Ramsey numbers of some bipartite graphs versus complete graphs

Transcription

1 Ramsey numbers of some bipartite graphs versus complete graphs Tao Jiang, Michael Salerno Miami University, Oxfor, OH 45056, USA Abstract. The Ramsey number r(h, K n ) is the smallest positive integer N such that every graph of orer N contains either a copy of H or an inepenent set of size n. The Turán number ex(m, H) is the maximum number of eges in a graph of orer m not containing a copy of H. We prove the following two results: () Let H be a graph obtaine from a tree F of orer t by aing a new vertex w an joining w to each vertex of F by a path of length k such that any two of these paths share only w. Then n r(h, K n ) c +/k k,t, where c ln /k n k,t is a constant epening only on k an t. This generalizes some results in [], [3], an [6]. (2) Let H be a bipartite graph with ex(m, H) = O(m γ ), where < γ < 2. Then r(h, K n ) c H ( n ln n) /(2 γ), where c H is a constant epening only on H. This generalizes a result in [4]. Key wors. Ramsey number, inepenence number. Introuction Let H, F be graphs without isolate vertices. The Ramsey number r(h, F ) is the smallest positive integer such that for each graph G on N vertices, either G contains H as a subgraph or its complement G contains F as a subgraph. For the purpose of this paper, we are only concerne with r(h, K n ), which can be interprete as the smallest positive integer N such that each graph on N vertices either contains H as a subgraph or has ) (m+)/2 ( inepenence number at least n. For m 3, the lower boun r(k m, K n ) c n m ln n was establishe by Spencer [4], where ln n is the natural logarithm function. For m = 3, Kim [9] obtaine the asymptotically sharp lower boun r(k 3, K n ) cn 2 / ln n. In general, Erős et al. [5] an later Krivelevich [0] showe that if H is a graph with p vertices an q eges, then r(h, K n ) (c o()) ( ) n (q )/(p 2), ln n as n. Li an Zang [2] generalize this lower boun by replacing K n with a sequence of ense graphs. For the upper bouns, Erős et al. prove that r(c m, K n ) c(m)n +/k, where k = m/2 an c(m) is a constant epening on m. This boun was improve by Caro et Research support in part by National Security Agency uner grant number H Dept. of Mathematics an Statistics, Miami University, Oxfor, OH 45056, USA. jiangt@muohio.eu. Dept. of Mathematics an Statistics, Miami University, Oxfor, OH 45056, USA. salernmj@muohio.eu.

2 2 Tao Jiang, Michael Salerno al [4] for even m to r(c 2k, K n ) c(k) (n/ ln n) k/(k ). Suakov [6] an inepenently Li an Zang [3] obtaine a similar improvement for o m, showing that r(c 2k+, K n ) c(k)n +/k / ln /k n. Li an Rousseau [] prove that if F is a tree with m eges then r(k + F, K n ) (m + o())n 2 / ln n, where K + F is the graph obtaine from F by aing a new vertex w an joining w to each vertex of F by an ege. Here we generalize these results by proving Theorem. Let n, k, t be positive integers. Let F be a tree with t vertices an H a graph obtaine from F by aing a new vertex w an joining w to each vertex of F by a path of length k such that any two of these paths share only w. Then r(h, K n ) c k,t n +/k / ln /k n, as n, where c k,t is a constant epening only on k an t. By taking F = K 2, Theorem yiels the above-mentione upper boun on r(c 2k+, K n ). By taking F to be any tree an k =, Theorem yiels the correct orer of magnitue in the above-mentione upper boun on r(k + F, K n ). So, Theorem irectly generalizes the result in [6] an [3], but it only generalizes the result in [] in a loose sense. 2. Proof of Theorem In this section we prove Theorem. We follow the approach of [6] an [3]. Another key ingreient of our proof is a simple but useful lemma (Lemma 2) which potentially can be applie elsewhere. We will assume whenever it is neee that n is sufficiently large. To keep the presentation concise we often choose multiplicative constants out of convenience, thus they are not optimal. We make no attempt to optimize these constants. We first nee the following well-known boun ([3], Chapter 2, Lemma 5). See also [6]. Proposition. ([3]) Let G be a graph on n vertices with average egree at most an let h be the number of triangles in G. Then G contains an inepenent set of orer at least 0. n (ln /2 ln(h/n)). It is well known that the inepenence number α(g) of a graph G with n vertices an maximum egree is lower boune by Ω(n/). Proposition can be applie to locally sparse graphs to give an improve lower boun of Ω(n ln /), where loosely speaking a graph G is locally sparse if on average the neighborhoo of a vertex spans o( 2 ) eges. See [] for a etaile stuy of locally sparse graphs. Also, see [8] for an application to coing theory. The following simple fact is frequently use in the literature. We inclue a proof for completeness. Lemma. Let T be a tree with t eges. If L is a n-vertex graph that oes not contain T as a subgraph then e(l) tn(l). Proof. Suppose otherwise that e(l) > tn(l). Let L enote a smallest subgraph of L satisfying e(l ) > tn(l ). Then the minimality of L implies that L has minimum egree

