The Threshold for the Erdos, Jacobson and Lehel Conjecture to Be True

Let σ（k, n） be the smallest even integer such that each n-term positive graphic sequence with term sum at least σ（k, n） can be realized by a graph containing a clique of k ＋ 1 vertices. Erdos et al. （Graph Theory, 1991, 439-449） conjectured that σ（k, n） = （k - 1）（2n- k） ＋ 2. Li et al. （Science in China, 1998, 510-520） proved that the conjecture is true for k 〉 5 and n ≥ （k2） ＋ 3, and raised the problem of determining the smallest integer N（k） such that the conjecture holds for n ≥ N（k）. They also determined the values of N（k） for 2 ≤ k ≤ 7, and proved that [5k-1/2] ≤ N（k） ≤ （k2） ＋ 3 for k ≥ 8. In this paper, we determine the exact values of σ（k, n） for n ≥ 2k＋3 and k ≥ 6. Therefore, the problem of determining σ（k, n） is completely solved. In addition, we prove as a corollary that N（k） -= [5k-1/2] for k ≥6.     

Inclusion-exclusion inequalities

Ifμ is a positive measure, andA ₂, ...,A _n are measurable sets, the sequencesS ₀, ...,S _n andP _[0], ...,P _[n] are related by the inclusion-exclusion equalities. Inequalities among theS _i are based on the obviousP _[k]≧0. Letting =the average average measure of the intersection ofk of the setsA _i , it is shown that (−1) ^k Δ ^k M _i ≧0 fori+k≦n. The casek=1 yields Fréchet’s inequalities, andk=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involvingk-th order divided differences. Using convexity arguments, it is shown that forS ₀=1, whenS ₁≧N−1, and for 1≦k<N≦n andv=0, 1, .... Asymptotic results asn → ∞ are obtained. In particular it is shown that for fixedN, for all sequencesM ₀, ...,M _n of sufficiently large length if and only if for 0<t<1.     

On the sum of the reciprocals of cycle lengths in sparse graphs

For a graphG let ℒ(G)=Σ{1/k contains a cycle of lengthk}. Erdős and Hajnal [1] introduced the real functionf(α)=inf {ℒ (G)|E(G)|/|V(G)|≧α} and suggested to study its properties. Obviouslyf(1)=0. We provef (k+1/k)≧(300k logk)⁻¹ for all sufficiently largek, showing that sparse graphs of large girth must contain many cycles of different lengths.     

Graphs whose every transitive orientation contains almost every relation

Given a graphG onn vertices and a total ordering ≺ ofV(G), the transitive orientation ofG associated with ≺, denotedP(G; ≺), is the partial order onV(G) defined by settingx<y inP(G; ≺) if there is a pathx=x ₁ x ₂…x _r=y inG such thatx ₁ ≺x _j for 1≦i<j≦r. We investigate graphsG such that every transitive orientation ofG contains ₂ ⁿ −o(n ²) relations. We prove that almost everyG _n,p satisfies this requirement if , but almost noG _n,p satisfies the condition if (pn log log logn)/(logn log logn) is bounded. We also show that every graphG withn vertices and at mostcn logn edges has some transitive orientation with fewer than ₂ ⁿ −δ(c)n ² relations. Partially supported by MCS Grant 8104854.     

On the Upper Bounds of the Numbers of Perfect Matchings in Graphs with Given Parameters

Let φ(G),κ(G),α(G),χ(G),cl(G),diam(G)denote the number of perfect matchings,connectivity,independence number,chromatic number,clique number and diameter of a graph G,respectively.In this note,by constructing some extremal graphs,the following extremal problems are solved:1.max{φ(G):|V(G)|=2n,κ(G)≤k}=k[(2n-3)!!],2.max{φ(G):|V(G)|=2n,α(G)≥k}=[multiply from i=0 to k-1(2n-k-i)[(2n-2k-1)!!],3.max{φ(G):|V(G)|=2n,χ(G)≤k}=φ(T_(k,2n))T_(k,2n)is the Turán graph,that is a complete k-partite graphon 2n vertices in which all parts are as equal in size as possible,4.max{φ(G):|V(G)|=2n,cl(G)=2}=n1,5.max{φ(G):|V(G)|=2n,diam(G)≥2}=(2n-2)(2n-3)[(2n-5)!!],max{φ(G):|V(G)|=2n,diam(G)≥3}=(n-1)~2[(2n-5)!!].     

