Constraining the clustering transition for colorings of sparse random graphs

TL;DR

This paper investigates the structure of proper q-colorings in sparse random graphs, showing that above certain density thresholds, almost all colorings form a single giant component in the associated graph.

Contribution

It establishes a new threshold for the clustering transition in the space of graph colorings, linking graph density and number of colors to the emergence of a giant component.

Findings

A giant component exists in the coloring graph for large enough average degree d.

The threshold for the clustering transition is characterized by q ≥ (c*d)/log d with c > 3/2.

Almost all colorings are contained within a single giant component above this threshold.

Abstract

Let $\Omega_q$ denote the set of proper $q$-colorings of the random graph $G_{n,m}, m=dn/2$ and let $H_q$ be the graph with vertex set $\Omega_q$ and an edge $\{\sigma,\tau\}$ where $\sigma,\tau$ are mappings $[n]\to[q]$ iff $h(\sigma,\tau)=1$. Here $h(\sigma,\tau)$ is the Hamming distance $|\{v\in [n]:\sigma(v)\neq\tau(v)\}|$. We show that w.h.p. $H_q$ contains a single giant component containing almost all colorings in $\Omega_q$ if $d$ is sufficiently large and $q\geq \frac{cd}{\log d}$ for a constant $c>3/2$.

Equations

$$\sum _{i=1}^q\left(\genfrac{}{}{0ex}{}{|V_i|}{2}\right)\le \left(\genfrac{}{}{0ex}{}{n}{2}\right)-m.$$

$$q_0=\frac{q}{q-1}\cdot \frac{d}{\log d-7\log \log d}\approx \frac{d}{\log d}.$$

$$N_q=\left(\genfrac{}{}{0ex}{}{n}{2}\right)-(1+o(1))q\left(\genfrac{}{}{0ex}{}{n/q}{2}\right)=(1-o(1))\frac{n^2}{2}\left(1-\frac{1}{q}\right)$$

$$u_{t+1}\text{ stochastically dominates }{u}_t-Bin(u_t,\hat{p})-1.$$

$${\tilde{u}}_{t+1}={\tilde{u}}_t-Bin({\tilde{u}}_t,\hat{p})-1.$$

$$\frac{\log (\hat{d}/{\log }^2\hat{d})-3\log \log (\hat{d}/{\log }^2\hat{d})}{\hat{d}}n\ge \frac{q-1}{q}\cdot \frac{\log d-6\log \log d}{d}n$$

$$\frac{q}{q-1}\cdot \frac{d}{\log d-6\log \log d}+\frac{\hat{d}}{{\log }^2\hat{d}}+2\le \frac{q}{q-1}\cdot \frac{d}{\log d-7\log \log d}=q_0$$

Full text

1 Introduction

In this short note, we will discuss a structural property of the set $\Omega_{q}$ of proper $q$-colorings of the random graph $G_{n,m}$, where $m=dn/2$ for some large constant $d$. For the sake of precision, let us define $H_{q}$ to be the graph with vertex set $\Omega_{q}$ and an edge $\left\{\sigma,\tau\right\}$ iff $h(\sigma,\tau)=1$ where $h(\sigma,\tau)$ is the Hamming distance $|\left\{v\in[n]:\sigma(v)\neq\tau(v)\right\}|$. In the Statistical Physics literature the definition of $H_{q}$ may be that colorings $\sigma,\tau$ are connected by an edge in $H_{q}$ whenever $h(\sigma,\tau)=o(n)$. Our theorem holds a fortiori if this is the case.

Heuristic evidence in the statistical physics literature (see for example [15]) suggests there is a clustering transition $c_{d}$ such that for $q>c_{d}$, the graph $H_{q}$ is dominated by a single connected component, while for $q<c_{d}$, an exponential number of components are required to cover any constant fraction of it; it may be that $c_{d}\approx\frac{d}{\log d}$. (Here $A(d)\approx B(d)$ is taken to mean that $A(d)/B(d)\to 1$ as $d\to\infty$. We do not assume $d\to\infty$, only that $d$ is a sufficiently large constant, independent of $n$.) Recall that $G_{n,m}$ for $m=dn/2$ becomes $q$-colorable around $q\approx\frac{d}{2\log d}$ or equivalently when $d\approx 2q\log q$, [3, 7]. In this note, we prove the following:

Theorem 1.1**.**

If $q\geq\frac{cd}{\log d}$ for constant $c>3/2$, and $d$ is sufficiently large, then w.h.p. $H_{q}$ contains a giant component that contains almost all of $\Omega_{q}$.

In particular, this implies that the clustering transition $c_{d}$, if it exists, must satisfy $c_{d}\leq\frac{3}{2}\frac{d}{\log d}$.

