5-Regular Multi-Graphs are 3-Colorable with ... - Semantic Scholar

5-Regular Multi-Graphs are 3-Colorable with Independent of their Size Positive Probability

?

J. D´ıaz1 , G. Grammatikopoulos2,3 , A.C. Kaporis2 L.M. Kirousis2,3 , X. Pérez1 , and D.G. Sotiropoulos4 1

Universitat Politècnica de Catalunya, Departament de Llenguatges i Sistemes Inform` atics, Campus Nord – Ed. Omega, 240, Jordi Girona Salgado, 1–3, E-08034 Barcelona, Catalunya, {diaz,xperez}@lsi.upc.edu 2 University of Patras, Department of Computer Engineering and Informatics, GR-265 04 Patras, Greece, {grammat,kaporis,kirousis}@ceid.upatras.gr 3 Research Academic Computer Technology Institute, P.O. Box 1122, GR-261 10 Patras, Greece 4 University of Patras, Department of Mathematics, GR-265 04 Patras, Greece, [email protected]

Abstract. We show that uniformly random 5-regular multi-graphs of n vertices, n even, are 3-colorable with probability that is positive independently of n.

1

Informal Presentation of the Problem, Previous Work and the Method of Solution

The chromatic number of random regular graphs has attracted much interest in recent years. All necessary background with respect to random regular graphs as well as the pioneering results about their chromatic number and other parameters can be found in the comprehensive review paper by Wormald [11]. Molloy [9] proved that 6-regular graphs have chromatic number at least 4, asymptotically almost surely (a.a.s.) with respect to the number of their vertices n. The basic ingredient of the proof was to compute the expected number of 3-colorings of a 6-regular graph and show that it converges to zero (first moment method). Achlioptas and Moore [3] proved that 4-regular graphs have chromatic number 3 with uniformly positive probability (w.u.p.p.). The proof was algorithmic in the sense that a backtracking-free algorithm based on Brelaz’ heuristic was designed and shown to produce a 3-coloring w.u.p.p. Subsequently, Achlioptas and Moore [4] showed that a.a.s., the chromatic number of a d-regular graph (d ≥ 3 ) is k or k + 1 or k + 2, where k is the smallest integer such that d < 2k log k. They also showed that if furthermore d > (2k − 1) log k, then a.a.s. the chromatic number is either k + 1 or k + 2. This result however gives no information for the chromatic number of either 4-regular or 5-regular graphs (apart from the known fact that a.a.s. the former have chromatic number either 3 or 4 and the later either 3 or 4 or 5). Shi and Wormald [10] showed that a.a.s. the chromatic number of a 4-regular graph is 3, that a.a.s. the chromatic number of a 6-regular graph is 4 and that a.a.s. the chromatic number of a 5-regular graph is either 3 or 4. (In addition, they showed that a.a.s. the chromatic number of a ?

The 1st, 2nd, 4th and 5th authors are partially supported by Future and Emerging Technologies programme of the EU under contract 001907 “Dynamically Evolving, Large-Scale Information Systems (DELIS)”. The 1st author was partially suported by the Distinci´ o de la Generalitat de Catalunya per a la promoci´ o de la recerca, 2002. The 3rd and 4th authors are partially supported by European Social Fund (ESF), Operational Program for Educational and Vacational Training II (EPEAEK II), and particularly Pythagoras. Part of the research of the 4th author was conducted while visiting on a sabbatical the Departament de Llenguatges i Sistemes Inform` atics of the Universitat Politècnica de Catalunya.

2

J. D´ıaz, G. Grammatikopoulos, A. Kaporis, L. Kirousis, X. Pérez, and D. Sotiropoulos

d-regular graph, for all other d up to 10, is restricted to a range of two integers.) Their proofs were algorithmic. The above results leave open the question of whether the chromatic number of a 5-regular graph can take the value 3 w.u.p.p. (or perhaps even a.a.s.). Previous attempts by some of the authors of this paper (and independently by other researchers) to answer negatively the above question, using refinements of the first moment method, failed. These attempts computed the expected number of successively more restricted types of 3-colorings (such that whenever a generic 3-coloring exists, at least one of the restricted type exists as well), and aimed at proving that it is a.a.s. equal to zero. All attempts however gave expected values that were a.a.s. equal to ∞. To failure as well led several attempts to design an algorithm that would be amenable to rigorous mathematical analysis and that would at least w.u.p.p. produce a 3-coloring for 5-regular graphs. Both the above failures were given a well founded empirical explanation by the work of physicists. Building on a statistical mechanics analysis of the space of truth assignments of the 3-SAT problem, which has not been shown yet to be mathematically rigorous, and on the Survey Propagation (SP) algorithm for 3SAT inspired by this analysis (see e.g. [8] and the references therein), Krz¸akala et al. [7] provided strong evidence that 5-regular graphs are a.a.s. 3-colorable by an SP algorithm. They also showed that the space of assignments of three colors to the vertices (legal or not, i.e. with no two adjacent vertices with the same color or not) consists of clusters of legal color assignments inside which one can move from point to point by steps of small Hamming distance. However, to go from one cluster to another by such small steps, it is necessary to go through assignments of colors that grossly violate the requirement of legality (high-energy color assignments). Also, the number of clusters that contain points with energy that is a local, but not global, minimum is exponentially large. As a result, local search algorithms are easily trapped into such local minima (metastable states). The above considerations left as the only plausible alternative (besides attempting to rigorously analyze SP or a similar algorithm, a rather ambitious goal) to try prove that 5-regular graphs are 3-colorable w.u.p.p. in a non-algorithmic (i.e. analytic) way. A technique that has been used towards similar ends is the Second Moment Method. The basic ingredient of this method is the fact that if X is a non-negative random variable (r.v.) then the probability that X is positive is bounded from below from the ratio of the square of its first moment to its second moment. We call this ratio the Moment Ratio. Symbolically:

