Unfriendly colorings of graphs with finite average degree

TL;DR

This paper proves that certain classes of graphs with finite average degree or subexponential growth admit Borel unfriendly colorings, advancing understanding of graph colorings in measure-theoretic and descriptive set theory contexts.

Contribution

It establishes the existence of Borel unfriendly colorings for graphs with finite average degree and subexponential growth, extending previous results to broader graph classes.

Findings

Graphs with finite average degree admit Borel unfriendly colorings.

Graphs of subexponential growth admit Borel unfriendly colorings.

Results apply to probability measure preserving Borel graphs.

Abstract

In an unfriendly coloring of a graph the color of every node mismatches that of the majority of its neighbors. We show that every probability measure preserving Borel graph with finite average degree admits a Borel unfriendly coloring almost everywhere. We also show that every bounded degree Borel graph of subexponential growth admits a Borel unfriendly coloring.

Equations

$$E=\{(x,y):y=sx\text{ for some }s\in S\}$$

$$M(H)={\int }_X|H_x|d\mu$$

$$M(G_n)-M(G_{n+1})$$

$$\ge {\int }_{B_n}(2-\epsilon )|(G_n{)}_x|-(2+\epsilon )|(G_{n+1}{)}_x|d\mu$$

$$=\sum _{r,s}{\int }_{A_{r,s}}(2-\epsilon )r-(2+\epsilon )sd\mu$$

$$=\sum _{r,s}{\int }_{A_{r,s}}2(r-s)-\epsilon (r+s)d\mu$$

$$\ge \sum _{r,s}{\int }_{A_{r,s}}2-d\epsilon d\mu$$

$$=\mu (B_n)$$

Full text

Clinton T. Conley was supported by NSF grant DMS-1500906. Omer Tamuz was supported by a grant from the Simons Foundation (#419427).

1. Introduction

Suppose that $G$ is a locally finite graph on the vertex set $X$. We say that $c\colon X\to 2$ is an unfriendly coloring of $G$ if for all $x\in X$ at least half of $x$’s neighbors receive a different color than $x$ does. More formally, letting $G_{x}$ denote the set of $G$-neighbors of $x$, such a function $c$ is an unfriendly coloring if $|\{y\in G_{x}:c(x)\neq c(y)\}|\geq|\{y\in G_{x}:c(x)=c(y)\}|$. By a compactness argument unfriendly colorings exist for all locally finite graphs (see, e.g., [aharoni1990unfriendly]). There exist graphs with uncountable vertex sets that have no unfriendly colorings [shelah1990graphs]; it is not known if this is possible for graphs with countably many vertices.

A large and growing literature considers measure-theoretical analogues of classical combinatorial questions (see, e.g., a survey by Kechris and Marks [kechris2015descriptive]). Following [conley2014measure], we consider a measure-theoretical analogue of the question of unfriendly colorings. Suppose that $G$ is a locally finite Borel graph on the standard Borel space $X$, and that $\mu$ is a Borel probability measure on $X$. We say that $G$ is $\mu$-preserving if there are countably many $\mu$-preserving Borel involutions whose graphs cover the edges of $G$. Equivalently, $G$ is $\mu$-preserving if its connectedness relation $E_{G}$ is a $\mu$-preserving equivalence relation.

An important example of such graphs comes from probability measure preserving actions of finitely generated groups. Indeed let a group, generated by the finite symmetric set $S$, act by measure preserving transformations on a standard Borel probability space $(X,\mu)$. Then the associated graph $G=(X,E)$ whose edges are

[TABLE]

is a $\mu$-preserving graph.

In [conley2014measure] it is shown that every free probability measure preserving action of a finitely generated group is weakly equivalent to another such action whose associated graph admits an unfriendly coloring. Note that such graphs are regular: (almost) every node has degree $|G_{x}|=|S|$. Recall that the ($\mu$-)cost of a $\mu$-preserving locally finite Borel graph $G$ is simply half its average degree: $\operatorname{cost}(G)=\frac{1}{2}\int_{X}|G_{x}|\ d\mu$. Equivalently, using the Lusin-Novikov uniformization theorem (see, e.g., [kechris2012classical, Lemma 18.12]) one may circumvent this factor of $\frac{1}{2}$ by instead computing $\int_{X}|\vec{G}_{x}|d\mu$, where $\vec{G}$ is an arbitrary measurable orientation of $G$.