Theorem 1.1 falls into the area of “Structural Properties of Solutions to Random Constraint Satisfaction Problems”. This is a growing area with connections to Computer Science and Theoretical Physics. In particular, much of the research on the graph $H_{q}$ has been focussed on the structure near the colorability threshold, e.g. Bapst, Coja-Oghlan, Hetterich, Rassman and Vilenchik [4], or the clustering threshold, e.g. Achlioptas, Coja-Oghlan and Ricci-Tersenghi [2], Molloy [13]. Other papers heuristically identify a sequence of phase transitions in the structure of $H_{q}$, e.g., Krza̧kala, Montanari, Ricci-Tersenghi, Semerijan and Zdeborová [12], Zdeborová and Krza̧kala [15]. The existence of these transitions has been shown rigorously for some other CSPs. One of the most spectacular examples is due to Ding, Sly and Sun [8] who rigorously showed the existence of a sharp satisfiability threshold for random $k$-SAT.

An obvious target for future work is improving the constant in Theorem 1.1 to 1. We should note that Molloy [13] has shown that w.h.p. there is no giant component if $q\leq\frac{(1-\varepsilon_{d})d}{\log d}$, for some $\varepsilon_{d}>0$. Looking in another direction, it is shown in [9] that w.h.p. $H_{q},q\geq d+2$ is connected. This implies that Glauber Dynamics on $\Omega_{q}$ is ergodic. It would be of interest to know if this is true for some $q\ll d$.

Before we begin our analysis, we briefly explain the constant 3/2. We start with an arbitrary $q$-cloring and then re-color it using only approximately $\approx d/\log d$ of the given colors. We then use a disjoint set of approximately $d/2\log d$ colors to re-color it with a target $\chi\approx\frac{d}{2\log d}$ coloring $\tau$.

2 Greedily Re-coloring

Our main tool is a theorem from Bapst, Coja-Oghlan and Efthymiou [5] on planted colorings. We consider two ways of generating a random coloring of a random graph. We will let $Z_{q}=|\Omega_{q}|$. The first method is to generate a random graph and then a random coloring. In the second method, we generate a random (planted) coloring and then generate a random graph compatible with this coloring.

Random coloring of the random graph $G_{n,m}$: Here we will assume that $m$ is such that w.h.p. $Z_{q}>0$.

1. (a)

Generate $G_{n,m}$ subject to $Z_{q}>0$. 2. (b)

Choose a $q$-coloring $\sigma$ uniformly at random from $\Omega_{q}$. 3. (c)

Output $\Pi_{1}=(G_{n,m},\sigma)$.

Planted model:

Choose a random partition of $[n]$ into $q$ color classes $V_{1},V_{2},\ldots,V_{q}$ subject to

[TABLE] 2. 2.

Let $\Gamma_{\sigma,m}$ be obtained by adding $m$ random edges, each with endpoints in different color classes. 3. 3.

Output $\Pi_{2}=(\Gamma_{\sigma,m},\sigma)$.

We will use the following result from [5]:

Theorem 2.1**.**

Let $d=2m/n$ and suppose that $d\leq 2(q-1)\log(q-1)$. Then $\operatorname{\bf Pr}(\Pi_{2}\in\mathcal{P})=o(1)$ implies that $\operatorname{\bf Pr}(\Pi_{1}\in\mathcal{P})=o(1)$ for any graph+coloring property $\mathcal{P}$.

Consequently, we will use the planted model in our subsequent analysis. Let

[TABLE]

The property $\mathcal{P}$ in question will be: “the given $q$-coloring can be reduced via single vertex color changes to a $q_{0}$ coloring” where $\alpha>1$ is constant.

In a random partition of $[n]$ into $q$ parts, the size of each part is distributed as $Bin(n,q^{-1})$ and so the Chernoff bounds imply that w.h.p. in a random partition each part has size $\frac{n}{q}\left(1\pm\frac{\log n}{n^{1/2}}\right)$.

We let $\Gamma$ be obtained by taking a random partition $V_{1},V_{2},\ldots,V_{q}$ and then adding $m=\frac{1}{2}dn$ random edges so that each part is an independent set. These edges will be chosen from

[TABLE]

possibilities. So, let $\widehat{d}=\frac{mn}{N_{q}}\approx\frac{dq}{q-1}$ and replace $\Gamma$ by $\widehat{\Gamma}$ where each edge not contained in a $V_{i}$ is included independently with probability $\widehat{p}=\frac{\widehat{d}}{n}$. $V_{1},V_{2},\ldots,V_{q}$ constitutes a coloring which we will denote by $\sigma$. Now $\widehat{\Gamma}$ has $m$ edges with probability $\Omega(n^{-1/2})$ and one can check that the properties required in Lemmas 2.2 and 2.3 below all occur with probability $1-o(n^{-1/2})$ and so we can equally well work with $\widehat{\Gamma}$.