Pr [X > 0] ≥

(E (X))2 . E (X 2 )

(1)

So if the Moment Ratio of a counting r.v. is asymptotically positive, then the set whose elements are counted by X is non-empty w.u.p.p. This technique was first used to obtain analytic lower bounds for the threshold of k-SAT (and actually compute it within a factor of 2) in [2]. It was later used in the context of the colorability problem (in [5]) to solve the long-standing open problem of computing the two possible values of a random regular graph. In the later work, the authors considered as X the r.v. that gives the number of 3-colorings of a graph. Achlioptas and Moore (privately communicated unpublished work) then used the same r.v. in an attempt to prove that 5-regular graphs are 3-colorable. However, unfortunately, they got that the Moment Ratio of this r.v. was asymptotically equal to zero (exponentially fast). A possible way out of this impasse is to consider a r.v. that counts some special 3-colorings and has an asymptotically positive second Moment Ratio. The question of course is how to select such an r.v. so that on one hand it is feasible to compute the asymptotic value of its Moment Ratio and on the other this computation yields a non-zero value.


3

We consider here the r.v. that counts what we call the stable 3-colorings. These are 3-colorings with the property that for no single vertex v can we change its color without the appearance of an edge with the same color at its endpoints. It turned out that for stable 3-colorings the variance diminishes (as expected) but the square of the expectation is diminished in a lesser degree and as a result the Moment Ratio becomes asymptotically positive. Actually, for this r.v. the dominant terms of both sums that express the first and the second moment, respectively, occur at central points of the domains over which the variables of the terms range (we call such points barycenters). As a result of this central location of the dominant terms of both moments, the square of the first moment is asymptotically equal to the second moment within a polynomial factor. As was shown by Achlioptas and Moore (privately communicated unpublished work), this is not true for the not necessarily stable 3-colorings (generic 3-colorings) and consequently, the Moment Ratio of the r.v. of generic 3-colorings is asymptotically zero (exponentially fast). This “dislocation” of the dominant term of the second moment in relation to the dominant term of the first moment in the case of generic 3-colorings is, perhaps, surprising given the symmetry inherent in the notion of a generic 3-coloring. Fortunately, the stable 3-colorings, due to the “geographically fair distribution” of the colors, do not permit such a dislocation. The computation of the second moment amounts essentially to counting number of pairs of 3-colorings on 5-regular graphs. To give an exact expression for the second moment of the stable 3-colorings we had to use a large number of variables (9 × 36). These variables express the number of vertices that have a given pair of colors (out of the nine possible pairs) and also have a given distribution of their five edges with respect to the pair of colors on the other endpoint of these edges (as we will see there are 36 possible distributions). The computation of the asymptotic value of this expression turned out to be a non-trivial, we believe, task. It was carried out in three successive phases, at each one of which we had to find the dominant term of a sum. The terms of the sum at each phase were exponential expressions, all with the same exponent. Therefore finding the dominant one entailed finding the maximum base. The first two phases were designed in a way that the functions to be maximized were convex (convex in the sense of a water-repellent shape, concave for some authors). Thus the maximizations in the first two phases were done in an analytically exact way. In the last phase, we had to maximize a non-convex function of only four variables ranging in [0, 1/3] . This function was an implicit one given in terms of the solution of a polynomial 45 × 45 system. We knew that the system had a unique solution, because its solutions corresponded to unique critical points of strictly convex functions of the previous phases. Actually, our task was not to locate the maximizer ex nihilo but rather to show that it occurs at a specific point of the domain of the function under consideration, i.e. the point corresponding to the barycenter. Our approach was to compute the value of the function at all points of a 4-dimensional grid covering the domain [0, 1/3]4 and compare them with the value of the function at the barycenter. We selected the step-size of the grid to be 1/600 (200 steps per dimension). The steps avoided a thin layer around boundary points, were we knew that the function had derivative approaching infinity, thus the max was certainly not there (this safety layer was taken with width 1/800). Recall that at every step of computation of the last phase, we had to solve a polynomial 45 × 45 system (which we knew had a unique solution). To solve these systems efficiently enough in order to go over all grid points in a reasonable time, we designed a fast search algorithm based on an algorithm by Byrd et al. [6]. We also devised a way to select good starting points for each system, whose basic principle was to select for each successive system a starting point that belonged to the convex hull of the solutions to the previous systems. The algorithm was implemented in Fortran and run on the IBM’s supercomputer in the Barcelona Supercomputing Center, which consists of 2.268 dual 64-bit processor blade nodes with a total of 4.536 2.2 GHz PPC970FX processors.

