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1.
Let X be an n-element set and T a family of k-subsets of X. Let r be an integer, k > r ? 2. Suppose that T does not contain r + 1 members having empty intersection such that any r of them intersect non-trivially. Chvátal and Erdös conjectured that for (r + 1) k ? rn we have |F|?n?1k?1. In this paper we first prove that This conjecture holds asymptotically (Theory 1). In Theorems 4 and 5 we prove it for r = 2, K ? 5, n > no(k); k ? 3r, n > no(k, r), respectively.  相似文献   

2.
Paul Erd?s conjectured that every K 4-free graph of order n and size ${k + \lfloor n^2/4\rfloor}$ contains at least k edge disjoint triangles. In this note, we prove that such a graph contains at least 32k/35 + o(n 2) edge disjoint triangles and prove his conjecture for k ≥  0.077n 2.  相似文献   

3.
If the simplicial complex formed by the neighborhoods of points of a graph is (k ? 2)-connected then the graph is not k-colorable. As a corollary Kneser's conjecture is proved, asserting that if all n-subsets of a (2n ? k)-element set are divided into k + 1 classes, one of the classes contains two disjoint n-subsets.  相似文献   

4.
In 1973, P. Erdös conjectured that for eachkε2, there exists a constantc k so that ifG is a graph onn vertices andG has no odd cycle with length less thanc k n 1/k , then the chromatic number ofG is at mostk+1. Constructions due to Lovász and Schriver show thatc k , if it exists, must be at least 1. In this paper we settle Erdös’ conjecture in the affirmative. We actually prove a stronger result which provides an upper bound on the chromatic number of a graph in which we have a bound on the chromatic number of subgraphs with small diameter.  相似文献   

5.
The Kneser graph K(n, k) is the graph whose vertices are the k-element subsets of an n-element set, with two vertices adjacent if the sets are disjoint. The chromatic number of the Kneser graph K(n, k) is n–2k+2. Zoltán Füredi raised the question of determining the chromatic number of the square of the Kneser graph, where the square of a graph is the graph obtained by adding edges joining vertices at distance at most 2. We prove that (K2(2k+1, k))4k when k is odd and (K2(2k+1, k))4k+2 when k is even. Also, we use intersecting families of sets to prove lower bounds on (K2(2k+1, k)), and we find the exact maximum size of an intersecting family of 4-sets in a 9-element set such that no two members of the family share three elements.This work was partially supported by NSF grant DMS-0099608Final version received: April 23, 2003  相似文献   

6.
Let G(n,k) be a graph whose vertices are the k-element subsets of an n-set represented as n-tuples of “O's” and “1's” with k “1's”. Two such subsets are adjacent if one can be obtained from the other by switching a “O” and a “1” which are in adjacent positions, where the first and nth positions are also considered adjacent. The problem of finding hamiltonian cycles in G(n,k) is discussed. This may be considered a problem of finding “Gray codes” of the k-element subsets of an n-set. It is shown that no such cycle exists if n and k are both even or if k=2 and n?7 and that such a cycle does exist in all other cases where k?5.  相似文献   

7.
Bollobás, Erdös, Simonovits, and Szemerédi conjectured [1] that for each positive constantc there exists a constantg(c) such that ifG is any graph which cannot be made 3-chromatic by the omission ofcn 2 edges, thenG contains a 4-chromatic subgraph with at mostg(c) vertices. Here we establish the following generalization which was suggested by Erdös [2]: For each positive constantc and positive integerk there exist positive integersf k(c) andn o such that ifG is any graph with more thann o vertices having the property that the chromatic number ofG cannot be made less thank by the omission of at mostcn 2 edges, thenG contains ak-chromatic subgraph with at mostf k(c) vertices.  相似文献   

8.
Let Fk denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollobás conjectured that for every ?>0 and positive integer k there is n(k,?) such that every 2-edge-coloring of the complete graph of order n?n(k,?) which has at least edges in each color contains a member of Fk. This conjecture was proved by Cutler and Montágh, who showed that n(k,?)<4k/?. We give a much simpler proof of this conjecture which in addition shows that n(k,?)<?−ck for some constant c. This bound is tight up to the constant factor in the exponent for all k and ?. We also discuss similar results for tournaments and hypergraphs.  相似文献   

9.
Erdoes and Soes conjectured in 1963 that every graph G on n vertices with edge number e(G) 〉 1/2(k - 1)n contains every tree T with k edges as a subgraph. In this paper, we consider a variation of the above conjecture, that is, for n 〉 9/ 2k^2 + 37/2+ 14 and every graph G on n vertices with e(G) 〉 1/2 (k- 1)n, we prove that there exists a graph G' on n vertices having the same degree sequence as G and containing every tree T with k edges as a subgraph.  相似文献   

10.
Following a conjecture of P. Erdös, we show that if F is a family of k-subsets of and n-set no two of which intersect in exactly l elements then for k ? 2l + 2 and n sufficiently large |F| ? (k ? l ? 1n ? l ? 1) with equality holding if and only if F consists of all the k-sets containing a fixed (l + 1)-set. In general we show |F| ? dknmax;{;l,k ? l ? 1};, where dk is a constant depending only on k. These results are special cases of more general theorems (Theorem 2.1–2.3).  相似文献   

11.
Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n ? 2. These conditions are best possible. A related conjecture also is proposed.  相似文献   

