The chromatic number of graph powers

Transcription

1 Combinatorics, Probability an Computing (19XX) 00, c 19XX Cambrige University Press Printe in the Unite Kingom The chromatic number of graph powers N O G A A L O N 1 an B O J A N M O H A R 1 Department of Mathematics, Tel Aviv University Tel Aviv, Israel Department of Mathematics, University of Ljubljana 1111 Ljubljana, Slovenia It is shown that the maximum possible chromatic number of the square of a graph with maximum egree an girth g is (1+o(1)) if g = 3, 4, 5 or 6, an is Θ( / log ) if g 7. Extensions to higher powers are consiere as well. 1. Introuction The square G of a graph G = (V, E) is the graph whose vertex set is V in which two istinct vertices are ajacent if an only if their istance in G is at most. What is the maximum possible chromatic number of G, as G ranges over all graphs with maximum egree an girth g? Our (somewhat surprising) answer is that for g = 3, 4, 5 or 6 this maximum is (1 + o(1)) (where the o(1) term tens to 0 as tens to infinity), whereas for all g 7, this maximum is of orer / log. To state this result more precisely, efine, for every two integers an g 3, f (, g) to be the maximum possible value of χ(g ) over all graphs with maximum egree an girth g. Since the maximum egree of G is at most + ( 1) =, it follows that for every g f (, g) + 1. (1) Equality hols in (1) for = an g 5, as shown by the 5-cycle, for = 3 an g 5, as shown by the Petersen graph, an for = 7 an g 5, as shown by the Hoffman-Singleton graph, see [HS]. Moreover, by Brooks Theorem (c.f., e.g., [Bo1]) it follows that equality can hol in (1) only for g 5 an only if there exists a -regular graph of iameter on + 1 vertices. As prove in [HS] if such a graph exists then Research supporte in part by a USA-Israel BSF grant an by a grant from the Israel Science Founation. Supporte in part by the Ministry of Science an Technology of Slovenia, Research Project J

2 N. Alon an B. Mohar {, 3, 7, 57}, an it is known to exist for {, 3, 7}, whereas the case = 57 is still open. It is also not ifficult to see that f (, g) = 4 for all g 6 as shown, for example, by the isjoint union of a g-cycle an a cycle of length l g where l is not ivisible by 3. In this short paper we prove the following. Theorem 1.1. (i) There exists a function ε() that tens to 0 as tens to infinity such that for all g 6 (1 ε()) f (, g) + 1. (ii) There are absolute positive constants c 1, c such that for every an every g 7 c 1 log f (, g) c log. Theorem 1.1 exhibits an interesting phase transition: As g grows from 3 to 6, f (, g) stays roughly the same, while it rops significantly when g increases from 6 to 7, an then it stays essentially the same as g keeps increasing. The rest of the paper contains the proof of the theorem an an extension of Theorem 1.1 for higher powers of graphs. Throughout the paper, all logarithms are in base e.. The upper bouns The proof of the upper bouns in parts (i) an (ii) of Theorem 1.1 are rather simple (given the main result in [AKS]). The upper boun in part (i) follows from (1). To prove the upper boun in (ii), we nee the following result of [AKS]. Theorem.1 ([AKS]). There is an absolute constant c such that for every integer an every real number t, t, the chromatic number of any graph H with maximum egree at most in which for every vertex v, the inuce subgraph on the set of all neighbors of v spans at most /t eges, satisfies χ(h) c log t. We can now prove that there is a constant c such that for all g 7 f (, g) c log. () Let G = (V, E) be a graph with maximum egree an girth g 7. Let H = G. Then the maximum egree of H is at most + ( 1) =. It is not ifficult to check that if u, v V are ajacent in H, then they have less than common neighbors in H. It follows that each neighborhoo of a vertex of H spans less than = 3 eges. (In fact, the inuce subgraph on each neighborhoo consists of at most + 1 ege-isjoint cliques of size at most each, implying it spans at most ( + 1)( 1)/ < 3 / eges.) Applying Theorem.1 with = an t =, we conclue that () hols.