Our first result shows that every measure preserving graph with finite cost admits an (almost everywhere) unfriendly coloring.

Theorem 1**.**

Suppose that $(X,\mu)$ is a standard probability space and that $G$ is a $\mu$-preserving locally finite Borel graph on $X$ with finite cost. Then there is a $\mu$-conull $G$-invariant Borel set $A$ such that $G\restriction A$ admits a Borel unfriendly coloring.

We next explore how the invariance assumption can be weakened. Recall that a Borel probability measure is $G$-quasi-invariant if the $G$-saturation of every $\mu$-null set remains $\mu$-null. Such measures admit a Radon-Nikodym cocycle $\rho\colon G\to\mathbb{R}^{+}$ so that whenever $A\subseteq X$ is Borel and $f\colon A\to X$ a Borel partial injection whose graph is contained in $G$, then $\mu(f[A])=\int_{A}\rho(x,f(x))\ d\mu$.

Theorem 2**.**

Suppose that $(X,\mu)$ is a standard probability space, that $G$ is a Borel graph on $X$ with bounded degree $d$, and that $\mu$ is $G$-quasi-invariant, with corresponding Radon-Nikodym cocycle $\rho$. Suppose also that for all $(x,y)\in G$, $1-\frac{1}{d}\leq\rho(x,y)\leq 1+\frac{1}{d}$. Then there is a $\mu$-conull $G$-invariant Borel set $A$ such that $G\restriction A$ admits a Borel unfriendly coloring.

The proofs of Theorems 1 and 2 build on a potential function technique used in [tamuz2015majority] (see also [benjamini2016convergence]) to study majority dynamics on infinite graphs; in the context of finite graphs, these techniques go back to Goles and Olivos [goles1980periodic]. Indeed, we show that in our settings (anti)-majority dynamics converge to an unfriendly coloring. The combinatorial nature of this technique allows us to extend our results to the Borel setting.

Theorem 3**.**

Suppose that $G$ is a bounded-degree Borel graph of subexponential growth. Then $G$ admits a Borel unfriendly coloring.

A natural question remains open: is there a locally finite Borel graph that does not admit a Borel unfriendly coloring? To the best of our knowledge this is not known, even with regards to the restricted class of bounded degree graphs. In contrast, Theorem 1 shows that for this class unfriendly colorings exist in the measure preserving case. Still, we do not know if the finite cost assumption in Theorem 1 is necessary, or whether every locally finite measure preserving graph admits an almost everywhere unfriendly coloring.

2. Proofs

Proof of Theorem 1.

By Kechris-Solecki-Todorcevic [kechris1999borel, Proposition 4.5], there exists a repetitive sequence of independent sets: a sequence $(X_{n})_{n\in\mathbb{N}}$ of $G$-independent Borel sets so that each $x\in X$ is in infinitely many $X_{n}$. We will recursively build for each $n\in\mathbb{N}$ a Borel function $c_{n}\colon X\to 2$ which converge $\mu$-almost everywhere to an unfriendly coloring of $G$.

The choice of $c_{0}$ is arbitrary, but we may as well declare it to be the constant [math] function.

Suppose now that $c_{n}$ has been defined. We build $c_{n+1}$ by “flipping” the color of vertices in $X_{n}$ with too many neighbors of the same color, and leaving everything else unchanged. More precisely, $c_{n+1}(x)=1-c_{n}(x)$ if $x\in X_{n}$ and $|\{y\in G_{x}:c_{n}(x)\neq c_{n}(y)\}|<|\{y\in G_{x}:c_{n}(x)=c_{n}(y)\}|$; otherwise, $c_{n+1}(x)=c_{n}(x)$.

