On an antiramsey type result


 Neil Collins
 1 years ago
 Views:
Transcription
1 On an antiramsey type result Noga Alon, Hanno Lefmann and Vojtĕch Rödl Abstract We consider antiramsey type results. For a given coloring of the kelement subsets of an nelement set X, where two kelement sets with nonempty intersection are colored differently, let inj (k, n) be the largest size of a subset Y X, such that the kelement subsets of Y are colored pairwise differently. Taking the minimum over all colorings, i.e. inj(k, n) = min {inj (k, n)}, it is shown that for every positive integer k there exist positive constants c k, c k > 0 such that for all integers n, n large, the following inequality holds c k (ln n) k 2k n 2k inj(k, n) c k (ln n) k 2k n 2k. Introduction In recent years some interest was drawn towards the study of antiramsey type results, cf. for example [ESS 75], [Ra 75], [Al 83], [SS 84], [GRS 89]. In particular, they include topics like Canonical Ramsey Theory and spectra of colorings. Some of these results show a close connection to the theory of Sidonsequences, cf. [BS 85], [So 90], and recently they turned out to be also fruitful in determining bounds for canonical Ramsey numbers, cf. [LR 90]. The general problem is: given a graph or hypergraph, where the edges are colored with certain restrictions on the coloring like, for example, edges of the same color cannot intersect nontrivially, one is interested in the largest size of some totally multicolored subgraph of a given type. In this paper we study this question for colorings of complete uniform hypergraphs. Let : [X] k ω, where ω = {0,,...}, be a coloring of the kelement subsets of X. A subset Y X is called totally multicolored if the restriction of to [Y ] k is a onetoone coloring. For coloring edges in complete graphs, where the coloring is such that edges of the same color are not incident, Babai gave bounds for the largest totally multicolored complete subgraph K k : Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel Fakultät für Mathematik, SFB 343, Universität Bielefeld, W Bielefeld, Germany Emory University, Department of Mathematics and Computer Science, Atlanta, Georgia 30322, USA
2 Theorem [Ba 85] Let n be a positive integer. Let X be an nelement set and let : E(K n ) ω, where ω = {0,, 2,...}, be a coloring of the edges of the complete graph K n on n vertices, such that incident edges get different colors. Then there exists a totally multicolored complete subgraph K k, which satisfies k (2 n) 3. () Moreover, for n large, there exists a coloring : E(K n ) ω, where incident edges are colored differently, such that the largest totally multicolored complete subgraph K k satisfies k 8 (n ln n) 3. (2) Here we improve the lower bound in Theorem () and show that the upper bound is tight, up to a constant factor. Moreover, we generalize Babai s result to colorings of kuniform complete hypergraphs. Specifically, we prove Theorem 2 Let k be a positive integer with k 2. Then there exist positive constants c k, c k > 0 such that for all positive integers n with n n 0 (k) the following holds. Let X be a set with X = n and let : [X] k ω be a coloring, where (S) (T ) for all sets S, T [X] k with S T and S T. Then there exists a totally multicolored set Y X of size Y c k (ln n) 2k n k 2k. (3) Moreover, there exists a coloring : [X] k ω, where (S) (T ) for all different S, T [X] k with S T, such that every totally multicolored subset Y X satisfies Y c k (ln n) 2k n k 2k. (4) The proof relies heavily on probabilistic arguments. The lower bound is proved is Section 2 and the upper bound is established in Section 3. 2 The proof of the lower bound Let F = (V, E) be a hypergraph with vertex set V and edge set E. For a vertex x V let deg F (x) denote the degree of x in F, i.e. the number of edges e E containing x, and let 2
3 (F) = max {deg F (x) x V}. Komlós, Pintz, Spencer and Szemerédi: Our lower bound is based on the following result of Ajtai, Theorem 3 [AKPSS 82] Let F be an (r + )uniform hypergraph on N vertices. Assume that (i) F is uncrowded ( i.e. contains no cycles of length 2, 3 or 4 ) and (ii) (F) t r, where t t 0 (r), and (iii) N N 0 (r, t), then the maximum independent set of F has size α(f) 0.98 e 0 5 r N t (ln t) r. (5) We want to apply this Theorem with the parameters r+ = 2 k and t = N δ, where δ r (2r ). The proof given in [AKPSS 82] suggests that N 0 (r, t) should be at least t 4r+O(). Indeed, these calculations can be avoided, as the statement of Theorem (3) is strong enough to imply the same statement, where condition (iii) is dropped. Theorem 3 Let G be an (r + )uniform hypergraph on n vertices. Assume that (i) G is uncrowded and (ii) (G) t r, where t t 0 (r), then α(g) 0.98 e 0 5 r n t (ln t) r. (6) To see that Theorem (3) implies (the seemingly stronger) Theorem (3 ) consider a hypergraph G on n vertices, which satisfies the assumptions of Theorem (3 ). Let m be a positive integer with m > N 0(r, t) n and consider the hypergraph F m consisting of m vertex disjoint copies of G. Then F m has N = m n vertices and satisfies the assumptions (i), (ii) and (iii) of Theorem (3). We infer that m α(g) = α(f m ) 0.98 e r N t (ln t) r