12.
Let f be a cusp form of weight k + 1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato–Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp 2)} p , where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it for the family {a(tn 2)} n where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato–Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of Dedekind–Dirichlet density.  相似文献   

13.
Let γ be a bounded convex curve on the plane. Then #(γ ∩ (?/n)2) = o(n 2/3). This strengthens the classical result due to Jarník [J] (the upper bound cn 2/3) and disproves the conjecture on the existence of a so-called universal Jarník curve.  相似文献   

14.
Let n and m be natural numbers, n ? m. The separation power of order n and degree m is the largest integer k = k(n, m) such that for every (0, 1)-matrix A of order n with constant linesums equal to m and any set of k 1's in A there exist (disjoint) permutation matrices P1,…, Pm such that A = P1 + … + Pm and each of the k 1's lies in a different Pi. Almost immediately we have 1 ? k(n, m) ? m ? 1, yet in all cases where the value of k(n, m) is actually known it equals m ? 1 (except under the somewhat trivial circumstances of k(n, m) = 1). This leads to a conjecture about the separation power, namely that k(n, m) = m ? 1 if m ? [n2] + 1. We obtain the bound k(n, m) ? m ? [n2] + 2, so that this conjecture holds for n ? 7. We then move on to latin squares, describing several equivalent formulations of the concept. After establishing a sufficient condition for the completion of a partial latin square in terms of the separation power, we can show that the Evans conjecture follows from this conjecture about the separation power. Finally the lower bound on k(n, m) allows us to show, after some calculations, that the Evans conjecture is true for orders n ? 11.  相似文献   

15.
A maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. Let i(G) denote the number of maximal independent sets of G. Here, we prove two conjectures, suggested by P. Erdös, that the maximum number of maximal independent sets among all graphs of order n in a family Φ is o(3n/3) if Φ is either a family of connected graphs such that the largest value of maximum degrees among all graphs of order n in Φ is o(n) or a family of graphs such that the approaches infinity as n → ∞.  相似文献   

16.
Asymptotic results are obtained for pA(k)(n), the kth difference of the function pA(n) which is the number of partitions of n into integers from A. Under certain restrictions on A it is shown that
PA(k+1)(n)PA(k)(n) = O(n?1/2) (n→ ∫)
thereby verifying for these A a conjecture of Bateman and Erdös.  相似文献   

17.
A subsetA of the positive integers ?+ is called sumfree provided (A+A)∩A=ø. It is shown that any finite subsetB of ?+ contains a sumfree subsetA such that |A|≥1/3(|B|+2), which is a slight improvement of earlier results of P. Erdös [Erd] and N. Alon-D. Kleitman [A-K]. The method used is harmonic analysis, refining the original approach of Erdös. In general, defines k (B) as the maximum size of ak-sumfree subsetA ofB, i.e. (A) k = $\underbrace {A + ... + A}_{k times}$ % MathType!End!2!1! is disjoint fromA. Elaborating the techniques permits one to prove that, for instance, $s_3 \left( B \right) > \frac{{\left| B \right|}}{4} + c\frac{{\log \left| B \right|}}{{\log \log \left| B \right|}}$ % MathType!End!2!1!as an improvement of the estimate $s_k \left( B \right) > \frac{{\left| B \right|}}{4}$ % MathType!End!2!1! resulting from Erdös’ argument. It is also shown that in an inequalitys k (B)>δ k |B|, valid for any finite subsetB of ?+, necessarilyδ k → 0 fork → ∞ (which seemed to be an unclear issue). The most interesting part of the paper are the methods we believe and the resulting harmonic analysis questions. They may be worthwhile to pursue.  相似文献   

18.
In this paper, we consider the problem of constructing a shortest string of {1,2,…,n} containing all permutations on n elements as subsequences. For example, the string 1 2 1 3 1 2 1 contains the 6 (=3!) permutations of {1,2,3} and no string with less than 7 digits contains all the six permutations. Note that a given permutation, such as 1 2 3, does not have to be consecutive but must be from left to right in the string.We shall first give a rule for constructing a string of {1,2,…,n} of infinite length and the show that the leftmost n2?2n+4 digits of the string contain all the n! permutations (for n≥3). We conjecture that the number of digits f(n) = n2?2n+4 (for n≥3) is the minimum.Then we study a new function F(n,k) which is the minimum number of digits required for a string of n digits to contain all permutations of i digits, ik. We conjecture that F(n,k) = k(n?1) for 4≤kn?1.  相似文献   

19.
n people have distinct bits of information. They can communicate via k-party conference calls. How many such calls are needed to inform everyone of everyone else's information? Let f(n,k) be this minimum number. Then we give a simple proof that f(n,k)= [(n?k)(k?1)]+[nk] for 1?n?k2, f(n,k)=2[(n?k)(k?1)] for n>k2.In the 2-party case we consider the case in which certain of the calls may permit information flow in only one direction. We show that any 2n-4 call scheme that conveys everone's information to all must contain a 4-cycle, each of whose calls is “two way”, along with some other results. The method follows that of Bumby who first proved the 4-cycle conjecture.  相似文献   

20.
Farthest-polygon Voronoi diagrams   总被引:2,自引:0,他引:2  
Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(nlog3n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k−1 connected components, but if one component is bounded, then it is equal to the entire region.  相似文献   

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