Now consider the following algorithm for going from $\sigma$ via a path in $\Omega_{q}$ to a coloring with significantly fewer colors. It is basically the standard greedy coloring algorithm, as seen in Bollobás and Erdős [6], Grimmett and McDiarmid [10] and in particular Shamir and Upfal [14] for sparse graphs.

In words, it goes as follows. At each stage of the algorithm, $U$ denotes the set of vertices that have not been re-colored. Having used $r-1$ colors to color some subset of vertices we start using color $r$. We let $W_{j}=V_{j}\cap U$ denote the uncolored vertices of $V_{j}$ for $j\geq 1$. We then let $k$ be the smallest index $j$ for which $W_{j}\neq\emptyset$. This is an independent set and so we can re-color the vertices of $W_{k}$, one by one, with the color $r$. We let $U_{r}\subseteq U$ denote the set of vertices that may possibly be re-colored $r$ by the algorithm i.e. those vertices with no neighbors in $C_{r}$, the current set of vertices colored $r$. Each time we re-color a vertex with color $r$, we remove its neighbors from $U_{r}$. We continue with color $r$, until $U_{r}=\emptyset$. After which, $C_{r}$ will be the set of vertices that are finally colored with color $r$.

At any stage of the algorithm, $U$ is the set of vertices whose colors have not been altered. The value of $L$ in line D is $n/\log^{2}\widehat{d}$.

[TABLE]

We first observe that each re-coloring of a singe vertex $v$ vertex in line C can be interpreted as moving from a coloring of $\Omega_{q}$ to a neighboring coloring in $H_{q}$. This requires us to argue that the re-coloring by greedy re-color is such that the coloring of $\widehat{\Gamma}$ is proper at all times. We argue by induction on $r$ that the coloring at line A is proper. When $r=1$ there have been no re-colorings. Also, during the loop beginning at line B we only re-color vertices with color $r$ if they are not neighbors of the set $U_{r}$ of vertices colored $r$. This guarantees that the coloring remains proper until we reach line D. The following lemma shows that we can then reason as in Lemma 2 of Dyer, Flaxman, Frieze and Vigoda [9], as will be explained subsequently.

Lemma 2.2**.**

Let $p=m/\binom{n}{2}=\Delta/n$ where $\Delta$ is some sufficiently large constant. With probability $1-o(n^{-1/2})$, every $S\subseteq[n]$ with $s=|S|\leq n/\log^{2}\Delta$ contains at most $s\Delta/\log^{2}\Delta$ edges.

The above lemma, is Lemma 7.7(i) of Janson, Łuczak and Ruciński [11] and it implies that if $\Delta=\widehat{d}$ then w.h.p. $\widehat{\Gamma}_{U}$ at line D contains no $K$-core, $K=\frac{2\widehat{d}}{\log^{2}\widehat{d}}+1$. Here $\widehat{\Gamma}_{U}$ denotes the sub-graph of $\widehat{\Gamma}$ induced by the vertices $U$. For a graph $G=(V,E)$ and $K\geq 0$, the $K$-core is the unique maximal set $S\subseteq V$ such that the induced subgraph on $S$ has minimum degree at least $K$. A graph without a $K$-core is $K$-degenerate i.e. its vertices can be ordered as $v_{1},v_{2},\ldots,v_{n}$ so that $v_{i}$ has at most $K-1$ neighbors in $\{v_{1},v_{2},\ldots,v_{i-1}\}$. To see this, let $v_{n}$ be a vertex of minimum degree and then apply induction.

We argue now that we can re-color the vertices in $U$ with $K+1$ new colors, all the time following some path in $H_{q}$. Let $v_{1},\dots,v_{n}$ denote an ordering of $U$ such that the degree of $v_{i}$ is less than $K$ in the subgraph $\widehat{\Gamma}_{i}$ of $\widehat{\Gamma}$ induced by $\{v_{1},v_{2},\dots,v_{i}\}$. We will prove the claim by induction. The claim is trivial for $i=1$. By induction there is a path $\sigma_{0},\sigma_{1},\ldots,\sigma_{r}$ from the coloring $\sigma_{0}$ of $U$ at line B, restricted to $\widehat{\Gamma}_{i-1}$ using only $K+1$ colors to do the re-coloring.