4


We used a similar approach (but with less accuracy) to show that the LPMD’s of the Hessian of a function, with a 3-dimensional domain and which we wanted to prove to be strictly convex, had alternating algebraic signs at all points of the function’s domain.

2

Exact Expressions for the First and Second Moments

Everywhere in this paper n will denote a positive integer divisible by 6 (the divisibility restriction is made for the standard model of regular multi-graphs to be meaningful and also to be meaningful to consider balanced 3-colorings —see definitions below). Asymptotics are always in terms of n. Let Pn be the probability space of 5-regular multi-graphs obtained by considering a set of n vertices, labelled 1, . . . , n, for each of these vertices a set of 5 semi-edges, labelled 1, . . . , 5, and a uniform random perfect matching of all 5n semi-edges. A pair of semi-edges according to the matching defines the edges of the multi-graph (see [11]). Definition 1. A 3-coloring of a multi-graph G ∈ Pn is called stable if for every vertex v and every color i = 0, 1, 2, either v itself or one of its neighbors are colored by i. Equivalently for no single vertex v can we change its color without the appearance of an edge with the same color at its endpoints. A 3-coloring is called balanced if for each i = 0, 1, 2, the number of vertices with color i is n/3. Let S be the class of balanced stable 3-colorings of a multi-graph G ∈ Pn and let X be the random variable that counts the number of balanced stable 3-colorings of multi-graphs in Pn . Then the following equations can be easily shown: |{G | G ∈ Pn }| = E (X) =

(5n)! 25n/2 (5n/2)!

,

|{(G, C) | G ∈ Pn , C ∈ S}| , |{G | G ∈ Pn }|

¡ ¢ |{(G, C1 , C2 ) | G ∈ Pn , C1 , C2 ∈ S}| E X2 = . |{G | G ∈ Pn }| 2.1

(2) (3) (4)

First Moment

Below we assume we are given a multi-graph G and a balanced stable 3-coloring C on G. The arithmetic in the indices is modulo 3. We start by giving some useful terminology and notation: Definition 2. A 1-spectrum s is an ordered pair of non-negative integers, s = (s−1 , s1 ), such that s−1 + s1 = 5 and s−1 , s1 > 0. Notice that there are four 1-spectra. The 1-spectra are intended to express the distribution of a the five edges stemming from a given vertex according to the color of their other endpoint. Formally, a vertex v of color i is said to have 1-spectrum s = (s−1 , s1 ), if s−1 out of its five edges are incident on vertices of color i − 1 and the remaining s1 edges are incident on vertices of color i + 1. The condition s−1 , s1 > 0 in the definition of 1-spectrum expresses the fact that the 3-coloring is a stable one. For each i = 0, 1, 2 and 1-spectrum s, we denote by d(i; s) the scaled (with respect to n) number of vertices of G which are colored by i and have 1-spectrum s. Then obviously: X X d(i; s) = 1/3 and therefore d(i; s) = 1. s

i,s


5

Let: N1 = |{(G, C) | G ∈ Pn , C ∈ S}|. Given any two colors i and j, observe that there are exactly 5n/6 edges connecting vertices with color i and j, respectively. a fixed ¡Given ¢ sequence (d(i; s)n)i,s that corresponds to a balanced stable 3-coloring, let us denote n (d(i;s)n)i,s the multinomial coefficient that counts the number of ways to distribute the n vertices ¡¢ ¡ 5 ¢ ¡5¢ into classes of cardinality d(i; s)n for all possible values of i and s. Let also 5s stand for s−1 = s1 .

by

By an easy counting argument, we have that N1 =

  ¶ Y µ ¶d(i;s)n µ ¶3 n 5   5n ! , (d(i; s)n)i,s 6 s i,s

X µ d(i;s)i,s

(5)

where the summation above is over all possible sequences (d(i; s))i,s that correspond to balanced stable 3-colorings.

2.2

Second Moment

Below we assume we are given a multi-graph G and two balanced stable 3-colorings C1 and C2 on G. For i, j = 0, 1, 2 let let Vij bet the set of vertices colored with i and j with respect to colorings C1 and C2 , respectively. Let nji = |Vij |/n and let Eij be the set of semi-edges whose starting vertex is in Vij . j,t Also, for r, t ∈ {−1, 1}, let Ei,r be the set of semi-edges in Eij which are matched with one in j+t j,t Ei+r . Let mj,t i,r = |Ei,r |/n.

The following obviously hold: X j mj,t i,r = 5ni , r,t∈{−1,1}

X

nji = 1

X

and therefore

i,j

mj,t i,r = 5.

i,j,r,t

Since matching sets of semi-edges should have equal cardinalities, we also have: j+t,−t mj,t i,r = mi+r,−r . 1 −1 1 Definition 3. A 2-spectrum s is an ordered quadruple of non-negative integers, s = (s−1 −1 , s−1 , s1 , s1 ), −1 −1 −1 −1 −1 −1 such that s−1 + s1−1 + s11 + s1 = 5 and (s−1 + s1−1 )(s11 + s1 )(s−1 + s1 )(s1−1 + s11 ) > 0.

