Chromatic number for a generalization of Cartesian product graphs

Transcription

1 Chromatic number for a generalization of Cartesian prouct graphs Daniel Král Douglas B. West Abstract Let G be a class of graphs. The -fol gri over G, enote G, is the family of graphs obtaine from -imensional rectangular gris of vertices by placing a graph from G on each of the lines parallel to one of the axes. Thus each vertex belongs to of these subgraphs. Let f(g; = max G G χ(g. If each graph in G is k-colorable, then f(g; k. We show that this boun is best possible by proving that f(g; = k when G is the class of all k-colorable graphs. We also show that f(g; at most one ege, an f(g; with maximum egree 1. 6log 6log when G is the class of graphs with when G is the class of graphs 1 Introuction The Cartesian prouct of graphs G 1,...,G is the graph with vertex set V (G 1 V (G in which two vertices (v 1,...,v an (v 1,...,v are Institute for Theoretical Computer Science, Faculty of Mathematics an Physics, Charles University, Prague, Czech Republic; kral@kam.mff.cuni.cz. The Institute for Theoretical Computer Science (ITI is supporte by the Ministry of Eucation of the Czech Republic as project 1M0545. Mathematics Department, University of Illinois, Urbana, IL; west@math.uiuc.eu. Research partially supporte by the National Security Agency uner Awar No. H

2 ajacent if they agree in all but one coorinate, an in the coorinate where they iffer the values are ajacent vertices in the corresponing graph. The prouct can be viewe as a rectangular gri with copies of G 1,..., G place on vertices forming lines parallel to the axes. It is well-known (an easy to show that the chromatic number of the Cartesian prouct of G 1,...,G is the maximum of the chromatic numbers of G 1,..., G. In this paper, we consier bouns on the chromatic number of graphs in a family resulting from a more general graph operation. Instea of placing copies of the same graph G i on all the lines parallel to the i-th axis, we may place ifferent graphs from a fixe class. Let [n] enote {1,..., n}. For a class G of graphs, the -fol gri over G, enote G, is the family of all graphs forme by choosing a vertex set of the form [n 1 ] [n ] an letting each set of vertices where all but one coorinate is fixe inuce a graph from G. For example, a Cartesian prouct of graphs in G belongs to G. The stuy of the chromatic number an inepenence number of graphs in G is motivate by an application in computational geometry. Frequency assignment problems for transmitters in the plane are moele by coloring an inepenence problems on certain graphs (see [, 6]. These graphs arise from sets of points using the Eucliean metric. When such problems are stuie using the Manhattan metric (see [4], results about gri structures give bouns for the number of frequencies neee. Motivate by such problems, Szegey [13] pose the following open problem at the workshop Combinatorial Challenges : What is the maximum chromatic number of a graph G G when G is the class B of all bipartite graphs or when G is the class S of graphs containing at most one ege? If each graph of G is k-colorable, then every graph in G has chromatic number at most k, since it is the union of subgraphs, each of which is k- colorable. In particular, all graphs in B are -colorable. We show that this boun is sharp, which is somewhat surprising since Cartesian proucts of bipartite graphs are bipartite. More generally, let f(g; = max G G χ(g. We show that if G is the class of all k-colorable graphs, then f(g; = k. We prove the existence of k -chromatic graphs in G probabilistically, but an

3 explicit construction can then be obtaine by builing, for each n, a graph in G that is universal in the sense that it contains all graphs in G with vertex set [n]. This settles the first part of Szegey s question. Determining f(s; is more challenging. Since the maximum egree of a graph in S oes not excee, an these graphs o not contain K +1, Brooks Theorem [3] implies that each graph in S is -colorable (when 3. Also graphs in S are bipartite, since cycles in such a graph alternate between horizontal an vertical eges. When is large, we can use a refinement of Brook s Theorem obtaine by Ree et al. [7, 10, 11, 1] to improve the upper boun. Theorem 1 (Molloy an Ree [11]. There exists a constant D 0 such that for all D D 0 an k satisfying k + k < D, every graph G with maximum egree D an χ(g > D k has a subgraph H with at most D + 1 vertices an χ(h > D k. Theorem 1 implies that f(s; + o(1. A still better upper boun follows from another result. Theorem (Johansson [9]. The chromatic number of a triangle-free graph with maximum egree D is at most O(D/ logd. This result, which was further strengthene by Alon et al. [1], implies that f(s; O(/ log. We show that though the graphs containe in S are very sparse, an it is natural to expect that the graphs containe in S can be colore just with a few colors, f(s; 6log. Our argument is again probabilistic. A similar argument yiels f(m;, where M is the class of all matchings 6log (i.e., graphs with maximum egree 1. This lower boun is asymptotically best possible, since the iscussion above yiels f(m; O(/log. Preliminaries In this section, we make several observations use in the proofs of our subsequent lower bouns on f(g; for various G. We start by recalling the 3