3 The chromatic number of graph powers 3 3. The lower bouns In this section we prove the lower bouns in Theorem 1.1. The boun in part (i) is simple. If = q + 1 for some prime power q, then the bipartite incience graph of the lines an points of the finite projective plane of orer q form a -regular bipartite graph G on (q + q + 1) vertices. The girth of G is 6 an its square contains two cliques of size q + q + 1 = + 1 each, an hence χ(g ) + 1. (It is, in fact, easy to check that χ(g ) = + 1.) Remark: It is not ifficult to check that if G has girth 6 an maximum egree, then the maximum clique in G is of size at most + 1. To see this, fix a vertex v in a maximum clique of G. Then, in G, v is connecte to the set N(v) of its neighbors in G, where N(v), an to the set N (v) of the vertices at istance from v, where N (v). Since the girth is 6, any member of N(v) is connecte in G to at most 1 members of N (v). It follows that if the clique contains more than 1 members of N (v) then it contains no member of N(v). Thus, the size of the clique is at most 1 +. The main result of Ree in [Re] asserts that there is some absolute positive constant ɛ such that for every graph H with maximum egree, maximum clique size ω an chromatic number χ, the inequality χ ɛω + (1 ɛ)( + 1) hols. This implies that if G has girth 6 an maximum egree > then χ(g ) Ω(), showing that the example above is nearly tight. Returning to the proof of the lower boun in part (i) of Theorem 1.1 for general, we simply choose the largest prime power q satisfying q + 1, an take the graph constructe above for q together with some extra penant eges to make sure that the maximum egree is precisely. By the known results on the istribution of primes (see, e.g., [Hu]), q + q + 1 (1 ε()), where ε() tens to 0 as tens to infinity. This proves the lower boun in Theorem 1.1 part (i) for g = 6. The bouns for g = 3, 4, 5 follow by simply aing to the example for g = 6 a vertex isjoint copy of a cycle of length g. It remains to prove the lower boun in Theorem 1.1(ii). This is one by a probabilistic construction. Note that we may assume, without loss of generality, that is sufficiently large (by choosing c 1 > 0 sufficiently small), an we thus assume, from now on, that is large (for example, will be enough). We also omit all floor an ceiling signs whenever they are not crucial, to simplify the presentation. Let n be a large integer (n >> g ), let V = {1,..., n}, efine p = n, an let G = (V, E ) be a ranom graph on V obtaine by choosing each pair of istinct elements of V, ranomly an inepenently, to be an ege with probability p. The expecte value of each vertex egree in G is n (n 1) <. However, G will have, with high probability, about n Θ() vertices of egree higher than. It will also have, with high probability, some cycles of length less than g (fewer than O( g ) of them). By omitting all vertices of egree > an by removing an arbitrarily chosen vertex from each cycle of length < g we get a graph of girth g an maximum egree at most. (If necessary, we can a a cycle of length g an some penant eges to make sure that the maximum egree is equal to an that the girth is precisely g.) As we show next, with high probability, for the graph G obtaine in this manner, G