3 Ramsey numbers of some bipartite graphs versus complete graphs 3 at least t. It is well-known that every graph with minimum egree at least t contains every tree on t eges. Thus, L contains T, contraicting L not containing T. Proposition an Lemma reaily imply the following. Corollary. Let H be a graph specifie in Theorem. Suppose G is a graph on n vertices with maximum egree which oes not contain H as a subgraph. Then the inepenence number α(g) of G satisfies α(g) 0.05 n (ln ln kt). Proof. Let v be any vertex in G. Consier the subgraph L v of G inuce by the neighborhoo N(v) of v. Note that n(l v ) = (v). Since G oes not contain H as a subgraph, L v oes not contain F = H w. By our efinition of H, F is a tree on t + t(k ) = kt vertices (an kt eges). By Lemma, e(l v ) kt n(l v ) kt. Now, if we sum up e(l v ) over all vertices v, we woul count each triangle in G three times, since in a triangle abc, we can esignate any one of a, b, c as w. Hence, the total number h of triangles in G satisfies h = v e(l v )/3 ktn/3. It follows from Proposition that α(g) 0. n (ln /2 ln(h/n)) 0.n (ln /2 ln(kt)) 0.05n (ln ln kt). The following lemma is key to our proof of Theorem. It generalizes the classical boun of Wei an inepenently of Caro on the inepenence number in terms of vertex egrees. A weaker version of the lemma was first prove by Haxell, Loh an Mubayi [7], followe by the first author of this paper. Suakov [5] observe an prove the lemma in its current form. Even though the weaker version woul suffice for our purpose, we present Suakov s version because it is stronger (best possible) an the proof is short an elegant. Lemma 2. ([5]) Let G be an n-vertex graph together with a proper vertex-coloring c. For each vertex v, let c (v) enote the number of ifferent colors appearing on the neighbors of v. Let D = max v V (G) c (v). Then α(g) v V (G) + c (v) n D +. Proof. Suppose the colors use by c are,..., p. Consier a ranom permutation σ of {,..., p}. Let X enote the set of vertices x whose color precees the colors of all of its neighbors in σ. Let E( X ) enote the expecte size of X. For each v V (G), the probability of v belonging to X is precisely + c(v) linearity of expectation, we have E( X ) = v V (G). Using inicator ranom variables an. So, there exists at least one + c(v) permutation of {,..., p} for which the corresponing X has size at least v V (G) Fix such an X. Suppose X contains two ajacent vertices u an v. Then by the efinition of X we must have both c(u) < c(v) an c(v) < c(u), which is impossible. So X is an inepenent set in G of size at least v V (G) + c(v). This proves the first inequality in the lemma. The secon inequality is trivial.. + c(v) If we take c to be the trivial coloring that gives each vertex of G a istinct color then Lemma 2 yiels α(g), which is a well-known lower boun on inepenence + (G)