4 and thus α(g) 0.98 e 0 5 r n t (ln t) r. We can now prove the lower bound in Theorem (2). The basic idea is to define a hypergraph on the set of vertices X whose edges are all unions of pairs of members of [X] k which have the same color. Then we choose an appropriate random subset of the set of vertices and show that with positive probability the induced hypergraph on this set does not contain too many edges, and contains only a small number of cycles of length 2,3 or 4. We can then delete these cycles and apply Theorem (3 ) to obtain the desired result. The actual calculations are somewhat complicated and are described below. A similar application of Theorem (3 ) is given in [RS 9]. Proof: Let : [X] k ω be a coloring, which satisfies the assumptions in Theorem (2). We may assume that X = n is divisible by k. This will not change the calculations asymptotically. Put r + = 2 k and consider an (r + )uniform hypergraph H with vertex set X and with U E(H), where E [X] r+, being an edge of H if and only if there exist different kelement subsets S, T U with (S) = (T ). Indeed, by assumption on the coloring the sets S and T are disjoint. Note that if Z X is an independent set of H, then Z is totally multicolored. Our aim is to find a large independent set in H. As there are at most n k of H satisfies kelement subsets of the same color, we infer that the number of edges E(H) ( n k) n k ( n k 2 For i = 2, 3, 4 we will further bound the number µ i of icycles in H. ) 2 k 2 (k )! nk+. (7) For distinct vertices x, y of H let deg H (x, y) be the number of edges of H containing both vertices x and y. Let {x, y} U E(H) (8) for some set U. Let S, T [X] k be such that S T = U and (S) = (T ). As S and T are by assumption disjoint sets, we may assume without loss of generality that one of the following possibilities happens: (i) either x S and y T 4
5 (ii) or {x, y} S. ( n 2 k The number of edges U E(H) satisfying (8) and (i) ((8) and (ii)) is bounded from above by ) (n 2 ( ( n k ) k 2) respectively). Summing, we obtain that n 2 deg H (x, y) k Using (9) one easily sees that n + k µ 3 c 3 n 3k µ 4 c 4 n 4k. n 2 < 2 k n k. (9) k 2 k! We will now discuss the 2cycles: for j = 2, 3,..., 2k let ν j be the number of (unordered) pairs of edges of H intersecting in a jelement set. We clearly have µ 2 = 2k j=2 ν j. Set p = n 2 4k. Let Y be a random subset of X with vertices chosen independently, each with probability p. Then P rob ( Y p n) > 0.9. (0) For i = 3, 4 let µ i (Y ) be the random variable counting the number of icycles no two of whose edges form a two cycle of the subgraph of H induced on a set Y. Similarily, for j = 2, 3,..., 2k let ν j (Y ) be the random variable counting the number of (unordered) pairs of edges of H induced on Y, which intersect in j vertices. Let E(µ i (Y )) and E(ν j (Y )) be the corresponding expected values. It is easy to see that for i = 3, 4 (and k 2). E(µ i (Y )) p (2k ) i c i n ki = o(p n) () In order to give an upper bound on E(ν j (Y )) we will first estimate ν j. Fix an edge S E(H) [X] 2k and fix nonnegative integers j 0 and j with j 0, j k. We will count the number of unordered pairs {T 0, T }, which satisfy (T 0 ) = (T ) (2) 5
6 and j 0 = S T 0, j = S T. (3) Assume first that j 0, j. Fixing the set T 0, there are at most 2k j0 j sets T satisfying (2) and (3). If we apply the same argument also when T is fixed we infer that the number of pairs {T 0, T }, which satisfy (2) and (3), is bounded from above by { } 2k 2k j0 n 2k 2k j n min, j 0 j k j 0 j j 0 k j Assume now that j 0 = 0 and j. Having fixed the set T there are at most ( n k 2) sets T 0, which satisfy (2) and (3). Therefore, the number of such pairs {T 0, T } is bounded from above by most 2k n j k j Setting j = j 0 + j 2 and summing this means, that for every edge S E(H) there are at n k. c j,k n k j 2 + c j,k n k j+. c j,k n k j 2 edges T E(H) such that S T = j, and hence with (7) we have ν j c j,k n k j 2 E(H) c j,k n 2k+ j 2, where c j,k is a constant depending on j and k only. Thus, for j = 2, 3,..., 2k it follows that E(µ 2 (Y )) = 2k j=2 2k j=2 E(ν j (Y )) p 4k j c j,k n 2k+ j 2 = o(p n). (4) 6