4 4 N. Alon an B. Mohar oes not contain an inepenent set of size bigger than c n log for an appropriately chosen constant c > 0, an hence χ(g ) Ω( / log ), as neee. The etaile proof is given in the following claims (where we make no attempt to optimize the absolute constants). Claim 3.1. The expecte number of vertices of egree > in G is at most n /10. Thus, with probability 0.9, there are at most 10n /10 such vertices. Proof. The egree of any fixe vertex is a binomial ranom variable with parameters n 1 an p = n, an hence its expecte egree is less than /. By the stanar estimates for binomial istributions (see, e.g., [AS, Appenix A]), the probability that the egree of such a vertex excees is smaller than /10. By linearity of expectation, the expecte number of vertices of egree > is thus at most n /10, an hence, by Markov s inequality, the probability that there are more than 10n /10 such vertices is at most 0.1. (In fact, this probability is exponentially small, but this is not neee here.) Claim 3.. The expecte number of cycles of length < g in G is g 1 i=3 n(n 1) (n i + 1) ( ) i 1 < i n g 1 ( ) i < g, an hence with probability 0.9, there are at most 10 g such cycles. i=3 Proof. Straightforwar. Claim 3.3. There exists a constant c, 0 < c < 1000, such that almost surely (that is, with probability that tens to 1 as n tens to infinity) the following hols: For every set of vertices U V of carinality U = c n 1 log, there are at least 30 c n log vertices outsie U that have exactly neighbors in U. Proof. Put x = c n log. Fix a set U, U = x, an fix a vertex w V \ U. Note that, as is large, V \ U 3n 4. The probability of the event E w that w has precisely two neighbors in U is ( ) x q = p (1 p) x x(x 1) ( ) ( 1 ) x n n x(x 1) ( ) ( n 1 x n ) 1 10 c log. The number of vertices w V \ U is at least 3n 4, an as the events E w are mutually inepenent, the number of vertices w V \ U that have neighbors in U is a binomial ranom variable with parameters V \ U 3n 4 an q 1 10 c log. The expecte value of this number is at least 3 40 c n log an hence, by the stanar estimates for binomial istributions (see, e.g., [AS, Appenix A]), the probability that it is smaller than 1 30 c n log is at most, say, e c n log.

5 The chromatic number of graph powers 5 Therefore, for each fixe set U, the probability that there are less than 1 30 c n log vertices w V \ U with precisely neighbors in U is at most e 1 number of sets U is ( ) ( ) ( ) n n e c n log = x c n e 3c n log c log log 300 c n log. As the total c it follows that if 300 > 3c, then with probability 1 o(1) every set U has at least that many vertices w with neighbors in U. This completes the proof of the claim. We can now complete the proof of Theorem 1.1. For any given g an (large), choose n >> g. By Claims , there exists a graph G on n vertices satisfying the conclusions of all three claims. Let G be the graph obtaine from G by omitting all vertices of egree > an an arbitrarily chosen vertex from each cycle of length < g. Then G has girth g, maximum egree, an more than n/ vertices. As G satisfies the conclusion of Claim 3.3, an as the total number of vertices omitte (which is at most 10n / g ) is smaller than c, it follows that G contains no inepenent 30 n log set of size x = c n log. Thus χ(g ) c log. By aing to G, if neee, penant eges an a isjoint copy of a cycle of length g we conclue that f (, g) c log for all, g, completing the proof of Theorem Higher powers The k th power G k of a graph G = (V, E) is the graph whose vertex set is V in which two istinct vertices are ajacent if an only if their istance in G is k. Note that G 1 = G. Let f k (, g) enote the maximum possible value of χ(g k ), as G ranges over all graphs of maximum egree an girth g. Trivially, f 1 (, 3) = + 1 for all. By the theorem of Johansson [Jo] an the known results about chromatic numbers of ranom graphs ([Bo, KM]), there are absolute constants c 1, c such that for every an every g 4 c 1 log f 1(, g) c log. The behavior of f k (, g) for k = is etermine by Theorem 1.1. For higher values of k, the situation is less clear, though the main parts of the proof of Theorem 1.1 can be extene. If G has maximum egree, then the maximum egree of G k is at most Therefore, for every fixe k, + ( 1) + + ( 1) k 1 = f k (, g) (( 1)k 1). (( 1)k 1) + 1 = O( k ).

