On the connectivity threshold for colorings of random graphs and hypergraphs

03/14/2018 ∙ by Michael Anastos, et al. ∙ 0

Let Ω_q=Ω_q(H) denote the set of proper [q]-colorings of the hypergraph H. Let Γ_q be the graph with vertex set Ω_q and an edge σ,τ} where σ,τ are colorings iff h(σ,τ)=1. Here h(σ,τ) is the Hamming distance |{v∈ V(H):σ(v)≠τ(v)}|. We show that if H=H_n,m;k, k≥ 2, the random k-uniform hypergraph with V=[n] and m=dn/k then w.h.p. Γ_q is connected if d is sufficiently large and q≳ (d/ d)^1/(k-1).

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1 Introduction

In this paper, we will discuss a structural property of the set of proper -colorings of the random hypergraph , where for some large constant . Here has vertex set and an edge set consisting of randomly chosen -sets from . Note that in the graph case where we have . A proper -coloring is a map such that for all i.e. no edge is mono-chromatic. Then let us define to be the graph with vertex set and an edge iff where is the Hamming distance . In the Statistical Physics literature the definition of may be that colorings are connected by an edge in whenever . Our theorem holds a fortiori if this is the case.

Notation: if for large and where and .

Then let

(1)

We prove the following.

Theorem 1.1.

Suppose that and and and that is sufficiently large. Then

  1. If then w.h.p. is connected.

  2. If then the diameter of is w.h.p.

Note that connected implies that “The Glauber Dynamics on

is ergodic”. At the moment we only know that Glauber Dynamics is rapidly mixing for

, see Efthymiou, Hayes, Štefankovič and Vigoda [12]. So, it would seem that the connectivity of is not likely to be a barrier to randomly sampling colorings of sparse random graphs.

We note that the lower bound for is close to where the greedy coloring algorithm succeeds w.h.p.

We should note that in the case that Molloy [18] has shown that w.h.p. there is no giant component in if . It is somewhat surprising therefore that w.h.p. jumps very quickly from having no giant to being connected. One might have expected that would simply imply the existence of a giant component.

Prior to this paper, it was shown in [11] that w.h.p. is connected. The diameter of the reconfiguration graph for graphs has been studied in the graph theory litrature, see Bousquet and Perarnau [8] and Feghali [13]. They show that if the maximum sub-graph density of a graph is at most and then has polynomial diameter. Using Theorem 1 of [8] we can show a linear bound on the diameter with a small increase in the number of colors.

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 has been focussed on the structure near the colorability threshold, e.g. Bapst, Coja-Oghlan, Hetterich, Rassman and Vilenchik [5], or the clustering threshold, e.g. Achlioptas, Coja-Oghlan and Ricci-Tersenghi [2], Molloy [18]

. Other papers heuristically identify a sequence of phase transitions in the structure of

, e.g., Krza̧kala, Montanari, Ricci-Tersenghi, Semerijan and Zdeborová [17], Zdeborová and Krza̧kala [20]. 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 [10] who rigorously showed the existence of a sharp satisfiability threshold for random -SAT.

Section 3 describes a property -colorability such that if has this property then implies that is connected. Section 4 proves that , is -colorable for defined in (1).

The paper uses some of the ideas from [4] which showed there is a giant component in w.h.p. when for .

2 Outline argument

We show that with the values given in (1) then w.h.p. has the property that any greedy coloring of will need at most maximal independent sets before being left with a graph without a -core. (See Lemma 4.4.) We call the colorings found in this way, good greedy colorings and we refer to this property as -colorability. It follows from this, basically using the argument from [4], that if and then there is a good path in to some good greedy coloring .

Suppose now that are good greedy colorings. If then there is a color that is not used by . From we move to by re-coloring vertices colored 1 in by . Then we move from to by coloring with color 1, all vertices that have color 1 in . At this point, and agree on color 1. may use colors and so we move by a good path from to a coloring that uses at most colors and does not change the color of any vertex currently with color 1. Here we use the fact that is -colorable. After this, it is induction that completes the proof.

3 -colorability

The degree of a vertex in a hypergraph is the number of edges such that . (For completeness, we will state several things in this short paper that one might think can be taken for granted.)