Let $(w_{j},c_{j})$ denote the $(vertex,color)$ change defining the edge $\left\{\sigma_{j-1},\sigma_{j}\right\}$. We construct a path (of length $\leq 2r$) that re-colors $\widehat{\Gamma}_{i}$. For $j=1,2,\ldots,r$, we will re-color $w_{j}$ to color $c_{j}$, if no neighbor of $w_{j}$ has color $c_{j}$. Failing this, $v_{i}$ must be the only neighbor of $w_{j}$ that is colored $c_{j}$. This is because $\sigma_{r}$ is a proper coloring of $\widehat{\Gamma}_{i-1}$. Since $v_{i}$ has degree less than $K$ in $\widehat{\Gamma}_{i}$, there exists a new color for $v_{i}$ which does not appear in its neighborhood. Thus, we first re-color $v_{i}$ to any new (valid) color, and then we re-color $w_{j}$ to $c_{j}$, completing the inductive step. Note that because the colors used in Step D have not been used in Steps A,B,C, this re-coloring does not conflict with any of the coloring done in Steps A,B,C.

We need to show next that each Loop B re-colors a large number of vertices. Let $\alpha_{1}(G)$ denote the minimim size of a maximal independent set of a graph $G$ i.e. an independent set that is not contained in any larger independent set. The round will re-color at least $\alpha_{1}(\Gamma_{U})$ vertices, where $U$ is as at the start of Loop B. The following result is from Lemma 7.8(i) of [11].

Lemma 2.3**.**

Let $p=m/\binom{n}{2}=\Delta/n$ where $\Delta$ is some sufficiently large constant. $\alpha_{1}(G_{n,m})\geq\frac{\log\Delta-3\log\log\Delta}{p}$ with probability $1-o(n^{-1/2})$. (see Lemma 7.8(i)).

Suppose now that we take $u_{0}$ to be the size of $U$ at the beginning of Step A and that $u_{t}$ is the size of $U$ after $t$ vertices have been finally colored $r$. Thus we assume that $u_{|W_{k}|}$ is the size of $U$ at the start of Step B. We observe that,

[TABLE]

This is because the edges inside $U$ are unconditioned by the algorithm and because $v\in V_{j}$ has no neighbors in $V_{j}$ for $j\geq 1$. On the other hand, if we apply Algorithm greedy re-color to $G_{n,\widehat{p}}$ then (2) is replaced by the recurrence

[TABLE]

(Putting $V_{j}=\left\{j\right\}$ means that greedy re-color is running on $G_{n,\widehat{p}}$.)

Comparing (2) and (3) we see that we can couple the two applications of greedy re-color so that $u_{t}\geq\tilde{u}_{t}$ for $t\geq 0$. Now the application of Loop B re-colors a maximal independent set of the graph $\widehat{\Gamma}_{U}$ induced by $U$ as it stands at the beginning of the loop. The size of this set dominates the size of a maximal independent set in the random graph $G_{|U|,p}$. So if we generate $G_{|U|,p}$ and then delete some edges, we see that every independent set of $G_{|U|,p}$ will be contained in an independent set of $\Gamma_{U}$. And so using Lemma 2.3 we see that w.h.p. each execution of Loop B re-colors at least

[TABLE]

vertices, for $d$ sufficiently large. We have replaced $\Delta$ of Lemma 2.3 by $\widehat{d}/\log^{2}\widehat{d}$ to allow for the fact that we hae replaced $n$ by $|U|\geq L$. Consequently, at the end of Algorithm greedy re-color we will have used at most

[TABLE]

colors. The term $\frac{\widehat{d}}{\log^{2}\widehat{d}}+2$ arises from the re-coloring of $U$ at line D.

Finishing the proof: Now suppose that $q\geq\frac{cd}{\log d}$ where $d$ is large and $c>3/2$. Fix a particular $\chi$-coloring $\tau$. We prove that almost every $q$-coloring $\sigma$ can be transformed into $\tau$ changing one color at a time. It follows that for almost every pair of $q$-colorings $\sigma$, $\sigma^{\prime}$ we can transform $\sigma$ into $\sigma^{\prime}$ by first transforming $\sigma$ to $\tau$ and then reversing the path from $\sigma^{\prime}$ to $\tau$.

We proceed as follows. The algorithm greedy re-color takes as input: (i) the coloring $\sigma$ and (ii) a specific subset of $q_{0}$ colors from $\left\{1,...,q\right\}$ that are not used in $\tau$. W.h.p. it transforms the input coloring into a coloring using only those $q_{0}$ colors. Then we process the color classes of $\tau$, re-coloring vertices to their $\tau$-color. When we process a color class $C$ of $\tau$, we switch the color of vertices in $C$ to their $\tau$-color $i_{C}$ one vertex at a time. We can do this because when we re-color a vertex $v$, a neighbor $w$ will currently either have one of the $q_{0}$ colors used by greedy re-color and these are distinct from $i_{C}$. Or $w$ will have already been been re-colored with its $\tau$-color which will not be color $i_{C}$. This proves Theorem 1.1. $\Box$