4 4 Tao Jiang, Michael Salerno number. If we take c to be a proper coloring of G using χ(g) colors, then trivially D χ(g) an Lemma 2 yiels α(g), which is another trivial lower boun on the χ(g) inepenence number. In general, we believe that Lemma 2 is of inepenent interest, an that it shoul fin applications elsewhere. Given a graph L together with a vertex coloring c, a subgraph L in it is polychromatic if all the vertices of L have ifferent colors. Lemma 3. Let T be a tree on t vertices. Let L be a graph together with a proper vertexcoloring c. Suppose that L contains no polychromatic copy of T. Then α(l) 2n(L)/t(t+ ). Proof. We use inuction on t. For the basis step, let t =. Then T = K 2. If L contains no polychromatic K 2, then L contains no ege an α(l) = n(l). So the claim hols. For the inuction step, let t 2 an suppose the claim hols for all trees with fewer than t eges. Let T be a tree on t eges. Let z be a leaf of T an y its unique neighbor in T. Let T = T y. For each vertex x in L let c (x) enote the number of ifferent colors appearing on its neighbors. Let S = {x V (L) : c (x) t} an S 2 = {x V (L) : c (x) t }. Suppose first that L[S ] contains a polychromatic copy T of T. Let y enote the image of y in T. Since y S, c (y ) t. Since only t colors are use on vertices of T, y has a neighbor z in L such that c(z ) is not use in T. In particular z / V (T ). Now we can exten T to a polychromatic copy of T in L via y z, contraicting L not containing a polychromatic copy of T. Hence L[S ] contains no polychromatic copy of T. By the inuction hypothesis, α(l[s ]) 2n(S )/(t )t. Now, since c (x) t for all x in L[S 2 ], by Lemma 2, we have α(l[s 2 ]) n(s 2 )/t. Let n = n(l), n = S, n 2 = S 2. Then n = n + n 2 an the iscussions above show that α(l) max{ 2n, n 2 t(t ) t }. It is easy to check that max{ 2n, n 2 t(t ) t } is minimize when n = t n an n t+ 2 = 2 n with minimum value 2n. Hence, α(l) 2n, completing t+ t(t+) t(t+) the proof. Lemma 4. Let H be efine as in Theorem. For all i [k] the following hols: if G is a graph not containing H as a subgraph an v is a vertex in G an S i (v) is a set of vertices at istance i from v in G then α(g[s i (v)]) S i (v) /8(kt) 2i. Proof. We use inuction on i. For the basis step, let i =. Let G be any graph not containing H as a subgraph, v a vertex in G, an S (v) a set of neighbors of v. Since G oes not contain H as a subgraph, G = G[S (v)] oes not contain F = H w. By our efinition of H, F is a tree on t+t(k ) = kt vertices. By Lemma, e(g ) kt n(g ). So, G has average egree at most 2kt. At most half of the vertices in G can have egree at least 4kt. By removing those vertices, we get an inuce subgraph G of G with at least n(g )/2 vertices an maximum egree at most 4kt. So α(g ) α(g ) n(g ) n(g ). + (G ) 8kt Thus, the lemma hols for i =. For the inuction step, let j 2 an suppose the claim hols for all i < j, we prove the claim for i = j. Let G be a graph not containing H as a subgraph, v a vertex in G, an S j (v) a set of vertices at istance j from v in G. For each vertex u in S j (v), let P u be a shortest v, u-path. The union of P u over all u in S j (v) contains a tree T roote at v containing S j (v) such that the istance from v to any vertex x in T is also the istance between v an x in G. (Note that T is just a subtree of the usual breath first search tree from v.) Let x,..., x p enote the chilren of v in T. For each i [p] let T i enote the