Degree and local connectivity in digraphs

It is shown that there is a digraphD of minimum outdegree 12m and μ(x, y; D)=11m, but every digraphD of minimum outdegreen contains verticesx ≠y withλ(x, y; D)≧n−1, whereμ(x, y; D) andλ(x, y; D) denote the maximum number of openly disjoint and edge-disjoint paths, respectively.     

Optimal strong parity edge-coloring of complete graphs

A parity walk in an edge-coloring of a graph is a walk along which each color is used an even number of times. Let p(G) be the least number of colors in an edge-coloring of G having no parity path (a parity edge-coloring). Let (G) be the least number of colors in an edge-coloring of G having no open parity walk (a strong parity edge-coloring). Always (G) ≥ p(G) ≥ χ′(G). We prove that (K _n ) = 2^⌈lgn⌉ − 1 for all n. The optimal strong parity edge-coloring of K _n is unique when n is a power of 2, and the optimal colorings are completely described for all n. Partially supported by NSF grant CCR 0093348. Work supported in part by the NSA under Award No. MDA904-03-1-0037.     

An extremal set theoretical characterization of some steiner systems

Letn, k, t be integers,n>k>t≧0, and letm(n, k, t) denote the maximum number of sets, in a family ofk-subsets of ann-set, no two of which intersect in exactlyt elements. The problem of determiningm(n, k, t) was raised by Erdős in 1975. In the present paper we prove that ifk≦2t+1 andk−t is a prime, thenm(n, k, t)≦( _t ⁿ )( _k ^2k-t-1 )/( _t ^2k-t-1 ). Moreover, equality holds if and only if an (n, 2k−t−1,t)-Steiner system exists. The proof uses a linear algebraic approach.     

Addendum To Schrijver's Work On Minimum Permanents

Let denote the set of n×n binary matrices which have each row and column sum equal to k. For 2≤k≤n→∞ we show that is asymptotically equal to (k−1)^k−1k^2−k. This confirms Conjecture 23 in Minc's catalogue of open problems. * Written while the author was employed by the Department of Computer Science at the Australian National University.     

Families of finite sets with three intersections

LetX be a finite set ofn elements and ℓ a family ofk-subsets ofX. Suppose that for a given setL of non-negative integers all the pairwise intersections of members of ℓ have cardinality belonging toL. Letm(n, k, L) denote the maximum possible cardinality of ℓ. This function was investigated by many authors, but to determine its exact value or even its correct order of magnitude appears to be hopeless. In this paper we investigate the case |L|=3. We give necessary and sufficient conditions form(n, k, L)=O(n) andm(n, k, L)≧O(n ²), and show that in some casesm(n, k, L)=O(n ^3/2), which is quite surprising.     

On matroid intersections

This paper exploits and extends results of Edmonds, Cunningham, Cruse and McDiarmid on matroid intersections. Letr ₁ andr ₂ be rank functions of two matroids defined on the same setE. For everyS ⊂E, letr ₁₂(S) be the largest cardinality of a subset ofS independent in both matroids, 0≦k≦r ₁₂(E)−1. It is shown that, ifc is nonnegative and integral, there is ay: 2 ^E →Z ⁺ which maximizes and , subject toy≧0, ∀j∈E, .     

Orthogonal vectors in then-dimensional cube and codes with missing distances

Fork a positive integer letm(4k) denote the maximum number of ±1-vectors of length 4k so that no two are orthogonal. Equivalently,m(4k) is the maximal number of codewords in a code of length 4k over an alphabet of size two, such that no two codewords have Hamming distance 2k. It is proved thatm(4k)=4 ifk is the power of an odd prime.     

Asymptotic behavior of the irrational factor

We study the irrational factor function I(n) introduced by Atanassov and defined by , where is the prime factorization of n. We show that the sequence {G(n)/n} _n≧1, where G(n) = Π _ν=1 ⁿ I(ν)^1/n , is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n). Research of the third author is supported in part by NSF grant number DMS-0456615.     