To show that this sequence $c_{n}$ converges $\mu$-a.e. to an unfriendly coloring, we introduce some auxiliary graphs. Let $G_{n}$ be the subgraph of $G$ containing exactly those edges between vertices of the same $c_{n}$-color, so $x\mathrel{G_{n}}y$ iff $x\mathrel{G}y$ and $c_{n}(x)=c_{n}(y)$. Certainly for all $n\in\mathbb{N}$, $\operatorname{cost}(G_{n})\leq\operatorname{cost}(G)$.

For $n\in\mathbb{N}$, let $B_{n}=\{x\in X:c_{n}(x)\neq c_{n+1}(x)\}$.

Claim**.**

$\operatorname{cost}(G_{n})-\operatorname{cost}(G_{n+1})\geq\mu(B_{n})$.

Proof of the claim.

Recall that, by the definition of $c_{n+1}$, $x\in B_{n}$ iff $x\in X_{n}$ and $|\{y\in G_{x}:c_{n}(x)\neq c_{n}(y)\}|<|\{y\in G_{x}:c_{n}(x)=c_{n}(y)\}|$. In particular, $B_{n}\subseteq X_{n}$ and hence is $G$-independent. Thus $G_{n+1}=G_{n}\mathrel{\triangle}\{(x,y):x\mathrel{G}y$ and $\{x,y\}\cap B_{n}\neq\emptyset\}$. But for each $x\in B_{n}$, the above characterization of membership in $B_{n}$ ensures that its $G_{n+1}$-degree is strictly smaller than its $G_{n}$-degree. The claim follows. ∎

In particular, since the sum telescopes we see $\sum_{n\in\mathbb{N}}\mu(B_{n})\leq\operatorname{cost}(G)<\infty$. Hence the set $C=\{x\in X:x\in B_{n}\mbox{ for infinitely many n}\}$ is $\mu$-null by the Borel-Cantelli lemma. Let $A=X\setminus[C]_{G}$, so $A$ is $\mu$-conull.

Claim**.**

$c$* is an unfriendly coloring of $G\restriction A$.*

Proof of the claim.

Fix $x\in A$ and fix $k\in\mathbb{N}$ sufficiently large so that $c_{n}$ has stabilized for $x$ and all its (finitely many) neighbors beyond $k$. Fix $n>k$ so that $x\in X_{n}$. Since $c_{n}(x)=c_{n+1}(x)$, the definition of $c_{n+1}$ implies that $|\{y\in G_{x}:c_{n}(x)\neq c_{n}(y)\}|\geq|\{y\in G_{x}:c_{n}(x)=c_{n}(y)\}|$. But $c_{n}=c$ on $G_{x}\cup\{x\}$, and hence $c$ is unfriendly as desired. ∎

This completes the proof of the theorem. ∎

We next analyze the extent to which the measure-theoretic hypotheses may be weakened in this argument. Note that the sequence $c_{n}$ of colorings is defined without using the measure at all (in fact it is determined by the graph $G$ and the sequence $(X_{n})$ of independent sets); the measure only appears in the argument that sequence converges to a limit coloring. And even in this convergence argument, invariance only shows up in the critical estimate $\operatorname{cost}(G_{n})-\operatorname{cost}(G_{n+1})\geq\mu(B_{n})$.

Definition 4**.**

Suppose that $G$ is a locally finite Borel graph on standard Borel $X$, that $(X_{n})_{n\in\mathbb{N}}$ is a sequence of $G$-independent Borel sets so that each $x\in X$ is in infinitely many $X_{n}$. We define the flip sequence $(c_{n})_{n\in\mathbb{N}}$ of Borel functions from $X$ to $2$ as follows:

• •

$c_{0}$ is the constant [math] function,

• •

$c_{n+1}(x)=1-c_{n}(x)$ if $x\in X_{n}$ and $|\{y\in G_{x}:c_{n}(x)\neq c_{n}(y)\}|<|\{y\in G_{x}:c_{n}(x)=c_{n}(y)\}|$; otherwise, $c_{n+1}(x)=c_{n}(x)$.