5 Ramsey numbers of some bipartite graphs versus complete graphs 5 subtree of T roote at x i an let A i enote the set of leaves of T i. Clearly, A,..., A p partition S j (v). For each i [p], let B i be a maximum inepenent set in G[A i ]. Since A i is a set of vertices at istance j from x i, by inuction hypothesis we have B i = α(g[a i ]) A i /8(kt) 2j 3. Let L = G[ p i= B i ]. Note that n(l) = p i= B i S j (v) /8(kt) 2j 3. Let c be a proper coloring of L with B,..., B p being its color classes. Recall that H is forme by joining a vertex w to vertices of a tree F via paths of length k such that every two of these paths share only w. Let H j enote the subgraph of H obtaine by eleting w an vertices at istance at most j from w in H. Claim. L contains no polychromatic copy of H j. Proof of Claim. Suppose otherwise that L contains a polychromatic copy H of H j. For each y V (H ) let Q y be the unique path in T of length j from y to the root v. If y lies in B i then Q y lies strictly insie T i vx i. By our assumption, vertices of H all lie in ifferent B i s. So every two of the Q y s share only v. Furthermore, these Q y s all have length j since V (L) S j (v). Now the union of these paths an H clearly contains a copy of H, contraicting G not containing H as a subgraph. Now, since L contains no polychromatic copy of H j, an H j is a tree on at most kt vertices, by Lemma 3, α(l) 2n(L)/kt(kt ) n(l)/(kt) 2 S 8(kt) 2j 3 j (v) /(kt) 2 S j (v) /8(kt) 2j. This completes the inuction step an the proof. With all these preparatory lemmas, we are reay to prove Theorem. Our proof follows closely that of [6]. As usual, given a graph G, a vertex v in G an a nonnegative integer i, we use N i (v) to enote the set of all vertices at istance i from v. Proof of Theorem : Let c = c k,t = 40(kt) 2k. Let G be a graph on m = (cn) +/k / ln k n vertices that oes not contain H as a subgraph. We prove that α(g) n. Let = c k n /k ln /k n. We start with G = G an I = an as long as G has a vertex of egree at least we perform the following iterative proceure. Pick a vertex v V (G ) with egree at least. If N k (v) in G has size at least cn, then by Lemma 4, α(g ) cn/8(kt) 2k n by our choice of c an hence we are one. So we may assume that N k (v) cn. Since N (v) / N 0 (v) = N (v), there exists an inex i, i k such that N i+ (v) N i (v) ( cn ) k = c k n k ln k n = β. () Pick the smallest i satisfying (). By Lemma 4, G [N i (v)] has an inepenent set I of size at least N i (v) /8(kt) 2i 5 N i (v) /c. Let I = I I an remove all vertices in N i (v), N i (v) an N i+ (v) from G. The number of vertices we have remove is N i (v) + N i (v) + N i+ (v) ( ) β + + β N i (v) 2β N i (v) 2cβ 5 I, (2) an they contain all the neighbors of vertices in I. Thus, uring the whole process I stays as an inepenent set. Also, by (2) the ratio between the total number of vertices which we remove an the orer of I is at most 2cβ 5.

6 6 Tao Jiang, Michael Salerno Consier the final G at the en of the process. Suppose first that there are at least m/2 vertices that remain. Since the process ens before we run out of vertices, it must be that no remaining vertex has egree at least. By Corollary, using also that c k kt an c 40k, we have α(g ) 0.05 m/2 c+ k n + k (ln ln kt) = 40 ln k n cn 40 ln n ln n cn k = 40k n. ln c k n k ln k n c k n k ln k n kt Since G is an inuce subgraph of G, we have α(g) n as well an we are one. Next, assume that fewer than m/2 vertices remain in G, which means at least m/2 vertices have been remove in the process. Then by earlier iscussions we have α(g) I m/2 2cβ/5 = 5 4 c+/k n +/k ln /k n This completes our proof of Theorem. c ln /k n c /k n = 5n /k 4 > n. 3. Ramsey numbers of general bipartite graphs versus complete graphs In this short section, we give a general boun on r(h, K n ) when H is bipartite. In [4], Caro et al. prove the following Theorem 2. ([4]) Suppose H is a subgraph of K + T where T is a tree, an ex(m, H) = O(m γ ) for some γ satisfying < γ < 2 (of necessity H is bipartite). Then there exists a positive constant c H such that r(h, K n ) c H ( n ) 2 γ ln n. Here, we generalize their result so that it applies to all bipartite graphs H instea of just bipartite graphs that are subgraphs of K + T for some tree T. Theorem 3. Let H be a bipartite graph with ex(m, H) = O(m γ ), where < γ < 2. Then there is a positive constant c H epening only on H such that r(h, K n ) c H ( n ln n ) 2 γ. Proof. By our assumption, there exists an absolute constant a such that ex(m, H) am γ for all m. Without loss of generality we may assume a. Let N = c H ( n ) 2 γ ln n, where c H is positive constant to be specifie later. Let G be a graph on N vertices that oes not contain H as a subgraph, we show that α(g) n. Since G oes not contain H as a subgraph, we have e(g) an γ. So the average egree of G is at most 2aN γ. At most half of the vertices can have egree exceeing 4aN γ. By removing those vertices from G we get an inuce subgraph G of G with n(g ) N/2 an maximum egree D = (G ) 4aN γ. Let m = n(g ) N/2. We may assume that D is a growing function of N; otherwise α(g ) m/(d + ) c N, for some constant c. With N = c H ( n ) 2 γ ln n, we woul alreay have α(g ) n by choosing c H to be large enough. For any v V (G ), its neighborhoo N G (v) in G contains no copy of H. Hence, N G (v) spanns at most a[ G (v)] γ ad γ eges. Hence the number h of triangles in G is at most