7 Summarizing (0), () and (4) and the fact that E( E(H) [Y ] 2k ) = p 2k E(H), (5) we infer that there exists a subset Y 0 X with Y 0 p n, µ i (Y 0 ) = o(p n) and ν j (Y 0 ) = o(p n) for i = 3, 4 and j = 2, 3,..., 2k and with E(H) [Y 0 ] 2k 2 p 2k E(H). We delete from Y 0 all vertices, which are contained in icycles of length i = 2, 3, 4 to obtain a subset Y Y 0 with Y p n, such that the subgraph induced on Y is uncrowded and has at most 2 p 2k E(H) edges. Finally, delete all vertices Y with degree bigger than 8 k p 2k E(H) p n. We obtain a subset Z Y with at least p n 2 ( o()) vertices such that the subgraph G of H induced on the set Z satisfies the assumptions of Theorem (3 ) with (G) t 2k = 8 k p2k E(H) p n 4 k! p2k n k, hence, t ( 4 k!) 2k p n k 2k. We apply Theorem (3 ) to the hypergraph G and obtain α(h) α(g) 0.98 e 0 5 r Z t ( o()) 0.49 e (ln t) r ( k! 2k This completes the proof of the lower bound. (4k ) (4k 2) ) 2k n k 2k (ln n) 2k. 3 The proof of the upper bound Let k, n be positive integers, where n is divisible by k. Let X be a set of vertices with X = n. A perfect kmatching on X is a collection of n k pairwise disjoint kelement subsets of X. 7
8 The number of perfect kmatchings of an nelement set, n divisible by k, is given by = ( n k) ( n k k n! (k!) n k n k!. ) ( n 2k ) (... k k) k n k! In the following we prove the upper bound given in Theorem (2). The basic idea is similar to the one used by Babai in [Ba 85], but there are several complications. Proof: Let X be a set with X = n, and assume that n is divisible by k. As long as k is fixed, this will not change the calculations asymptotically. Let m = cn k, where c = (k )! 8. Let M, M 2,..., M m be random perfect kmatchings, chosen uniformly randomly and independently from the set of all perfect kmatchings. Put H j = i<j M i, hence H j is the set of all kelement subsets occurring in one of the first (j ) perfect kmatchings M i, i < j. Define a coloring : [X] k ω in the following way: for j =, 2,..., m color the sets in M j \ H j with color j, and, in order to complete the coloring, color those sets in [X] k \ H m+ with new colors in an arbitrary way, compatible with the assumptions. (E.g., using a different color for each edge). Now let Y X be a subset of X with Y = l, where l > n k k+ but l = o(n k ). Our objective is to estimate the probability that Y is totally multicolored. Very roughly, this is done as follows. We show that with a high probability Y does not contain too many edges of any single matching, and hence it does not contain too many edges of H j for every j. If this occurs, then for each j, with a reasonably high probability Y does contain two edges of M j that do not lie in H j (and thus have the same color). This implies that with extremely high probability such two edges exist for some j, and hence Y is not totally multicolored. The actual proof is rather complicated, and is described in the following sequence of lemmas. Lemma For all integers j, t, j m and t 0, the following inequality holds Proof: l k Prob ( M j [Y ] k t) ( k n k )t. (6) The kmatchings M j above have been chosen at random and the set Y was fixed. In order to give an upper bound for the probability Prob ( M j [Y ] k t), view it the other way around: Let M j be fixed and let Y be chosen at random. This does not change the corresponding probabilities. Now, Y can be chosen in ( n) l ways. From Mj we can choose t kelement sets in ( n ) k t 8
9 ways and the remaining elements of Y in at most ( n kt l kt ) ways. This implies Prob ( M j [Y ] k t) ( n ) ( k t n kt ) l kt ( n ) l = n t ( l kt k n) l k t k n k. Let E i denote the event H i [Y ] k 6c k lk = 3 4 (k!) lk. (7) Notice, that Prob (E i ) Prob (E i+ ) for every i, i m. The following Lemma gives a lower bound for the probability that E m+ occurs: Lemma 2 For n large, Prob (E m+ ) 2 6c k lk. Proof: For i =, 2,..., m define random variables x i by x i = M i [Y ] k. Thus E m+ [Y ] k mi= x i. As the kmatchings are chosen independently, the x i s are independent random variables too. Therefore, m Prob ( H m+ [Y ] k t) Prob ( x i t) i= (t i ) m i=,t i 0, t i =t m Prob (x i t i ). (8) i= The number of sequences (t i ) m i= with t i 0 and m i= t i = t is given by the binomial coefficient ). By Lemma we know ( t+m t l k Prob (x i t i ) ( k n k )t i, hence with (8) we have Prob ( H m+ [Y ] k t) t + m l k ( t k n k )t e(t + m) t t t k nk e(t + m)lk ( tkn k ) t (9) l k 9