Let . A -core of is a maximal subgraph of in which every vertex has degree at least . For every , if the subgraph of induced by does not have a -core then there is an ordering of the vertices in such that every vertex in has at most neighbors that precede it in that ordering.

If a hypergraph that does not have a -core then we can color it with at most colors. Let be an ordering on where

(2)

Such an ordering must exist when there is no -core. We color the vertices in the order and assign to a color that is not blocked by the neighbors that precede it. A color is blocked for vertex by vertices if and have already been given color .

Next let be a sequence of independent sets of such that for each , is maximal in the sub-hypergraph induced by . We say that such a sequence is a maximally independent sequence of length . Note that we allow here, in order to make our sequences of length exactly .

Definition 3.1.

We say that a hypergraph is -colorable if there does not exist a maximally independent sequence of length such that has a -core.

The main result of this section is the following.

Theorem 3.2.

Let be -colorable and let . Then is connected.

Later, in Section 4 we will show that is -colorable, for a suitable values of .

Lemma 3.3.

Let be an -colorable hypergraph and be a maximal independent set of . Set and let be the subgraph of induced by . Then is -colorable.

Proof.

Assume that in not -colorable. Then there exists a partition of into such that for , is a maximal independent set of and has a -core. For set . Furthermore set . Then is a maximal independent sequence of length and has a -core which contradicts the fact that is -colorable. ∎

Lemma 3.4.

Let be a hypergraph, and . Let be such that the subgraph of induced by has no -core. Furthermore let and be two colorings of such that

  1. They agree on .

  2. They use only colors on the vertices in .

  3. uses at most colors on that are distinct from the ones it uses on .

Then there exists a path from to in .

Proof.

Without loss of generality we may assume that and use to color . The proof that follows is an adaptation to hypergraphs of the proof in [4] that is connected when a graph has no -core. Because has no -core there exists an ordering of its vertices, , such that for , has at most neighbors in . For let be the coloring that agrees with on and with on . On it agrees with both. Thus and .

We proceed by induction on to show that there is a sequence of colorings from to such that (i) going from one coloring to the next in only re-colors one vertex and (ii) all colorings in the sequence are proper for the hypergraph induced by . We do not claim that the colorings in are proper for . On the other hand, taking we get a sequence of -proper colorings that starts with , ends with , such that the consecutive pairs of proper colorings differ on a single vertex. Clearly, such a sequence corresponds to a path from to in .

The case is trivial as we have assumed that agree on and so we can give the color . Assume that the assertion is true for and let be a sequence of colorings promised by the inductive ssertion. Let denote the change defining the change from to . We construct a sequence of colorings of length at most that yields the assertion for . For , we will re-color to color , unless there exists a set such that and . The fact that is a proper coloring of implies that . Because has at most neighbors in and only uses colors in to color , there exists a color for in which is not blocked by a subset of and is diferent from its current color. We first re-color to and then we re-color to , completing the inductive step. At the very end, i.e. at step we give its color in . ∎

Definition 3.5.

A coloring with color sets is said to be a good greedy coloring if (i) is a maximally independent sequence of length and (ii) has no -core.

We prove Theorem 3.2 in two steps. In Lemma 3.6, we show that if and is then we can reach a good greedy coloring in starting from any coloring. Then in Lemma 3.8, we show that if then any good greedy coloring can be reached in from any other good greedy coloring .

Lemma 3.6.

Let be an -colorable hypergraph, and be a -coloring of . Then there exists a good greedy coloring of such that there exists a path in from to .

Proof.

We generate the coloring as follows. Let be the color classes of . Then let be a maximal independent set containing . In general, having defined we let and then we let be a maximal independent set in that contains . Thus is a maximal independent sequence of length . We now describe how we transform the coloring vertex by vertex into a coloring in which vertices in get color for . We first re-color the vertices in by giving them color 1, one vertex at a time. The coloring stays proper, as is an independent set. In general, having re-colored we re-color the vertices in with color . Again, the coloring stays proper, as is an independent set, containing all vertices in that have not been re-colored. We observe that each re-coloring of a vertex done while turning into can be interpreted as moving from a coloring in to a neighboring coloring.