4 Chernoff Boun, an upper boun on the probability that a sum of inepenent ranom variables eviates greatly from its expecte value (see [8] for more etails. Proposition 3. If X is a ranom variable equal to the sum of N inepenent ientically istribute 0, 1-ranom variables having probability p of taking the value 1, then the following hols for every 0 < δ 1: Prob(X (1 + δpn e δ pn 3 an Prob(X (1 δpn e δ pn. Next, we establish two technical claims. We begin with a stanar boun on the number of subsets of a certain size. Proposition 4. For α >, the number of N/α-element subset of an N- element set is at most N α (1+log α. Proof. An N-element set has ( N N/α subsets of size N/α. It is well known, see e.g. [5], that ( N N/α N H(1/α, where H(p = p log p (1 p log(1 p (H(p is calle the entropy function. A simple calculation yiels the upper boun: ( N N ( 1 α 1 log α+ α α log α α 1 N ( 1 α 1 log α+ α α 1 α 1 N (1+log α α N/α The secon claim is a straightforwar upper boun on a certain type of prouct of expressions of the form (1 ε: Proposition 5. If a 1,...,a m are nonnegative integers with sum n, then m ( ( ( ai n m x x i=1 for any positive real x such that x ( max i a i. Proof. Since a i = n, it suffices to show that ( ( ( a a 1 (1 x x 4

5 for every nonnegative integer a. If a 1, then the left sie of (1 is 1 an the right sie is at least 1. If a, then (1 follows (by setting k = a 1 from the well-known inequality which hols whenever 0 k x. 1 k x ( 1 1 x k, 3 Proucts of k-colorable graphs In this section, we prove that f(b; =. Note again that after the probabilistic proof of existence, we can construct such graphs explicitly as explaine in Introuction. Even so, the argument that they are not ( 1- colorable remains probabilistic. We prove the result in the more general setting of k-colorable graphs. Theorem 6. For, k N, the -fol gri over the class of k-colorable graphs contains a graph with chromatic number k. Proof. The claim hols trivially if = 1, so we assume. The integers k an are fixe, an N is a large integer to be chosen in terms of k an. Consier the set [N]. For each v [N], efine a ranom -tuple X(v such that X(v i takes each value in [k] with probability 1/k, an the coorinate variables are inepenent. Generate a graph G with vertex set [N] by making two vertices u an v ajacent if they iffer in exactly one coorinate an X(u l X(v l, where l is the coorinate in which u an v iffer. By construction, any set of vertices in G that all agree outsie a fixe coorinate inuce a complete multipartite graph with at most k parts. Hence G is in the -fol gri over the k-colorable graphs. It will suffice to show that almost surely (as N tens to infinity G oes not have an inepenent set with more than N vertices. This yiels χ(g k 0.5 k, since otherwise some color class woul be an inepenent set of size at least N. k 1 For an inepenent set A in G, let the shae of A be the function σ: [] [N] 1 [k] efine as follows. For z = (l; i 1,..., i l 1, i l+1,...,i [] 5

6 [N] 1, consier the vertices in A of the form (i 1,...,i l 1, j, i l+1,...,i. By the construction of G, the value of X(v l is the same for each such vertex v, since vertices of A are nonajacent. Let this value be σ(z. If there is no vertex of A with this form, then let σ(z = 1. The union of inepenent sets with the same shae is an inepenent set. Hence for each function σ there is a unique maximal inepenent set in G with shae σ; enote it by A σ. In orer to have v A σ, where v = (i 1,...,i, the ranom variables X(v 1,...,X(v must satisfy X(v l = σ(l; i 1,...,i l 1, i l+1,...,i. Hence each v lies in A σ with probability k. As a result, the expecte size of A σ is (N/k. Since the variables X(v l are inepenent for all v an l, we can boun the probability that A m N k 0.5 this yiels using the Chernoff Boun (Proposition 3. Applie with δ = 1 k 1, Prob ( A m N k 0.5 N e 3(k 1 k e N 1k 3. Since there are k N 1 possible shaes, the probability that some inepenent set has more than N k 0.5 vertices is at most kn 1 e N /1k 3, an we compute k N 1 e N /1k 3 = e log kn 1 N /1k 3 0. If N is sufficiently large in terms of k an, then the boun is less than 1, an there exists such a graph G with no inepenent set of size at least N k Proucts of single-ege graphs In this section, we prove the lower boun for the -fol gri over the class of graphs with at most one ege. Theorem 7. For, there exists G S such that χ(g. Proof. Let k = 6log 6log. For k, the conclusion is immeiate. Hence, we assume k 3. We generate a graph G with vertex set [k]. For 6