10 For t = 6c k lk and l = o(n k k ) the quotient e(t+m) lk tkn k Hence Prob (E m+ ) 2 6c k lk. occurring in (9) is less than /2 for n large. Next, define another random variable y j by y j = [M j ] 2 [[Y ] k \ H j ] 2. (20) Clearly, y j counts the number of those pairs of disjoint kelement sets i [Y ] k \ H j which have both elements in the kmatching M j (and hence have the same color). Let E(y j M) denote the conditional expected value of y j given M. Lemma 3 For positive integers n, n large, E(y j E j, M, M 2,..., M j ) > 30 k 2 l 2k. (2) n2k 2 Proof: As E j holds, (7) implies for n large that [Y ] k \ H j l 6c k k lk c l k, (22) where, say, c = 8 (k!). For every set S [Y ] at most k( l k ) kelement subsets of Y, which are not disjoint from S, hence the the number of sets {S, T } [[Y ] k \ H j ] 2, where S and T are disjoint, is for n large at least l 2 c l k (c l k k k ) > 30 (k!) 2 l2k. (23) Now, for given disjoint kelement sets S, T, the probability that both are in M j is given by Prob (S, T M j ) = = ( n k ( n k ( ) n ( k) n k ) k ) ( n k ) ( n k ( n k ) 2 k (k )! n k ) 2. 0
11 Therefore, E(y j E j, M, M 2,..., M j ) 30 k 2 l 2k n 2k 2. Lemma 4 For every positive integer j, j m, and n large Prob (y j = E j, M, M 2,..., M j ) > 3 k 2 l 2k. (24) n2k 2 Proof: First we claim that Prob (y j t E j, M, M 2,..., M j ) ( k n l k 2t+ ) (25) k for every positive integer t. This follows from the fact that t pairwise different twoelement sets imply that the underlying set has cardinality at least 2t +. Hence, y j t implies M j [Y ] k 2t +. By Lemma this gives Prob (y j t E j, M, M 2,..., M j ) Prob ( M j [Y ] k 2t + ) ( k n l k 2t+ ), k proving (25). For every positive integer i put p i = Prob (y j = i E j, M, M 2,..., M j ). Clearly, we have E(y j E j, M, M 2,..., M j ) = i<ω i p i and by (25) this implies E(y j E j, M, M 2,..., M j ) p + i ( k n i 2 l k 2i+ ) k p + O(( lk n k )3 ) l 2k p + o. (26) n 2k 2 The last inequality (26) follows from the fact that l = o(n k k ). By Lemma 3 this implies that, say, p > 3 k 2 l 2k n 2k 2 for n large. Let F j denote the event (y j = 0 and E j+ ). Then F F 2... F j is the event that E j and y i = 0 for i =, 2,..., j.
12 Let M be the set of all mutually exclusive events (E j, M, M 2,..., M j ) for which y i = 0, i =, 2,..., j, holds. Then clearly and thus Hence, by Lemma 4 we infer that Now we are ready to prove F F 2... F j = M(E j, M, M 2,..., M j ) Prob (y j = F F 2... F j ) M = Prob (y j = (E j, M M 2... M j )) M Prob (E j, M, M 2,..., M j ) Prob (y j = (E j, M, M 2,..., M j )) min M Prob (E j, M, M 2,..., M j ) = min M Prob (y j = E j, M, M 2,..., M j ). Prob (y j = F F 2... F j ) > Lemma 5 For every positive integer n, n large, Proof: we infer that 3 k 2 l 2k. (27) n2k 2 Prob (F F 2... F m ) < exp( 048 k (k!) l 2k ). (28) nk As m Prob (F F 2... F m ) = Prob (F ) Prob (F j F F 2... F j ), (29) j=2 Prob (F j F F 2... F j )) Prob (y j = 0 F F 2... F j ) Prob (y j F F 2... F j ) With (29) it follows that Prob (F F 2... F m ) < ( 3 k 2 l 2k )m n2k 2 exp( 3 k 2 m l 2k n 2k 2 ) = Prob (y j = F F 2... F j ) < 3 k 2 l 2k n 2k 2 by (27). exp( 048 k (k!) l 2k n k ) as m = 8 ((k )!) nk. 2
13 We finish the proof of Theorem 2 by bounding the probability that there exists a totally multicolored subset Y X with Y = l. If a fixed subset Y X is totally multicolored, then either F F 2... F m is true or for some j, j m, the event E j+ does not occur and therefore by (7) also not the event E m+. With Lemma 2 and Lemma 5 we infer that Prob (Y is totally multicolored ) < 2 6c k lk + exp( 048 k (k!) l 2k n k ), and doing this for all ( n l ) lelement subsets Y X implies < Prob (there exists Y X with Y = l and Y is totally multicolored ) n (exp( l 048 k (k!) l 2k 6c ) + 2 k lk ). (30) nk For l (049 k (k!)) 2k n k 2k (ln n) 2k (where the constant can be easily improved) expression (30) goes to 0 with n going to infinity. Thus for X = n and n large there exists a coloring : [X] k ω such that every totally multicolored subset Y X has size Y (049 k (k!)) 2k n k 2k (ln n) 2k. References [AKPSS 82] [Al 83] M. Ajtai, J. Komlós, J. Pintz, J. Spencer and E. Szemerédi, Extremal uncrowded hypergraphs, Journal of Combinatorial Theory Ser. A 32, 982, N. Alon, On a conjecture of Erdös, Simonovits and Sós concerning AntiRamsey Theorems, J. Graph Theory 7, 983, [Ba 85] L. Babai, An antiramsey Theorem, Graphs and Combinatorics, 985, [BS 85] [ESS 75] L. Babai and V. T. Sós, Sidon sets in groups and induced subgraphs of Cayley graphs, Europ. J. Combinatorics 6, 985, 04. P. Erdös, M. Simonovits and V. T. Sós, AntiRamsey theorems, in: Infinite and Finite Sets, ed. by Hajnal, Rado, Sós, Coll. Math. Soc. Janos Bolyai, NorthHolland, Amsterdam, 975,