6 6 N. Alon an B. Mohar The known constructions of large graphs with a given maximum egree an iameter at most k show that for every k there exists a constant c k > 0 such that for some small values of g f k (, g) c k k. For example, the DeBruijn graphs show that f k (, 3) / k, an the existence of generalize m-gons imply similar estimates for somewhat larger values of g. See also the constructions in [BDF], [LU] an their references. Finally, the proof in Section an the ranom construction escribe in Section 3 can be extene to prove the following Theorem 4.1. There exists an absolute constant c > 0 such that for all integers k 1, an g 3k + 1 f k (, g) c k k log. For every integer k 1 there exists a positive number b k such that for every an g 3 k f k (, g) b k log. The proof in Section easily extens to a proof of the upper boun in Theorem 4.1, an the etails are left to the reaer. The proof of the lower boun follows the basic approach in Section 3, but requires somewhat more sophisticate tools in orer to show that the ranom graph G satisfies the following property almost surely: For an appropriately chosen constant c k, for every set U V n of size U = c k log, there are at least Ω(c k k n log ) internally vertex k isjoint paths of length k, both whose enpoints lie in U, an whose other vertices lie in V \ U. This can be prove using Talagran s Inequality [Ta] (cf. also [AS], Chapter 7) an implies the assertion of Theorem 4.1 following the reasoning of the proof in Section 3. We procee with the etails. Throughout the proof we assume, whenever this is neee, that is sufficiently large as a function of k, an omit, as before, all floor an ceiling signs whenever these are not crucial. Let V = {1,,..., n}, suppose n >> g, an let G = (V, E ) be a ranom graph on V obtaine by choosing each pair of istinct elements of V, ranomly an inepenently, to be an ege with probability p = n. Define x = c k n log, where c k k > 0 will be chosen later. Lemma 4.1. For an appropriate choice of c k, the following hols almost surely. For every set of vertices U V of carinality U = x, there are at least c k n log internally k+5 k vertex isjoint paths of length k, both of whose enpoints lie in U, an whose other vertices lie in V \ U.

7 The chromatic number of graph powers 7 Proof. We shall apply Talagran s Inequality. To o so, fix a set U V, U = x. Call any path of length k, whose enpoints lie in U an whose internal vertices lie outsie U, a U-path. Let X = X(G ) be the ranom variable counting the maximum number of internally vertex isjoint U-paths. We first show that the expecte value of X satisfies E(X) c k n log k+ k. (3) Inee, the expecte number of U-paths is ( ) x µ = (n x)(n x 1) (n x k + )p k > 0.49 c k nk+1 k log p k = 0.49 c k n k k log, where here we use the fact that is sufficiently large. Let enote the expecte number of pairs of U-paths that share at least one common internal vertex. We claim that, as n >> >> k, < µ/3 (with room to spare). Inee, the expecte number of such pairs that share only one enpoint an its unique neighbor is at most µn k xp k 1 = µc k log k 1. It is not ifficult to check that for n >> >> k, the expecte number of pairs of internally intersecting U-paths of any other type is much smaller, an the number of types is boune by a function of k. By omitting an arbitrarily chosen path from each pair of internally intersecting U- paths we get a collection of internally pairwise vertex isjoint U-paths. It follows, by linearity of expectation, that E(X) µ, implying (3). To apply Talagran s Inequality (in the form presente, for example, in [AS, Chapter 7]), note that X is a Lipschitz function, i.e., X(G ) X(G ) 1 if G, G iffer in at most one ege. This is because X counts internally vertex isjoint paths, which are ege isjoint, an hence no single ege can change the value of X by more than 1. Note also that X is f-certifiable for f(s) = ks, that is, when X(G ) s there is a set of at most ks eges of G so that for every graph G that agrees with G on these eges, X(G ) s. By Talagran s Inequality we conclue that for every b an t P r[x b t kb] P r[x b] e t /4. (4) Let B enote the meian of X. Without trying to optimize the constants, we prove that B c k n log k+4 k. (5) Inee, assume this is false an apply (4) with b = c k n log an t = 1 k+3 k that b k to conclue P r[x b b ] P r[x b] e 16k. Since, by assumption, B b, it follows that P r[x b/] 1/, an hence P r[x b]