Symmetric products of surfaces and the cycle index

We study some of the combinatorial structures related to the signature ofG-symmetric products of (open) surfacesSP _G ^m (M)=M ^m/G whereG ⊂S _m.The attention is focused on the question, what information about a surfaceM can be recovered from a symmetric productSP ⁿ(M). The problem is motivated in part by the study of locally Euclidean topological commutative (m+k,m)-groups, [16]. Emphasizing a combinatorial point of view we express the signature Sign(SP _G ^m (M))in terms of the cycle index ofG, a polynomial which originally appeared in Pólya enumeration theory of graphs, trees, chemical structures etc. The computations are used to show that there exist punctured Riemann surfacesM _g,k,M _g′,k′such that the manifoldsSP ^m(M _g,k)andSP ^m(M)_g′,k′)are often not homeomorphic, although they always have the same homotopy type provided 2 _g +k=2 _g′ +k′ andk,k′≥1. Supported by the Serbian Ministry for Science and Technology, Grant No. 1643.     

On the vector space of 0-configurations

Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 ^x if for everyA εA we have α(B)=0. The 0-configurations form a vector space of dimension 2 ⁿ − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).     

On the Similarity of Analytic Matrix Functions

Consider a domain , and two analytic matrix-valued functions functions . Consider also points and positive integers n ₁, n ₂, . . . , n _N . We are interested in the existence of an analytic function such that X(ω) is invertible, and G(ω) coincides with X(ω)F(ω)X(ω)⁻¹ up to order n _j at the point ω _j . We will see that such a function exists provided that F(ω _j ),G(ω _j ) have cyclic vectors, and the characteristic polynomials of F,G coincide up to order n _j at ω _j . This allows one to give a short proof to a result of Huang, Marcantognini and Young concerning spectral interpolation in the unit disk. The author was partially supported by a grant from the National Science Foundation. Received: September 8, 2006. Accepted: January 11, 2007.     

Integration and Lipschitz functions

The aim of the paper is to prove that every f ∈ L ¹([0,1]) is of the form f = , where j _n,k is the characteristic function of the interval [k- 1 / 2 ⁿ , k / 2 ⁿ ) and Σ _n=0^∞Σ _k=1²ⁿ |a _n,k | is arbitrarily close to ||f|| (Theorem 2). It is also shown that if μ is any probabilistic Borel measure on [0,1], then for any ɛ > 0 there exists a sequence (b _n,k ) _n≧0 ^k=1,...,2n of real numbers such that and for each Lipschitz function g: [0,1] → ℝ (Theorem 3).      

The Existence of BIB Designs

Abstract Given any positive integers k≥ 3 and λ, let c(k, λ) denote the smallest integer such that v∈B(k, λ) for every integer v≥c(k, λ) that satisfies the congruences λv(v− 1) ≡ 0(mod k(k− 1)) and λ(v− 1) ≡ 0(mod k− 1). In this article we make an improvement on the bound of c(k, λ) provided by Chang in [4] and prove that . In particular, . Supported by NSFC Grant No. 19701002 and Huo Yingdong Foundation     

A bound on the <Emphasis Type="Italic">k</Emphasis>-domination number of a graph

Let G be a graph with vertex set V(G), and let k ⩾ 1 be an integer. A subset D ⊆ V(G) is called a k-dominating set if every vertex υ ∈ V(G)-D has at least k neighbors in D. The k-domination number γ _k (G) of G is the minimum cardinality of a k-dominating set in G. If G is a graph with minimum degree δ(G) ⩾ k + 1, then we prove that

Precise rates in the law of the iterated logarithm under dependence assumptions

Let {X, X1, X2,...} be a strictly stationaryφ-mixing sequence which satisfies EX = 0,EX^2（log2{X}）^2〈∞and φ（n）=O（1/log n）^Tfor some T〉2.Let Sn=∑k=1^nXk and an=O（√n/（log2n）^γ for some γ〉1/2.We prove that limε→√2√ε^2-2∑n=3^∞1/nP（｜Sn｜≥ε√ESn^2log2n＋an）=√2.The results of Gut and Spataru （2000） are special cases of ours.