_{k})\big{)}_{k\in\mathbb{N}}, but this is true due to Lemma 22 combined with Lemma 18. ∎

The equi-bi-Lipschitz embeddability of \big{(}\operatorname{La}(\mathrm{K}_{k})\big{)}_{k\in\mathbb{N}} into \big{(}\operatorname{La}(\mathrm{B}_{k})\big{)}_{k\in\mathbb{N}} can be made quantitatively more precise using Lemma 19.

Proposition 23**.**

Let $k\in\mathbb{N}$ and $\varepsilon>0$. Then $\operatorname{La}(\mathrm{K}_{k})$ embeds with distortion at most $1+\varepsilon$ into $\operatorname{La}(\mathrm{B}_{N})$ whenever $N>n+\log_{2}n+1$ and $n\geqslant\log_{2}(k)\cdot\frac{1+\varepsilon}{\varepsilon}$.

Proof.

For $k=1,2$, it is clear that $\mathrm{K}_{k}$ embeds isometrically into $\mathrm{B}_{k}$, and hence the same holds for the corresponding lamplighter graphs. We now assume that $k\geqslant 3$ and follow the notation from the proof of Lemma 22. We have $s\geqslant 2$, and hence $2(s+t)-(2t+2)=2s-2\geqslant 2$. Choose $r\in\mathbb{N}$ with $2^{r}>2(s+t)$, and let $Q$ be the pointed graph $(\mathrm{B}_{r},\emptyset)$. It follows from Lemma 19 and the subsequent remark that there is a subset $W$ of the vertices of $Q$ such that the map \tilde{f}\colon\operatorname{La}(\mathrm{K}_{k})\to\operatorname{La}\big{(}\widetilde{\mathrm{B}}_{n}\big{)}, induced by $f$, $Q$ and $W$, has distortion at most $\frac{s+t}{t+1}<1+\varepsilon$. Finally observe that, since the image of $f$ is contained in the set of leaves of $\mathrm{B}_{n}$, it follows that $\widetilde{\mathrm{B}}_{n}$ isometrically embeds into $\mathrm{B}_{n+r}$, which in turn implies that \operatorname{La}\big{(}\widetilde{\mathrm{B}}_{n}\big{)} isometrically embeds into $\operatorname{La}(\mathrm{B}_{n+r})$. ∎

7. Conclusions

Assume that a sequence $(G_{k})_{k\in\mathbb{N}}$ of graphs equi-bi-Lipschitzly contains $(K_{k})_{k\in\mathbb{N}}$. It follows then from Theorem C and Lemma 18, together with the remark thereafter, that the sequence $(\mathrm{H}_{k})_{k\in\mathbb{N}}$ of Hamming cubes equi-bi-Lipschitzly embeds into $(\operatorname{La}(G_{k}))_{k\in\mathbb{N}}$. We do not know if the converse holds.

Problem 24**.**

Given a sequence $(G_{k})_{k\in\mathbb{N}}$ of graphs, if the Hamming cubes $(\mathrm{H}_{k})_{k\in\mathbb{N}}$ equi-bi-Lipschitzly embed into $(\operatorname{La}(G_{k}))_{k\in\mathbb{N}}$, does it follow that $(\mathrm{K}_{k})_{k\in\mathbb{N}}$ equi-bi-Lipschitzly embeds into $(G_{k})_{k\in\mathbb{N}}$?

The following tree might be a counterexample to Problem 24.

Example*.*

We construct a tree which can be seen as a “binary tree with variable-size legs” as follows. Given $k\in\mathbb{N}$ and $\accentset{\rule{2.79996pt}{0.6pt}}{\ell}=(\ell_{1},\ell_{2},\dots,\ell_{k})\in\mathbb{N}^{k}$, replace each edge on the $j^{\text{th}}$ level of the binary tree of length $k$ by a path of length $\ell_{j}$, where by an edge on the $j^{\text{th}}$ level we mean an edge such that the distance from its farthest endpoint to the root is $j$. Denote by $\mathrm{B}_{\accentset{\rule{2.79996pt}{0.6pt}}{\ell}}$ the new tree, of length $\ell=\sum_{i=1}^{k}{\ell_{i}}$, thus obtained. The tree $\mathrm{B}_{\accentset{\rule{2.79996pt}{0.6pt}}{\ell}}$, with $\accentset{\rule{2.79996pt}{0.6pt}}{\ell}=(4,2,1)$ is the tree of length $\ell=7$ depicted in the illustration below.

If we choose for every $k\in\mathbb{N}$, the sequence \accentset{\rule{2.79996pt}{0.6pt}}{\ell}^{(k)}=\big{(}\ell_{1}^{(k)},\ell_{2}^{(k)},\dots,\ell^{(k)}_{k}\big{)} so that $\ell_{1}^{(k)}$ is chosen large enough compared to $\ell_{2}^{(k)}$, $\ell_{2}^{(k)}$ is chosen large enough compared to $\ell_{3}^{(k)}$, etc. it is not hard, but cumbersome, to prove that the sequence \big{(}\mathrm{B}_{\accentset{\rule{2.79996pt}{0.6pt}}{\ell}^{(k)}}\big{)}_{k\in\mathbb{N}} does not equi-bi-Lipschitzly contain $(\mathrm{K}_{k})_{k\in\mathbb{N}}$. So for this example to become a counterexample to Problem 24, we need a positive answer to the following question.

Problem 25**.**

Let \big{(}\mathrm{B}_{\accentset{\rule{2.79996pt}{0.6pt}}{\ell}^{(k)}}\big{)}_{k\in\mathbb{N}} constructed as in the description above. Does $(\mathrm{H}_{k})_{k\in\mathbb{N}}$ equi-bi-Lipschitzly embed into \big{(}\operatorname{La}(\mathrm{B}_{\accentset{\rule{2.79996pt}{0.6pt}}{\ell}^{(k)}})\big{)}_{k\in\mathbb{N}}?

Any counterexample to Problem 24 would be a counterexample to the following problem.

Problem 26**.**

If $(G_{k})_{k\in\mathbb{N}}$ is a sequence of graphs which does not equi-bi-Lipschitzly contain $(\mathrm{K}_{k})_{k\in\mathbb{N}}$, and if $X$ is a non-reflexive Banach space, does it follow that the sequence \big{(}\operatorname{La}(G_{k})\big{)}_{k\in\mathbb{N}} equi-bi-Lipschitzly embed into $X$?

Indeed, there are non-reflexive Banach spaces $X$ with non-trivial type (cf.  [james:74], [james:78] or [pisier-xu:87]). By the observation at the beginning of this section, these spaces cannot equi-bi-Lipschitzly contain sequences of graphs which equi-bi-Lipschitzly contain $(\mathrm{H}_{k})_{k\in\mathbb{N}}$.