We are using the same symbol s for both 1-and 2-spectra. The difference will be clear form the context. Notice that there are 36 2-spectra. Let v be a vertex in Vij . Vertex v is said to have 21 1 −1 t spectrum s = (s−1 −1 , s−1 , s1 , s1 ) if sr out of its five edges, r, t ∈ {−1, 1}, are incident on vertices in j+t −1 −1 −1 1 1 1 1 Vi+r . The condition (s−1 + s−1 )(s1 + s−1 1 )(s−1 + s1 )(s−1 + s1 ) > 0 in the definition of a 2-spectrum expresses the fact that both C1 and C2 are stable. For each i, j = 0, 1, 2 and 2-spectrum s, we denote by d(i, j; s) the scaled number of vertices which belong to Vij and have 2-spectrum s. The following equations obviously hold: X X d(i, j; s) = nji , str d(i, j; s) = mj,t and therefore i,r s

s

X i,j,s

d(i, j; s) = 1.

6


Throughout this paper we refer to the set of the nine numbers nji as the set of the overlap matrix variables, we also refer to the set of the thirty-six numbers mj,t i,r as the set of the matching variables. Finally, we refer to the 9 × 36 numbers d(i, j; s) as the spectral variables. Let: N2 = |{(G, C1 , C2 ) | G ∈ Pn , C1 , C2 ∈ S|. Given a fixed¡sequence (d(i, ¢ j; s)n)i,j,s that corresponds to a pair of balanced stable 3-colorings, let n us denote by (d(i,j;s)n) the multinomial coefficient that counts the number of ways to distribute i,j,s ¡¢ the n vertices into classes of cardinality d(i, j; s)n for all possible values of i, j and s. Let also 5s ¢ ¡ stand for s−1 ,s15 ,s1 ,s1 (the distinction from a similar notation for 1-spectra will be obvious from −1

the context).

−1

1

Now, by an easy counting argument we have:     ¶ Y µ ¶d(i,j;s)n  X µ Y 1 n 5   2 N2 = ((mj,t , i,r n)!)  (d(i, j; s)n)i,j,s  s i,j,s i,j,r,t

(6)

d(i,j;s)i,j,s

where the summation above is over all possible sequences (d(i, j; s))i,j,s that correspond to pairs of balanced stable 3-colorings.

3

Asymptotics

In this section we will show that E(X 2 ) = O((E(X))2 ). An immediate consequence of this is that 5-regular multi-graphs have a balanced stable 3-coloring (and hence a generic 3-coloring) w.u.p.p. √ ¡ ¢δn As a first step, observe that by applying the generalized Stirling approximation (δn)! ∼ 2πδn δn , e where δ is a positive real (see [1]), to the results obtained in the previous section (especially formulas (2), (3), (4), (5) and (6)), we easily get that:   µ ¶5/2 Y Ã ¡5¢ !d(i;s) n X 1 s  , f1 (n, d(i; s)i,s )  E (X) ∼ (7) 6 d(i; s) i,s d(i;s)i,s

¡ ¢ E X2 ∼ X d(i,j;s)i,j,s

  n µ ¶5/2 Y Ã ¡5¢ !d(i,j;s) Y j,t 1 m 1 i,r 2 s   , f2 (n, d(i, j; s)i,j,s )  (mj,t i,r ) 5 d(i, j; s) i,j,s i,j,r,t

(8)

where f1 and f2 are functions that are sub-exponential in n and also depend on the sequences d(i; s)i,s and d(i, j; s)i,j,s , respectively. By a result similar to Lemma 3 in [4], we can prove (essentially by using a Laplace-type integration technique) that the Moment Ratio is asymptotically positive if (E(x))2 ³ E(X 2 ), i.e. ln((E(x))2 ) ∼ ln(E(X 2 )).

(9)

The proof is considered standard but somewhat tedious and so we avoid giving it here. µ 5 ¶d(i;s) ¡ ¢5/2 Q (s) Let now M1 be the maximum base 16 as d(i; s)i,s ranges over all possible i,s d(i;s) sequences that correspond to balanced stable 3-colorings and let M2 be the maximum base    µ ¶5/2 Y Ã ¡5¢ !d(i,j;s) j,t 1 Y m 1 j,t s   (mi,r ) 2 i,r  5 d(i, j; s) i,j,s i,j,r,t


7

as d(i, j; s)i,j,s ranges over all possible sequences that correspond to pairs of balanced stable 3colorings. From the equations (7) and (8) one can immediately deduce that the relation (9) is true if (M1 )2 = M2 . Below we compute the exact values of M1 and M2 .

3.1

First Moment: Computing M1

Let f=

Y

Ã ¡ ¢ !d(i;s) 5 s

(10)

d(i; s)

i,s

be a real function of non-negative real variables d(i; s) , where P i = 0, 1, 2 and s runs over 1-spectra (i.e. f is a function of 12 variables), defined over the polytope s d(i; s) = 1/3, i = 0, 1, 2. We will prove the following Lemma: Lemma 1. The function ln f is strictly convex (water-repellent shape). Let D1 = The function f has a maximizer at the point where ¡ 5¢ d(i; s) =

s

D1

P

¡¢

5 i,s s

= 3 × 30.