7 (l; i 1,...,i l 1, i l+1,...,i [] [k] 1, choose a ranom pair of istinct integers j an j from [k], an make the vertices (i 1,...,i l 1, j, i l+1,...,i an (i 1,...,i l 1, j, i l+1,...,i ajacent in G. The choices of j an j are inepenent for all elements of [] [k] 1. Since G S, its chromatic number is at most. To show that χ(g is at least k with positive probability, it suffices to show that with positive probability, G has no inepenent set of size at least (k /k. Consier a set A in V (G with size (k /k; we boun the probability that A is an inepenent set in G. Again we think of an element z in [] [k] 1 as esignating a line in [k] parallel to some axis. Let A[z] be the intersection of A with this line. By the construction of G, the probability that some two vertices in A[z] are ajacent in G is ( A[z] 1 / ( k which is at most 1 ( A[z] 1. By applying Proposition 5 with x = 1/k, n = k A (k = (k 1, an m = (k 1, we conclue that the probability of k all subsets of A lying along lines in a particular irection being inepenent in G is at most z {l} [k] 1 ( 1 ( A[z] 1 k, ( 1 1 (k 1. k Let p be the probability that A is an inepenent set in G. Since the eges in each of the irections are ae to G inepenently, p ( 1 1 (k 1 k e (k 1 k e (k 3 (k 3. We want to show that with positive probability, G has no inepenent set of size (k /k. Let M be the number of subsets of V (G with size (k /k. By Proposition 4, M (k k (1+log k (k 1 (1+log k 3(k 1 log k. 7

8 Therefore, we boun the probability that G has an inepenent set of size (k /k by the following computation: 3(k 1 log k (k 3 = (k 3 (6k log k < 1. The last inequality uses the fact that 6k log k is negative, by the choice of k. We conclue that some such graph G has no inepenent set of size at least (k /k. 5 Proucts of matchings Finally, we consier the -fol gri over the class M of matchings. Theorem 8. For, there exists G M such that χ(g 6log Proof. As the proof is similar to the proof of Theorem 7, we will give less etail an focus on the ifferences between the proofs. Set k = an 6log assume k 3. We ranomly generate a graph G with vertex set [k]. As before an element z in [] [k] 1 esignates a line in [k] parallel to some axis. We place a ranom perfect matching on the k vertices in each such line. Hence, the resulting graph G is -regular. It suffices to show that with positive probability, G has no inepenent set of size at least (k /k. Consier a set A in V (G with size (k /k; we boun the probability that A is an inepenent set in G. Let A[z] be the intersection of A with a line esignate by z [] [k] 1. Let X be the ranom variable that is the number of eges in G inuce by A[z]. By the construction of G, we ( A[z] have E(X = 1 k 1 variable, Prob[X 1]. When X is a nonnegative integer-value ranom E(X. Since A[z] cannot inuce more than A[z] / max(x eges, we boun the probability p that A[z] contains an ege by computing p ( A[z] A[z] / = A[z] 1 k 1 1 k 1 A[z] 1. k Let q l enote the probability that all subsets of A lying along lines in irection l are inepenent (each such line consists of -tuples that agree outsie the. 8

9 lth coorinate. We compute ( q l 1 A[z] 1 k z {l} [k] 1 = e (k /k (k 1 k = e (k. z {l} [k] 1 e A[z] 1 k The probability P that A is inepenent can now be boune as follows: P e (k (k. Finally, an upper boun on the probability that G has an inepenent set of size (k /k is obtaine by multiplying the boun on P an the boun on the number of (k /k-element subsets of vertices from Proposition 4. 3(k 1 log k (k = (k (6k log k < 1. Acknowlegement This research was conucte uring the DIMACS, DIMATIA an Renyi tripartite workshop Combinatorial Challenges hel in DIMACS in April 006; we thank the DIMACS center for its hospitality. We also thank Stephen Hartke an Hemanshu Kaul for participating in lively early iscussions at DIMACS about the problem. References [1] N. Alon, M. Krivelevich, B. Suakov: Coloring graphs with sparse neighborhoos, J. Combinatorial Theory Ser. B 77(1 (1999, [] M. e Berg, M. van Krevel, M. Overmars, O. Schwarzkopf: Computational geometry, algorithms an applications, Springer-Verlag, Berlin,

10 [3] R. L. Brooks: On coloring of the noes of a network, Proc. Cambrige Phil. Soc. 37 (1941, [4] X. Chen, J. Pach, M. Szegey, G. Taros: Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles, in: Proc. 19th annual ACM-SIAM symposium on Discrete algorithms (SODA 008, [5] T. M. Cover an J. A. Thomas: Elements of information theory, John Wiley & Sons, [6] G. Even, Z. Lotker, D. Ron, S. Smoroinsky: Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks, SIAM J. Comput. 33 (003, [7] B. Farza, M. Molloy, B. Ree: ( k-critical graphs, J. Combinatorial Theory Ser. B 93 (005, [8] T. Hagerup, Ch. Rüb: A guie tour of Chernoff bouns. Inform. Process. Letters 33 ( [9] A. R. Johannson: Asymptotic choice number for triangle-free graphs, manuscript. [10] M. Molloy, B. Ree: Colouring graphs whose chromatic number is near their maximum egree, in: Proc. 3r Latin American Symposium on Theoretical Informatics (LATIN 1998, [11] M. Molloy, B. Ree: Colouring graphs when the number of colours is nearly the maximum egree, in: Proc. 33r annual ACM symposium on Theory of computing (STOC 001, [1] B. Ree: A strengthening of Brooks theorem, J. Combinatorial Theory Ser. B 76 (1999, [13] M. Szegey, presentation at open problem session of DIMACS, DIMA- TIA an Renyi tripartite workshop Combinatorial Challenges, DI- MACS, Rutgers, NJ, April