8 8 N. Alon an B. Mohar e b 16k. However, as X(G ) < n for every G, this implies that E(X) b + ne b 16k contraicting (3) an thus proving (5). = b + ne c k n log k+7 k k = b + o(1) = c k n log k+3 k + o(1), We can now apply (4) again with b = c k n log k+4 k an t = 1 By (5), P r[x b] 1/, an hence P r[x b b ] P r[x b] e 16k. P r[x b ] e b 16k b k = e c k n log k+8 k k. to conclue that We have thus prove that for every fixe U, the probability that there are less than b = c k n log internally vertex isjoint U-paths is at most e c k n log k+5 k k+8 k k. Since the total number of sets U is at most ( ) n ( en ) ( ) n x e k ck k log e c kk n x x c k log k log it follows that if, say c k k+8 k > kc k then with probability 1 o(1) for every set U there are at least c k n log k+5 k pairwise internally vertex isjoint U-paths. This completes the proof of the lemma. We can now complete the proof of Theorem 4.1. For any given g an k, an any (large) an sufficiently large n there exists a graph G satisfying the conclusions of Claim 3.1, Claim 3. an Lemma 4.1. Let G be the graph obtaine from G by omitting all vertices of egree greater than an an arbitrarily chosen vertex from each cycle of length < g. Then G has girth g, maximum egree, an more than n/ vertices. As G satisfies the conclusion of Lemma 4.1, an as the total number of vertices omitte from it to create G is at most 10n / g < c k n log k+5 k, it follows that G k n log contains no inepenent set of size x = c k. Therefore χ(g k ) k k c k log. By aing to G, if neee, penant eges an a isjoint copy of a cycle of length g this implies that for an appropriately efine b k > 0, k f k (, g) b k log for all, g, completing the proof of Theorem 4.1. Note that for k = 1 an, the function f k (, g) exhibits a rastic change when g

9 The chromatic number of graph powers 9 changes from 3k to 3k + 1. It woul be interesting to ecie if this is the case for higher values of k as well. The results mentione in this section o suggest that for every k there is some integer g k such that f k (, g) = Θ k ( k ), for g g k an f k (, g) = Θ k ( k / log ), for g > g k. At the moment, however, we are unable to prove the existence of such a g k. Acknowlegement. Part of this work was one uring a visit at the workshop on Graph Coloring organize by Pavol Hell in the PIMS Institute at the Simon Fraser University in July, 000. We woul like to thank Pavol an the other hosts at PIMS for their hospitality. References [AS] N. Alon an J. H. Spencer, The Probabilistic Metho, Secon Eition, Wiley, New York, 000. [AKS]N. Alon, M. Krivelevich, an B. Suakov, Coloring graphs with sparse neighborhoos, J. Combin. Theory Ser. B 77 (1999) [BDF] J.-C. Bermon, C. Delorme, an G. Farhi, Large graphs with given egree an iameter, II, J. Combin. Theory Ser. B 36 (1984) [Bo1]B. Bollobás, Extremal Graph Theory, Acaemic Press, Lonon an New York, [Bo]B. Bollobás, Chromatic number, girth an maximal egree, Discrete Math. 4 (1978), [HS] A. J. Hoffman an R. R. Singleton, On Moore graphs with iameters an 3, IBM J. Res. Develop. 4 (1960) [Hu] M. N. Huxley, On the ifference between consecutive primes, Invent. Math. 15 (197) [Jo] A. R. Johansson, Asymptotic choice number for triangle free graphs, to appear. [KM]A. V. Kostochka an N. P. Mazurova, An inequality in the theory of graph coloring (in Russian), Metoy Diskret. Analiz. 30 (1977) 3 9. [LU]F. Lazebnik an V. A. Ustimenko, New examples of graphs without small cycles an of large size, European J. Combin. 14 (1993) [Re] B. Ree, ω,, an χ, J. Graph Theory 7 (1998) [Ta] M. Talagran, Concentration of measure an isoperimetric inequalities in prouct spaces, Inst. Hautes Étues Sci. Publ. Math. 81 (1995)