, ∀i, s.

Proof The strict convexity of ln f is easy to check. Using Lagrange multipliers, we will find the maximizer of ln f , which obviously coincides with the maximizer of f . µ ¶ ¶ Xµ 5 ln f = d(i; s) ln − d(i; s) ln d(i; s) s i,s P P We relax the three constraints s d(i; s) = 1/3, i = 0, 1, 2 to the single one i,s d(i; s) = 1. Obviously, this causes no harm if the maximizer turns out to satisfy all constraints. It is µ ¶ ∂ ln f 5 = ln − 1 − ln d(i; s), ∂d(i; s) s so that we have

¡ 5¢ s

d(i; s)

= x,

where x is the Lagrange multiplier introduced by the constraint follows.

P i,s

d(i; s) = 1. The required easily ¤

By direct substitution, we get: Lemma 2.

  r µ ¶5/2 Y µ ¶5/2 25 1 1 d(i,s)   . M1 = D1 D1 = = 6 6 24 i,s

From the above we immediately get: Theorem 1. The expected number of balanced stable 3-colorings of a random 5-regular multi-graph approaches infinity as n grows large.

8

3.2


Second Moment: Computing M2

Let

 F =

Y

i,j,s

Ã

¡5¢ s

d(i, j; s)

 !d(i,j;s)   j,t 1 Y m i,r 2   (mj,t i,r )

(11)

i,j,r,t

be a real function of non-negative real variables d(i, j; s) , where i, j = 0, 1, 2 and s runs over 2-spectra (i.e. f is a function of 9 × 36 variables), defined over the polytope determined by: X

d(i, j; s) = 1/3, ∀i;

j,s

where mj,t i,r =

P

t s sr

X

j+t,−t d(i, j; s) = 1/3, ∀j and mj,t i,r = mi+r,−r ,

(12)

i,s

d(i, j; s).

We will maximize F in three phases. In the first one, we will maximize F assuming the matching variables mj,t i,r are fixed constants such that their values are compatible with the polytope over which F is defined. Thus we will get a function Fm of the 36 matching variables mj,t i,r . At the second phase P j,t we will maximize Fm assuming that the nine overlap matrix variables nji = r,t mi,r are fixed constants compatible with the domain of the matching variables. Thus we will get a function Fn of the overlap matrix variables. The preceding two maximizations will be done in an analytically exact way. Observe now that since we consider balanced 3-colorings, Fn depends only on the values of four n’s. We will maximize Fn by going through its 4-dimensional domain over a fine grid. Let us point out that the maximizations above will not be done ex nihilo. Actually, we know (see below) the point where we would like the maximizer to occur. Therefore all we do is not find the maximizer but rather prove that it is where we want it to be. Lemma 3. Let D2 =

P

¡¢

5 i,j,s s

= 9 × 900 and let ¡5¢ d(i, j; s) =

s

D2

, ∀i, j, s.

Then the value of the base    µ ¶5/2 µ ¶5/2 Y Ã ¡5¢ !d(i,j;s) j,t 1 Y m 1 1 j,t i,r 2 s    F = (mi,r ) 5 5 d(i, j; s) i,j,s i,j,r,t at the above values of d(i, j; s)i,j,s is equal to (M1 )2 = 25/24. Proof

By direct substitution.

¤ ¡ 5¢ We call the sequence d(i, j; s) = s /D2 the barycenter. Barycenter as well we call the corresponding point in the domain of Fm , i.e. the point mj,t i,r = 5/36, ∀i, j = 0, 1, 2, r, t = −1, 1. Finally, barycenter also we call the corresponding point in the domain of Fn , i.e. the point nji = 1/9, ∀i, j = 0, 1, 2. We will see, by direct substitutions, that the functions (1/5)5/2 Fm and (1/5)5/2 Fn as well take the n value (M1 )2 = (25/24) at their corresponding barycenters. Therefore, after computing Fm and Fn , all that will remain to be proved is that Fn has a maximizer at its barycenter nji = 1/9, i, j = 0, 1, 2. Below, we compute first the function Fm (which, by the definition of Fm , amounts to maximizing F when the matching variables are fixed) and then we compute Fn (which, by the definition of Fn , amounts to maximizing Fm when the overlap matrix variables are fixed) and finally we show that the barycenter is a maximizer for Fn by sweeping its 4-dimensional domain.