14 [GRS 89] [LR 90] [Ra 75] [RS 9] R. L. Graham, B. L. Rothschild and J. Spencer, Ramsey theory, 2nd edition, John Wiley, New York, 989. H. Lefmann and V. Rödl, On canonical Ramsey numbers for complete graphs versus paths, 990, to appear in J. Comb. Theory (Series B). R. Rado, AntiRamsey theorems, in: Infinite and Finite Sets, ed. by Hajnal, Rado, Sós, Coll. Math. Soc. Janos Bolyai, NorthHolland, Amsterdam, 975, V. Rödl and E. Si najová, Note on independent sets of Steiner systems, preprint, 99. [SS 84] M. Simonovits and V. T. Sós, On restricted colorings of K n, Combinatorica 4, 984, 00. [So 90] V. T. Sós, An additive problem in different structures, 990, preprint. 4
The degree, size and chromatic index of a uniform hypergraph
The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a kuniform hypergraph in which no two edges share more than t common vertices, and let D denote the
More informationMean RamseyTurán numbers
Mean RamseyTurán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρmean coloring of a graph is a coloring of the edges such that the average
More informationStrong Ramsey Games: Drawing on an infinite board
Strong Ramsey Games: Drawing on an infinite board Dan Hefetz Christopher Kusch Lothar Narins Alexey Pokrovskiy Clément Requilé Amir Sarid arxiv:1605.05443v2 [math.co] 25 May 2016 May 26, 2016 Abstract
More informationOn Integer Additive SetIndexers of Graphs
On Integer Additive SetIndexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A setindexer of a graph G is an injective setvalued function f : V (G) 2 X such that
More informationRamsey numbers for bipartite graphs with small bandwidth
Ramsey numbers for bipartite graphs with small bandwidth Guilherme O. Mota 1,, Gábor N. Sárközy 2,, Mathias Schacht 3,, and Anusch Taraz 4, 1 Instituto de Matemática e Estatística, Universidade de São
More informationON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction
ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove
More informationHigh degree graphs contain largestar factors
High degree graphs contain largestar factors Dedicated to László Lovász, for his 60th birthday Noga Alon Nicholas Wormald Abstract We show that any finite simple graph with minimum degree d contains a
More informationA Turán Type Problem Concerning the Powers of the Degrees of a Graph
A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of HaifaORANIM, Tivon 36006, Israel. AMS Subject Classification:
More informationBest Monotone Degree Bounds for Various Graph Parameters
Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer
More informationPartitioning edgecoloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edgecoloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political
More informationInduced subgraphs of Ramsey graphs with many distinct degrees
Journal of Combinatorial Theory, Series B 97 (2007) 62 69 www.elsevier.com/locate/jctb Induced subgraphs of Ramsey graphs with many distinct degrees Boris Bukh a, Benny Sudakov a,b a Department of Mathematics,
More informationCMSC 858T: Randomized Algorithms Spring 2003 Handout 8: The Local Lemma
CMSC 858T: Randomized Algorithms Spring 2003 Handout 8: The Local Lemma Please Note: The references at the end are given for extra reading if you are interested in exploring these ideas further. You are
More informationCycles in a Graph Whose Lengths Differ by One or Two
Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More informationOn three zerosum Ramseytype problems
On three zerosum Ramseytype problems Noga Alon Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel and Yair Caro Department of Mathematics
More informationEmbedding nearlyspanning bounded degree trees
Embedding nearlyspanning bounded degree trees Noga Alon Michael Krivelevich Benny Sudakov Abstract We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of
More informationThe chromatic spectrum of mixed hypergraphs
The chromatic spectrum of mixed hypergraphs Tao Jiang, Dhruv Mubayi, Zsolt Tuza, Vitaly Voloshin, Douglas B. West March 30, 2003 Abstract A mixed hypergraph is a triple H = (X, C, D), where X is the vertex
More informationOnline Ramsey numbers
Online Ramsey numbers David Conlon Abstract Consider the following game between two players, Builder and Painter Builder draws edges one at a time and Painter colours them, in either red or blue, as each
More informationPh.D. Thesis. Judit NagyGyörgy. Supervisor: Péter Hajnal Associate Professor
Online algorithms for combinatorial problems Ph.D. Thesis by Judit NagyGyörgy Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai
More informationStationary random graphs on Z with prescribed iid degrees and finite mean connections
Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the nonnegative
More informationMinimum degree condition forcing complete graph immersion
Minimum degree condition forcing complete graph immersion Matt DeVos Department of Mathematics Simon Fraser University Burnaby, B.C. V5A 1S6 Jacob Fox Department of Mathematics MIT Cambridge, MA 02139
More informationOn end degrees and infinite cycles in locally finite graphs
On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel
More informationThe positive minimum degree game on sparse graphs
The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University
More informationFinding and counting given length cycles
Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationNONCANONICAL EXTENSIONS OF ERDŐSGINZBURGZIV THEOREM 1
NONCANONICAL EXTENSIONS OF ERDŐSGINZBURGZIV THEOREM 1 R. Thangadurai StatMath Division, Indian Statistical Institute, 203, B. T. Road, Kolkata 700035, INDIA thanga v@isical.ac.in Received: 11/28/01,
More informationA 2factor in which each cycle has long length in clawfree graphs
A factor in which each cycle has long length in clawfree graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science
More informationLecture 7: Approximation via Randomized Rounding
Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining
More informationAll trees contain a large induced subgraph having all degrees 1 (mod k)
All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New
More informationLarge induced subgraphs with all degrees odd
Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order
More informationZachary Monaco Georgia College Olympic Coloring: Go For The Gold
Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationGraphs without proper subgraphs of minimum degree 3 and short cycles
Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract
More informationEvery tree contains a large induced subgraph with all degrees odd
Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University
More informationA Sublinear Bipartiteness Tester for Bounded Degree Graphs
A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublineartime algorithm for testing whether a bounded degree graph is bipartite
More informationLecture 22: November 10
CS271 Randomness & Computation Fall 2011 Lecture 22: November 10 Lecturer: Alistair Sinclair Based on scribe notes by Rafael Frongillo Disclaimer: These notes have not been subjected to the usual scrutiny
More information2.3 Scheduling jobs on identical parallel machines
2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed
More informationResearch Statement. Andrew Suk
Research Statement Andrew Suk 1 Introduction My research interests are combinatorial methods in discrete geometry. In particular, I am interested in extremal problems on geometric objects. My research
More informationSEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov
Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices
More informationONLINE VERTEX COLORINGS OF RANDOM GRAPHS WITHOUT MONOCHROMATIC SUBGRAPHS
ONLINE VERTEX COLORINGS OF RANDOM GRAPHS WITHOUT MONOCHROMATIC SUBGRAPHS MARTIN MARCINISZYN AND RETO SPÖHEL Abstract. Consider the following generalized notion of graph colorings: a vertex coloring of
More informationRegular Induced Subgraphs of a Random Graph*
Regular Induced Subgraphs of a Random Graph* Michael Krivelevich, 1 Benny Sudaov, Nicholas Wormald 3 1 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel; email: rivelev@post.tau.ac.il
More informationERDOS PROBLEMS ON IRREGULARITIES OF LINE SIZES AND POINT DEGREES A. GYARFAS*
BOLYAI SOCIETY MATHEMATICAL STUDIES, 11 Paul Erdos and his Mathematics. II, Budapest, 2002, pp. 367373. ERDOS PROBLEMS ON IRREGULARITIES OF LINE SIZES AND POINT DEGREES A. GYARFAS* Problems and results
More informationThe Online Set Cover Problem
The Online Set Cover Problem Noga Alon Baruch Awerbuch Yossi Azar Niv Buchbinder Joseph Seffi Naor ABSTRACT Let X = {, 2,..., n} be a ground set of n elements, and let S be a family of subsets of X, S
More information136 CHAPTER 4. INDUCTION, GRAPHS AND TREES
136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics
More informationLabeling outerplanar graphs with maximum degree three
Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationNotes on counting finite sets
Notes on counting finite sets Murray Eisenberg February 26, 2009 Contents 0 Introduction 2 1 What is a finite set? 2 2 Counting unions and cartesian products 4 2.1 Sum rules......................................