Let . Because is -colorable, we find that has no -core. Because has no -core there exists a proper coloring of the subgraph of induced by that uses only colors in . Set to be the coloring that agrees with on and with on .

Lemma 3.4 implies that there is a path from to . Hence there is a path from to . ∎

Remark 3.7.

In the proof of Lemma 3.6 we see that each vertex is re-colored at most twice before we apply Lemma 3.4. Thus this part of the proof yields at most distinct sub-paths of length .

Lemma 3.8.

Let be an -colorable hypergraph, and let be two good greedy colorings. Then there exists a path from to in .

Proof.

We proceed by induction on . For , is colorable and so it does not have a -core. Thus the base case follows directly from Lemma 3.4 by taking .

Assume that the statement of the Lemma is true for and let . There exists a maximal independent sequence of length such that if then (i) for , assigns the color to and (ii) assigns only colors in to vertices in .

Let be a color not assigned by . There is one as . Starting from we recolor all vertices that are colored 1 by color to create a coloring . Then we continue from by recoloring all the vertices in by color 1 and we let be the resulting coloring. Clearly there is a path from to in .

We now set , and set to be the restrictions of on . Observe that since is a maximal independent set, Lemma 3.3 implies that is colorable and in addition that is a good greedy coloring of . Lemma 3.6 implies that in there is a path from to some good greedy coloring that uses only colors from . The induction hypothesis implies that in that there is a path from to .

Color 1 is not used in or in any of colorings found in the path . Thus the path corresponds to a path in from to . Consequently the colorings are connected in by the path . ∎

Proof of Theorem 3.2: Let be colorable, , and let be two colorings of . Lemma 3.6 implies that in , there ia path from to a good greedy coloring for . Lemma 3.8 implies that there is a path in from to . ∎

4 Random Hypergraphs

Theorem 1.1 follows from

Lemma 4.1.

Let and suppose that and that is sufficiently large. If and then w.h.p. is connected.

In the following we will assume for simplicity of notation that , so that . We do not know if there is an upper bound needed for the growth rate of , but we doubt it.

To prove Lemma 4.1 we use Lemmas 4.2, 4.4, 4.5 (below) in order to deduce that w.h.p. is colorable. Then we apply Theorem 3.2. (Lemmas 4.2 and 4.5 are hardly new or best possible, but we prove them here for completeness.)

We will do our calculations on the random graph and use the fact for any hypergraph property , we have

(3)
Lemma 4.2.

Let and and sufficiently large. Then, w.h.p. does not contain an independent set of size .

Proof.

Let

. The probability that there exists an independent set of size

in is bounded by

(4)

Notation 4.3.

We let

Furthermore, for we let

Lemma 4.4.

If and is sufficiently large then, w.h.p. there does not exist and disjoint sets such that:

  1. is a maximal independent sequence of length in .

  2. .

Proof.

Fix , and let . Since we have that . There are ways to pick disjoint sets of sizes respectively. So satisfy condition (i) of Lemma 4.4 only if for every and every , there exist such that . So, given the probability that we have (i) is at most

(5)

Now let and set . We consider 2 cases.

Case 1: .
Now and so , which implies that . Then,

Now

Thus the probability that for some there exist satisfying conditions (i), (ii) of Lemma 4.4 is bounded by

Case 2: .
Thus . Observe that from Lemma 4.2 we can assume that

(6)

For (6) we are using Lemma 4.2 to argue that we need at least this many independent sets to partition a set of size . The -1 comes from the fact that the upper bound in the definition of may not be tight.

Thus,

(7)

and now with as defined in (5) we have

Thus the probability that for some there exist satisfying conditions (i), (ii) of Lemma 4.4 is bounded by

For sufficiently large , (7) implies and we also have that . Therefore

Furthermore, Lemma 4.2 implies that . Thus

So,

Lemma 4.5.

If and is sufficiently large then w.h.p. every set of size at most spans fewer than edges in . Hence no subset of size at most contains a core.

Proof.

Let . The probability that there exists of size at most that spans at least edges is bounded by