9

From the Spectral to the Matching Variables: Everywhere below we assume that the 36 matching variables mj,t i,r are non-negative and moreover take only values for which there exist 9 × 36 spectral non-negative variables d(i, j, s) such that the equations in (12) hold and moreover X mj,t str d(i, j; s), i, j = 0, 1, 2, r, t = −1, 1. (13) i,r = s

It is not hard to see that the above restrictions on the matching variables are equivalent to assuming that ∀i, j = 0, 1, 2, ∀r, t = −1, 1, j,−t j,t j,−t mj,t i,r + mi,r ≤ 4(mi,−r + mi,−r ) j,t j,−t j,−t mj,t i,r + mi,−r ≤ 4(mi,r + mi,−r ) j+t,−t mj,t i,r = mi+r,−r

mj,t i,r ≥ 0,

(14)

Fix such values for the mj,t i,r . It is easy to see that to maximize the function F given by equation (11) over the polytope described in (12) for such fixed values of the matching variables mj,t i,r , i, j = 0, 1, 2, r, t = −1, 1, it is sufficient to maximize the function F subject to the 36 constraints given by the equations in (13). Since for different pairs of (i, j), i, j = 0, 1, 2, neither the variables d(i, j; s) neither the constraints in (13) have anything in common, and since the matching variables are fixed, it is necessary and sufficient to maximize for each i, j = 0, 1, 2 the functions Ã ¡ ¢ !d(i,j;s) 5 Y s Fi,j = (15) d(i, j; s) s subject to the four constraints:

X

str d(i, j; s) = mj,t i,r , r, t = −1, 1.

(16)

s

Actually, for each i, j = 0, 1, 2, we will maximize the functions ln Fi,j under the four constraints (16), using multiple Lagrange multipliers (one for each constraint). Notice that the functions ln Fi,j are strictly convex. We define the following function: Ftricol (x, y, z, w) = (x + y + z + w)5 − (x + w)5 − (x + y)5 − (y + z)5 − (z + w)5 + x5 + y 5 + z 5 + w5 . Also for each of the nine possible pairs (i,j), i,j=0,1,2, consider the 4 × 4 system: j,−1 j,1 j,−1 ∂ Ftricol (µi,−1 , µj,1 i,−1 , µi,1 , µi,1 )

∂µj,t i,r

j,t µj,t i,r = mi,r ,

r, t = −1, 1,

(17)

where µj,t i,r , i, j = 0, 1, 2, r, t = −1, 1 denote the 36 unknowns of these nine 4 × 4 systems. Then after the application of the method of the Lagrange multipliers, we get Lemma 4. Each of the nine systems in (17) has a unique solution. Moreover in terms of the solutions of these systems !mj,t Ã 1 i,r Y (mj,t 2 ) i,r . Fm = µj,t i,r i,j,r,t Proof

See the Appendix.

¤

By the above Lemma, we have computed in an analytically exact way the function Fm . Notice that the function Fm is a function of the 36 matching variables mj,t i,r , i, j = 0, 1, 2, r, t = −1, 1, over the domain given by (14), however its value is given through the solutions of the systems in (17), which provably have a unique solution.

10


From the Matching to the Overlap MatrixPVariables: We assume P now that we fix nine nonnegative overlap matrix variables nji such that i nji = 1/3, ∀j and j nji = 1/3, ∀i. Using again multiple Lagrange multipliers, we will find the maximum, call it Fn , of the function Fm given in Lemma (4) under the constraints: X j,t j+t,−t mi,r = 5nji , i, j = 0, 1, 2 and mj,t (18) i,r = mi+r,−r , i, j = 0, 1, 2, r, t = −1, 1, r,t

assuming in addition that the mj,t i,r satisfy the inequalities in (14) (we consider the later inequality restrictions not as constraints to be dealt with Lagrange multipliers, but as restrictions of the domain of Fm that must be satisfied by the maximizer to be found). We will need that the function ln Fm over the polytope determined by the constraints (18) (for fixed values of the variables nji ) is strictly convex. To show this it is sufficient to fix an arbitrary i, j = 0, 1, 2 and show that the 4-variable function   Ã !mj,t 1 i,r Y (mj,t )2 i,r , ln  µj,t i,r r,t P j subject to the single linear constraint r,t mj,t i,r = ni , is strictly convex. To show the later, we computed the Hessian and its LPMD’s after solving the single linear constraint for one of its variables (thus we obtained a function of three variables). Notice that the value of the function under examination is given through the unique solutions of a 4 × 4 system. The Hessian and the LPMD’s were analytically computed in terms of these solutions by implicit differentiation of the equations of the system. The strict convexity then would follow if we showed that at every point of the domain of this 3-variable function, the LPMD’s were non-zero and of alternating sign. We demonstrated this by going over this domain over a fine grid and computing at all its points the LPMD’s. The values of the LPMD’s that we got were safely away from zero and with the desired algebraic sign. Notice that although to prove the convexity of the function ln Fm , subject to the constraints in (18), we j+t,−t relaxed the constraints mj,t i,r = mi+r,−r , the later ones are essential for correctly computing Fn . To apply Lagrange multipliers, we have to find the partial derivatives of the function: X 1 j,t j,t j,t ( mi,r ln mj,t ln Fm = i,r − mi,r ln µi,r ). 2 i,j,r,t

(19)

As a first step, we immediately have: ∂ ln Fm ∂mj,t i,r

0

X j,t0 ∂ ln µj,t 1 1 i,r 0 j,t − ln µ − = + ln mj,t mi,r0 . i,r i,r j,t 2 2 ∂mi,r r 0 ,t0

(20)

We now claim: Lemma 5.