More informationABSTRACT. For example, circle orders are the containment orders of circles (actually disks) in the plane (see [8,9]).
Degrees of Freedom Versus Dimension for Containment Orders Noga Alon 1 Department of Mathematics Tel Aviv University Ramat Aviv 69978, Israel Edward R. Scheinerman 2 Department of Mathematical Sciences
More informationCOMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the HigmanSims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
More informationMarkov random fields and Gibbs measures
Chapter Markov random fields and Gibbs measures 1. Conditional independence Suppose X i is a random element of (X i, B i ), for i = 1, 2, 3, with all X i defined on the same probability space (.F, P).
More informationCrossing the Bridge at Night
A1:01 Crossing the Bridge at Night Günter Rote Freie Universität Berlin, Institut für Informatik Takustraße 9, D14195 Berlin, Germany rote@inf.fuberlin.de A1:02 August 21, 2002 A1:03 A1:04 Abstract We
More informationThe Independence Number in Graphs of Maximum Degree Three
The Independence Number in Graphs of Maximum Degree Three Jochen Harant 1 Michael A. Henning 2 Dieter Rautenbach 1 and Ingo Schiermeyer 3 1 Institut für Mathematik, TU Ilmenau, Postfach 100565, D98684
More informationCycles and cliqueminors in expanders
Cycles and cliqueminors in expanders Benny Sudakov UCLA and Princeton University Expanders Definition: The vertex boundary of a subset X of a graph G: X = { all vertices in G\X with at least one neighbor
More informationGame Chromatic Index of Graphs with Given Restrictions on Degrees
Game Chromatic Index of Graphs with Given Restrictions on Degrees Andrew Beveridge Department of Mathematics and Computer Science Macalester College St. Paul, MN 55105 Tom Bohman, Alan Frieze, and Oleg
More informationSOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts
More informationAbout the inverse football pool problem for 9 games 1
Seventh International Workshop on Optimal Codes and Related Topics September 61, 013, Albena, Bulgaria pp. 15133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute
More informationUniform Multicommodity Flow through the Complete Graph with Random Edgecapacities
Combinatorics, Probability and Computing (2005) 00, 000 000. c 2005 Cambridge University Press DOI: 10.1017/S0000000000000000 Printed in the United Kingdom Uniform Multicommodity Flow through the Complete
More information8.1 Min Degree Spanning Tree
CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree
More informationDegree Hypergroupoids Associated with Hypergraphs
Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated
More informationOdd induced subgraphs in graphs of maximum degree three
Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A longstanding
More informationRandom graphs with a given degree sequence
Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.
More informationMINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES IN 3UNIFORM HYPERGRAPHS., where
MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES IN UNIFORM HYPERGRAPHS ENNO BUSS, HIỆP HÀN, AND MATHIAS SCHACHT Abstract. We investigate minimum vertex degree conditions for uniform hypergraphs
More informationRemoving Even Crossings
EuroComb 2005 DMTCS proc. AE, 2005, 105 110 Removing Even Crossings Michael J. Pelsmajer 1, Marcus Schaefer 2 and Daniel Štefankovič 2 1 Department of Applied Mathematics, Illinois Institute of Technology,
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationFinite Sets. Theorem 5.1. Two nonempty finite sets have the same cardinality if and only if they are equivalent.
MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we
More informationComputer Science Department. Technion  IIT, Haifa, Israel. Itai and Rodeh [IR] have proved that for any 2connected graph G and any vertex s G there
 1  THREE TREEPATHS Avram Zehavi Alon Itai Computer Science Department Technion  IIT, Haifa, Israel Abstract Itai and Rodeh [IR] have proved that for any 2connected graph G and any vertex s G there
More informationThe Open University s repository of research publications and other research outputs
Open Research Online The Open University s repository of research publications and other research outputs The degreediameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:
More informationBrendan D. McKay Computer Science Dept., Vanderbilt University, Nashville, Tennessee 37235
139. Proc. 3rd Car. Conf. Comb. & Comp. pp. 139143 SPANNING TREES IN RANDOM REGULAR GRAPHS Brendan D. McKay Computer Science Dept., Vanderbilt University, Nashville, Tennessee 37235 Let n~ < n2
More informationIntersecting Families
Intersecting Families Extremal Combinatorics Philipp Zumstein 1 The ErdsKoRado theorem 2 Projective planes Maximal intersecting families 4 Hellytype result A familiy of sets is intersecting if any two
More informationA simpler and better derandomization of an approximation algorithm for Single Source RentorBuy
A simpler and better derandomization of an approximation algorithm for Single Source RentorBuy David P. Williamson Anke van Zuylen School of Operations Research and Industrial Engineering, Cornell University,
More informationCatalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni versity.
