Constraining the clustering transition for colorings of sparse random graphs

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 $c>3/2$.


Introduction
In this short note, we will discuss a structural property of the set Ω 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 Ω q and an edge {σ, τ } iff h(σ, τ ) = 1 where h(σ, τ ) is the Hamming distance | {v ∈ [n] : σ(v) = τ (v)} |. In the Statistical Physics literature the definition of H q may be that colorings σ, τ are connected by an edge in H q whenever h(σ, τ ) = 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 ≈ d log d . (Here A(d) ≈ B(d) is taken to mean that A(d)/B(d) → 1 as d → ∞. We do not assume d → ∞, only that d is a sufficiently large constant, independent of n.) Recall that G n,m for m = dn/2 becomes qcolorable around q ≈ d 2 log d or equivalently when d ≈ 2q log q, [3,7]. In this note, we prove the following: In particular, this implies that the clustering transition c d , if it exists, must satisfy c d ≤ 3 2 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., Krzakala, Montanari, Ricci-Tersenghi, Semerijan and Zdeborová [12], Zdeborová and Krzakala [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 ≤ (1−ε d )d log d , for some ε d > 0. Looking in another direction, it is shown in [9] that w.h.p. H q , q ≥ d + 2 is connected. This implies that Glauber Dynamics on Ω q is ergodic. It would be of interest to know if this is true for some q ≪ 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 ≈ 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 χ ≈ d 2 log d coloring τ .

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 = |Ω 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.
(a) Generate G n,m subject to Z q > 0.
(b) Choose a q-coloring σ uniformly at random from Ω q .
Planted model: 2. Let Γ σ,m be obtained by adding m random edges, each with endpoints in different color classes.
We will use the following result from [5]: for any graph+coloring property P.
Consequently, we will use the planted model in our subsequent analysis. Let The property P in question will be: "the given q-coloring can be reduced via single vertex color changes to a q 0 coloring" where α > 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 n q 1 ± log n n 1/2 .
We let Γ be obtained by taking a random partition V 1 , V 2 , . . . , V q and then adding m = 1 2 dn random edges so that each part is an independent set. These edges will be chosen from possibilities. So, let d = mn Nq ≈ dq q−1 and replace Γ by Γ where each edge not contained in a V i is included independently with probability p = d n . V 1 , V 2 , . . . , V q constitutes a coloring which we will denote by σ. Now Γ has m edges with probability Ω(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 Γ. Now consider the following algorithm for going from σ via a path in Ω 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 ∩ U denote the uncolored vertices of V j for j ≥ 1. We then let k be the smallest index j for which W j = ∅. 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 ⊆ 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 = ∅. 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 d.
algorithm greedy re-color begin Initialise: r = 0, U = [n], C 0 ← ∅; repeat; r ←r + 1, C r ← ∅; Let W j = V j ∩ U for j ≥ 1 and let k = min {j : W j = ∅}; A: C r ← W k , U ← U \ C r , U r ← U \ neighbors of C r in Γ ; If r < k, re-color every vertex in C r with color r; B: repeat (Re-color some more vertices with color r); C: until U r = ∅; D: until |U| ≤ L; Re-color U with d log 2 d + 2 unused colors from our initial set of q 0 colors; end 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 Ω 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 Γ 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.
The above lemma, is Lemma 7.7(i) of Janson, Luczak and Ruciński [11] and it implies that if ∆ = d then w.h.p. Γ U at line D contains no K-core, K = 2 d log 2 d + 1. Here Γ U denotes the sub-graph of Γ induced by the vertices U. For a graph G = (V, E) and K ≥ 0, the K-core is the unique maximal set S ⊆ 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 , . . . , v n so that v i has at most K − 1 neighbors in {v 1 , v 2 , . . . , 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 , . . . , v n denote an ordering of U such that the degree of v i is less than K in the subgraph Γ i of Γ induced by {v 1 , v 2 , . . . , v i }. We will prove the claim by induction. The claim is trivial for i = 1. By induction there is a path σ 0 , σ 1 , . . . , σ r from the coloring σ 0 of U at line B, restricted to Γ 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 {σ j−1 , σ j }. We construct a path (of length ≤ 2r) that re-colors Γ i . For j = 1, 2, . . . , 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 σ r is a proper coloring of Γ i−1 . Since v i has degree less than K in Γ 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 α 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 α 1 (Γ U ) vertices, where U is as at the start of Loop B. The following result is from Lemma 7.8(i) of [11]. 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, This is because the edges inside U are unconditioned by the algorithm and because v ∈ V j has no neighbors in V j for j ≥ 1. On the other hand, if we apply Algorithm greedy re-color to G n, p then (1) is replaced by the recurrencẽ (Putting V j = {j} means that greedy re-color is running on G n, p .) Comparing (1) and (2) we see that we can couple the two applications of greedy re-color so that u t ≥ũ t for t ≥ 0. Now the application of Loop B re-colors a maximal independent set of the graph Γ 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 Γ U . And so using Lemma 2.3 we see that w.h.p. each execution of Loop B re-colors at least vertices, for d sufficiently large. We have replaced ∆ of Lemma 2.3 by d/ log 2 d to allow for the fact that we hae replaced n by |U| ≥ L. Consequently, at the end of Algorithm greedy re-color we will have used at most colors. The term d log 2 d + 2 arises from the re-coloring of U at line D.
Finishing the proof: Now suppose that q ≥ cd log d where d is large and c > 3/2. Fix a particular χ-coloring τ . We prove that almost every q-coloring σ can be transformed into τ changing one color at a time. It follows that for almost every pair of q-colorings σ, σ ′ we can transform σ into σ ′ by first transforming σ to τ and then reversing the path from σ ′ to τ .
We proceed as follows. The algorithm greedy re-color takes as input: (i) the coloring σ and (ii) a specific subset of q 0 colors from {1, ..., q} that are not used in τ . W.h.p. it transforms the input coloring into a coloring using only those q 0 colors. Then we process the color classes of τ , re-coloring vertices to their τ -color. When we process a color class C of τ , we switch the color of vertices in C to their τ -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 τ -color which will not be color i C . This proves Theorem 1.1. ✷