X r 0 ,t0

and therefore:

∂ ln Fm ∂mj,t i,r

=

0

j,t0 mi,r 0

∂ ln µj,t i,r 0 ∂mj,t i,r

=

1 , 5

1 1 1 j,t + ln mj,t i,r − ln µi,r − . 2 2 5

Proof We consider a fixed pair of values for i, j. By adding the four equations of the corresponding 4X4 system (17), after performing the differentiations, we get that: X j,t j,−1 j,1 j,−1 mi,r , 5 Ftricol (µi,−1 , µj,1 i,−1 , µi,1 , µi,1 ) = r,t


therefore:

j,−1 j,1 j,−1 ∂ Ftricol (µi,−1 , µj,1 i,−1 , µi,1 , µi,1 )

∂ mj,t i,r

=

1 5

11

(21)

To simplify the notation, we will denote the four variables µj,t i,r , r, t = −1, 1 by x, y, z, w and the j,t corresponding mi,r by mx , my , mz , mw . Then by applying the Chain Rule to equation (21) we have that ∂Ftricol ∂x ∂Ftricol ∂y ∂Ftricol ∂z ∂Ftricol ∂w 1 + + + = ∂x ∂mx ∂y ∂mx ∂z ∂mx ∂w ∂mx 5 or that

mx ∂x my ∂y mz ∂z mw ∂w 1 + + + = x ∂mx y ∂mx z ∂mx w ∂mx 5

or that mx

∂ ln x ∂ ln y ∂ ln z ∂ ln w 1 + my + mz + mw = ∂mx ∂mx ∂mx ∂mx 5

which concludes the proof.

¤

By applying now the technique of multiple Lagrange multipliers, we get (for the details see the Appendix): j Lemma 6. Consider the 45 × 45 system with unknowns µj,t i,r , i, j = 0, 1, 2, r, t = −1, 1 and xi , i, j = 0, 1, 2: j,−1 j,1 j,−1 ∂ Ftricol (µi,−1 , µj,1 i,−1 , µi,1 , µi,1 ) j,t j+t,−t j j+t µj,t i,r = µi,r µi+r,−r xi xi+r , ∂µj,t i,r X j,t j+t,−t j j+t nji = µi,r µi+r,−r xi xi+r . r,t

This system has a unique solution. Moreover in terms of the solutions of this system: Y j j Fn = (xi )5ni . i,j

So we have computed in an analytically exact way the function Fn . Since solving the 45 × 45 in Lemma (6) when nji = 1/9, i, j = 0, 1, 2 is trivial, we get by direct substitution: Lemma 7. The value of

µ ¶5/2 µ ¶5/2 Y j 1 1 (xji )5ni Fn = 5 5 i,j

at the barycenter ni , j = 1/9, i, j = 0, 1, 2 is equal to (M1 )2 = 25/24. Therefore the value of Fn at the barycenter is > 58.2309. Therefore all that it remains to be proved is that the function Fn maximizes at the barycenter. From the Overlap Matrix Variables to the Conclusion We have to prove the the function P Fn maximizes at the barycenter. Since we have assumed that the 3-coloring is balanced, i.e. ∀i, j nji = P 1/3 and ∀j, i nji = 1/3, the domain of Fn has four degrees of freedom, all in the range [0, 1/3]. We swept over this domain going over the points of a grid with 200 steps per dimension. The sweeping avoided a thin layer (of width 1/1000) around the boundary (the points in the domain where at least one of the nji = 0), because at the boundary the derivative of the original function F is infinity, thus no maximum occurs there. We also avoided a small neighborhood around the barycenter (of radius 1/1000), as the barycenter is a local maximum (easy to see by computing the LPMD’s of the

12


Hessian of the function Fn there). At all points where we got a value for Fn greater than 58, we made an additional sweep at their neighborhood of step-size 1/1000 (all these points where close the barycenter). Nowhere did we get a value greater than the value at the barycenter (about the algorithm we used to find the unique solution of the 4 × 45 system, the language of implementation and the hardware we used, see the last paragraphs of the Introduction). Therefore we conclude: Theorem 2. The chromatic number of random 5-regular multi-graph is 3 with uniformly positive probability.

4

Open Issues

There still remain two puzzling questions that we are working on: (i) Can the result be extended to be a.a.s?, and (ii) Can the result be extended to simple graphs? By well known results (see e.g. [11]), an affirmative answer to the first question implies an affirmative answer to the second one. It is plausible that an affirmative answer to the first question can be obtained by concentration results similar to those in [4].

Acknowledgement We wish to thank D. Achlioptas and C. Moore for their essential help at all phases of this research, without which we would not have obtained the results of this work. We are also thankful to the Barcelona Supercomputing Center and in particular to David Vicente for the help in running the optimization programs on the Mare Nostrum supercomputer.