7 Ramsey numbers of some bipartite graphs versus complete graphs 7 3 m(adγ ) < amd γ. By Proposition, α(g) α(g ) m (ln D ) 0D 2 ln(h/m) m 20D (2 ln D ln adγ ) c N ln D (for some positive constant c ) D c γ ln 4aN N (since ln D ecreases in D for D e) D 4aN γ c N 2 γ ln N ( for some positive constant c ). With N = c H ( n 2 γ, one can ensure c N 2 γ ln N n by choosing c H to be large enough. This completes the proof. ln n ) A careful reaer may notice that in the proof above, the subgraph inuce by the neighborhoo of a vertex v in fact shoul not contain any subgraph of H of the form H x, where x V (H). So one might expect to get a better boun. However, in this case, such improvement woul not affect the asymtotics. As shown in [4], since ex(m, C 2k ) = O(m +/k ), Theorem 3 immeiately implies r(c 2k, K n ) c(k)( n ) k k ln n for some constant c(k) epening on k, as mentione in the introuction. Since ex(m, K p,q ) = O(m 2 /p ), where p q, Theorem 3 also yiels Corollary 2. Let p, q, n be positive integers where p q. There exists a constant c p,q such that r(k p,q, K n ) c p,q ( n ln n )p. Alon, Krivelevich an Suakov [2] prove that if H is a bipartite graph in which vertices in one partite set all have egree at most p then ex(m, H) = O(m 2 /p ). By Theorem 3 we have Corollary 3. Let p be a positive integer. Let H be a bipartite graph with a bipartition (X, Y ) such that vertices in X all have egree at most p. Then there exists a positive constant c H epening on H such that r(h, K n ) c H ( n ln n )p. References. N. Alon, M. Krivelevich, B. Suakov: Coloring graphs with sparse neighborhoos, J. Combin. Theory Ser B, 77 (999), N. Alon, M. Krivelevich, B. Suakov: Turán numbers of bipartite graphs an relate Ramsey-type questions, Combinatorics, Probability, an Computing, 2, B. Bollobás: Ranom Graphs, Acaemic Press, Lonon, Y. Caro, Y. Li, C. Rousseau, Y. Zhang: Aysmptotic bouns for some bipartite graph: complete graph Ramsey numbers, Discrete Mathematics, 220 (2000), P. Erős, R. Fauree, C. Rousseau, R. Schelp: On cycle-complete graph Ramsey numbers, J. Graph Theory, 2 (978), R. Fauree, M. Simonovits: On a class of egenerate extremal graph problems, Combinatorica, 3(983), P. Haxell, D. Mubayi an P-S Loh: personal communications.

8 8 Tao Jiang, Michael Salerno 8. T. Jiang, A. Vary: Asymptotic improvement of the Gilbert-Varshamov boun on the size of binary coes, IEEE Transactions on Information Theory, 50 (2004), J.H. Kim: The Ramsey number R(3, t) has orer of magnitue t 2 / log t, Ranom Struct. Algorithms, 7 (995), M. Krivelevich: Bouning Ramsey numbers through large eviation inequalities, Ranom Struct. Algorithms, 7 (995), Y. Li, C.C. Rousseau: Fan-complete graph Ramsey numbers, J. Graph Theory, 23 (996), Y. Li, W. Zang: Ramsey numbers involving large ense graphs an bipartite Turán numbers, J. Combin. Theory Ser. B, 87 (2003), Y. Li, W. Zang: The inepenence number of graphs with forbien cycle an Ramsey numbers, J. Combin. Optimization, 7 (2003), J. Spencer: Asymptotic lower bouns for Ramsey functions, Discrete Mathematics, 20 (977), B. Suakov: personal communications. 6. B. Suakov: A note on o cycle-complete graph Ramsey numbers, Electronic Journal of Combinatorics, 9 (2002), N, 4pp. 7. D. B. West: Introuction to graph theory, secon eition, Prentice Hall, 200.