7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,
More informationTesting Hereditary Properties of NonExpanding BoundedDegree Graphs
Testing Hereditary Properties of NonExpanding BoundedDegree Graphs Artur Czumaj Asaf Shapira Christian Sohler Abstract We study graph properties which are testable for bounded degree graphs in time independent
More informationOn the crossing number of K m,n
On the crossing number of K m,n Nagi H. Nahas nnahas@acm.org Submitted: Mar 15, 001; Accepted: Aug 10, 00; Published: Aug 1, 00 MR Subject Classifications: 05C10, 05C5 Abstract The best lower bound known
More informationLecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs
CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like
More informationExtreme Intersection Points in Arrangements of Lines
Extreme Intersection Points in Arrangements of Lines Rom Pinchasi August 1, 01 Abstract We provide a strengthening of the classical theorem of Erdős [3] saing that n lines in the plane, no two of which
More informationMax Flow, Min Cut, and Matchings (Solution)
Max Flow, Min Cut, and Matchings (Solution) 1. The figure below shows a flow network on which an st flow is shown. The capacity of each edge appears as a label next to the edge, and the numbers in boxes
More informationExponential time algorithms for graph coloring
Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A klabeling of vertices of a graph G(V, E) is a function V [k].
More informationTHE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok
THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan
More informationTreerepresentation of set families and applications to combinatorial decompositions
Treerepresentation of set families and applications to combinatorial decompositions BinhMinh BuiXuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no
More informationComplexity Theory. IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar
Complexity Theory IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar Outline Goals Computation of Problems Concepts and Definitions Complexity Classes and Problems Polynomial Time Reductions Examples
More information8. Matchings and Factors
8. Matchings and Factors Consider the formation of an executive council by the parliament committee. Each committee needs to designate one of its members as an official representative to sit on the council,
More informationCollinear Points in Permutations
Collinear Points in Permutations Joshua N. Cooper Courant Institute of Mathematics New York University, New York, NY József Solymosi Department of Mathematics University of British Columbia, Vancouver,
More informationWeek 5 Integral Polyhedra
Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory
More informationChapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twentyfold way
Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or pingpong balls)
More informationSum of squares of degrees in a graph
Sum of squares of degrees in a graph Bernardo M. Ábrego 1 Silvia FernándezMerchant 1 Michael G. Neubauer William Watkins Department of Mathematics California State University, Northridge 18111 Nordhoff
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationCONTRIBUTIONS TO ZERO SUM PROBLEMS
CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg
More informationLecture 17 : Equivalence and Order Relations DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion
More informationOnline Ramsey Theory for Bounded Degree Graphs
Online Ramsey Theory for Bounded Degree Graphs Jane Butterfield Tracy Grauman William B. Kinnersley Kevin G. Milans Christopher Stocker Douglas B. West University of Illinois Urbana IL, U.S.A. Submitted:
More informationSubsets of Euclidean domains possessing a unique division algorithm
Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness
More informationNearoptimum Universal Graphs for Graphs with Bounded Degrees (Extended Abstract)
Nearoptimum Universal Graphs for Graphs with Bounded Degrees (Extended Abstract) Noga Alon 1, Michael Capalbo 2, Yoshiharu Kohayakawa 3, Vojtěch Rödl 4, Andrzej Ruciński 5, and Endre Szemerédi 6 1 Department
More informationRecent progress in additive prime number theory
Recent progress in additive prime number theory University of California, Los Angeles Mahler Lecture Series Additive prime number theory Additive prime number theory is the study of additive patterns in
More informationTotal colorings of planar graphs with small maximum degree
Total colorings of planar graphs with small maximum degree Bing Wang 1,, JianLiang Wu, SiFeng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More informationThe UnionFind Problem Kruskal s algorithm for finding an MST presented us with a problem in datastructure design. As we looked at each edge,
The UnionFind Problem Kruskal s algorithm for finding an MST presented us with a problem in datastructure design. As we looked at each edge, cheapest first, we had to determine whether its two endpoints
More informationGROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.
Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the
More informationUniversal hashing. In other words, the probability of a collision for two different keys x and y given a hash function randomly chosen from H is 1/m.
Universal hashing No matter how we choose our hash function, it is always possible to devise a set of keys that will hash to the same slot, making the hash scheme perform poorly. To circumvent this, we
More informationApproximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai
Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationSingle machine parallel batch scheduling with unbounded capacity
Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NPCompleteness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More information