References 1. M. Abramowitz and I. A. Stegun (eds.). Handbook of Mathematical Functions. (National Bureau of Standards, Applied Math. Series 55, 10th Printing, 1972). 2. D. Achlioptas and C. Moore. The asymptotic order of the random k-SAT threshold. In: Proc. 43th Annual Symp. on Foundations of Computer Science (FOCS), 126–127, 2002. 3. D. Achlioptas and C. Moore. Almost all graphs with degree 4 are 3-colorable. Journal of Computer and Systems Sciences 67(2), 441–471, 2003. 4. D. Achlioptas and C. Moore. The chromatic number of random regular graphs. In: Proc. 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) and 8th International Workshop on Randomization and Computation (RANDOM) (Springer, LNCS, 2004) 219–228. 5. D. Achlioptas and A. Naor. The two possible values of the chromatic number of a random graph. In: 36th Symposium on the Theory of Computing (STOC), 587–593, 2004. 6. R.H. Byrd, P. Lu and J. Nocedal. A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific and Statistical Computing 16(5), 1190–1208, 1995. 7. F. Krz¸akala, A. Pagnani and M. Weigt. Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs. Phys. Rev. E 70, 046705 (2004) 8. Random K-satisfiability: from an analytic solution to a new efficient algorithm. Phys.Rev. E 66, 056126 (2002) 9. M. Molloy. The Chromatic Number of Sparse Random Graphs. Master’s Thesis, University of Waterloo, 1992. 10. L. Shi and N. Wormald. Colouring random regular graphs Research Report CORR 2004-24, Faculty of Mathematics, University of Waterloo, 2004. 11. N.C. Wormald. Models of random regular graphs. In: J.D. Lamb and D.A. Preece, eds., Surveys in Combinatorics (London Mathematical Society Lecture Notes Series, vol. 267, Cambridge U. Press, 1999) 239–298.


13

Appendix Proof of Lemma (4) From the remarks preceding equation (15) we conclude that it is sufficient to maximize µ ¶ ¶ Xµ 5 ln Fi,j = d(i, j; s)(ln − ln d(i, j; s)) , (22) s s subject to the four constraints (for each fixed i, j): X Li,j,r,t = str d(i, j; s) − mj,t i,r = 0, r, t = −1, 1.

(23)

s

We have that

µ ¶ ∂ ln Fi,j 5 = ln − 1 − ln d(i, j; s), ∂d(i, j; s) s

and

∂Li,j,r,t = str , ∂d(i, j; s)

∀r, t.

Notice first that the function ln Fi,j is strictly convex. For each one of the four constraints in (23) a Langrange multiplier is introduced which will be denoted by µj,t i,r . Then: µ ¶ X j,t 5 ln − 1 − ln d(i, j; s) = µi,r str , s r,t which results (after some elementary calculations and harmless renaming of the Langrange multipliers) into: !str Ã ¡ 5¢ Y 1 s = . (24) d(i, j; s) µj,t i,r r,t Plugging (24) into (22) gives: Fi,j =

Y r,t

Ã

1 µj,t i,r

!mj,t i,r .

(25)

Finally, substituting the values obtained for each d(i, j; s) in (24) into the constraints in (23), we get the following 4 × 4 system of equations which provides the four variables µj,t i,r as functions of the four mj,t : i,r Ã µ ¶ ! X Y j,t str j,t t 5 mi,r = sr (µ ) . s r,t i,r s This system has a unique solution because of the strict convexity of ln Fi,j . The required follows by elementary computations. ¤ Proof of Lemma (6) We have computed in Lemma (5) the values of the derivatives: ∂ ln Fm ∂mj,t i,r

=

1 1 1 j,t + ln mj,t i,r − ln µi,r − . 2 2 5

The constraints that the variables of Fm are subject to are on one hand the 9 equations: X j,t mi,r − 5nji = 0, i, j = 0, 1, 2, Li,j =

(26)

(27)

r,t

for which we introduce the Lagrange multipliers xji , i, j = 0, 1, 2, and on the other the 18 equations: j+t,−t Li, j, t = mj,t i,1 − mi+r,−1 = 0, i, j = 0, 1, 2, t = −1, 1,

(28)

14


for which we introduce the Lagrange multipliers yij,t , i, j = 0, 1, 2, t = −1, 1. Observe that:

∂Li,j ∂mj,t i,r

= 1,

∂Li,j,t ∂mj,t i,1

∂L( i − 1, j + t, −t)

= 1,

∂mj,t i,−1

= −1.

(29)

Therefore by equations (26) and (29) and Langrange’s technique we get that: 1 1 1 j,t + ln mj,t = xji + yij,t , i,1 − ln µi,1 − 2 2 5 1 1 1 j,t j+t,−t + ln mj,t = xji − yi−1 . i,−1 − ln µi,−1 − 2 2 5 Getting rid of the logarithms and after some harmless renaming of the Langrange multipliers, we get that: 1 2 (mj,t i,1 ) = xji yij,t , j,t µi,1 2 (mj,t i,−1 ) 1

µj,t i,−1

j+t,−t −1 = xji (yi−1 ) .

or equivalently that:

2

2

2

j,t j j,t mj,t i,1 = (µi,1 ) (xi ) (yi ) , 2

2

−2

j,t j j+t,−t mj,t ) i,−1 = (µi,−1 ) (xi ) (yi−1

.

Combining the equations above with the constraints (27) and (28) finally yields: j,t j+t,−t j j+t mj,t i,r = µi,r µi+r,−r xi xi+r

(30)

Substituting the above expressions for the matching variables mj,t i,r to the equations (17) and (27) yields a 45 × 45 system which by the strict convexity of the function ln Fm has a unique solution. Then the maximum of Fm in terms of the solutions of this system can be easily shown to be Y j j (xi )5ni , i,j

which concludes the proof.

¤