Definition 5**.**

Given a locally finite Borel graph $G$ on $X$ and a sequence $(X_{n})_{n\in\mathbb{N}}$ of repetitive independent sets as above, we say that a Borel measure $\mu$ on $X$ is compatible with $G$ and $(X_{n})$ if the corresponding flip sequence $c_{n}$ converges on a $\mu$-conull set.

The proof of Theorem 1 shows that whenever $\mu$ is a $G$-invariant Borel probability measure with respect to which the average degree of $G$ is finite, then $\mu$ is compatible with every sequence of independent sets. We seek to weaken the invariance assumption when $G$ has bounded degree.

Proposition 6**.**

Suppose that $G$ is a Borel graph on $X$ with bounded degree $d$, and that $\mu$ is a $G$-quasi-invariant Borel probability measure with corrsponding Radon-Nikodym cocycle $\rho$. Suppose further that for all $(x,y)\in G$, $1-\frac{1}{d}\leq\rho(x,y)\leq 1+\frac{1}{d}$. Then $\mu$ is compatible with every repetitive sequence of independent sets.

Theorem 2 is an immediate consequence of this proposition.

Proof of Proposition 6.

Put $\varepsilon=\frac{1}{d}$. Define a measure $M$ on $G$ by putting for all Borel $H\subseteq G$,

[TABLE]

This new measure $M$ will replace the occurrences of cost in the proof of Theorem 1.

Consider the flip sequence $c_{n}$, and define corresponding graphs $G_{n}\subseteq G$ by $x\mathrel{G_{n}}y$ iff $x\mathrel{G}y$ and $c_{n}(x)=c_{n}(y)$. As before, let $B_{n}$ denote those $x\in X_{n}$ for which $c_{n+1}(x)\neq c_{n}(x)$. Note that the “double counting” that occurred in the proof of Theorem 1 may no longer be true double counting, but the bound on $\rho$ ensures that each edge is counted at most $(2+\varepsilon)$ times and at least $(2-\varepsilon)$ times.

Claim**.**

$M(G_{n})-M(G_{n+1})\geq\mu(B_{n})$**

Proof of the claim.

Partition $B_{n}$ into finitely many Borel sets $A_{r,s}$ where $x\in A_{r,s}$ iff $x$ has $r$-many $G_{n}$ neighbors and $s$-many $G_{n+1}$ neighbors (so $r>s$ and $r+s\leq d$). We compute

[TABLE]

as required. ∎

The remainder of the argument is as in the proof of Theorem 1. ∎

Given Proposition 6, the proof of Theorem 3 is straightforward.

Proof of Theorem 3.

Fix a degree bound $d$ for $G$ and put $\varepsilon=\frac{1}{d}$. It suffices to construct for each $x\in X$ a $G$-quasi-invariant Borel probability measure $\mu_{x}$ whose Radon-Nikodym cocycle is $\varepsilon$-bounded on $G$ such that $\mu_{x}(\{x\})>0$. If we do so, Proposition 6 ensures that the flip sequence $c_{n}$ converges $\mu_{x}$-everywhere for each $x$, and thus it converges everywhere. The limit is then an unfriendly coloring by the same argument as in the final claim in the proof of Theorem 1.

To construct $\mu_{x}$, first define a purely atomic measure $\nu_{x}$ supported on the $G$-component of $x$ by declaring $\nu_{x}(\{y\})=(1+\varepsilon)^{-\delta(x,y)}$, where $\delta$ denotes the graph metric. Subexponential growth of $G$ ensures that $K=\sum_{y\in[x]_{G}}\nu_{x}(\{y\})<\infty$. Finally, put $\mu_{x}=\frac{1}{K}\nu_{x}$. ∎

Acknowledgments

We thank the anonymous referee for insightful comments. We also thank Alekos Kechris for organizing the seminar that inspired this paper.