A characterization for a sequence to be potentially K r+1 − e-graphic

Let n 〉 r, let lr --- （dl,d2,-,dn） be a non-increasing sequence of nonnegative integers and let Kr＋l - e be the graph obtained from Kr＋l by deleting one edge. If zr has a realization G containing Kr＋l - e as a subgraph, then r is said to be potentially Kr＋l - e-graphic. In this paper, we give a characterization for a sequence π to be potentially Kr＋l - e-graphic.     

Hajek-Renyi-type Inequality for a Class of Random Variable Sequences and Its Applications

In this paper, we obtain the Hejek-Renyi-type inequality for a class of random variable sequences and give some applications for associated random variable sequences, strongly positive dependent stochastic sequences and martingale difference sequences which generalize and improve the results of Prakasa Rao and Soo published in Statist. Probab. Lett., 57（2002） and 78（2008）. Using this result, we get the integrability of supremum and the strong law of large numbers for a class of random variable sequences.     

蕴含K4＋P2-可图序列的刻划

对于给定的图H,若存在可图序列π的一个实现包含H作为子图,则称π为蕴含H-可图的．Gould等人考虑了下述极值问题的变形：确定最小的偶整数σ（H,n）,使得每个满足σ（π）≥σ（H,n）的n项可图序列π=（d1,d2,…,dn）是蕴含H-可图的,其中σ（π）=∑di．本文刻划了蕴含K4＋P2-可图序列,其中K4＋P2是向致的一个顶点添加两条悬挂边后构成的简单图．这一刻划导出σ（K4＋P2,n）的值．     

Generalized statistical convergence and statistical core of double sequences

In this paper we extend the notion of A-statistical convergence to the （λ,μ）statistical convergence for double sequences x =（xjk）. We also determine some matrix transformations and establish some core theorems related to our new space of double sequences Sλ,μ.     

A NON-TRIVIAL PRODUCT OF FILTRATION s＋6 IN THE STABLE HOMOTOPY GROUPS OF SPHERES

By a method improving that of [1], the authors prove the existence of a non-trivial product of filtration, s ＋ 6, in the stable homotopy groups of sphere, πt-6S, which is represented up to non-zero scalar by β^-s＋2ho（hmbn-1 -hnbm-1） ∈ ExtA^s＋6,t＋s（Zp, Zp） in the Adams spectral sequence, where p ≥ 7, n ≥ m ＋ 2 ≥ 5, q = 2（p- 1）, 0 ≤ s 〈 p - 2, t= （s ＋ 2 ＋ （s ＋ 2）p ＋ p^m ＋ p^n）q. The advantage of this method is to extend the range of s without much complicated argument as in [1].     

Bubble tree convergence for the harmonic sequence of harmonic surfaces in ℂℙ n

We show that any harmonic sequence determined by a harmonic map from a compact Riemannian surface M to CP^n has a terminating holomorphic （or anti-holomorphic） map from M to CP^n, or a ＂bubble tree limit＂ consisting of a harmonic map f： M → CP^n and a tree of bubbles hλ^μ： S^2 --→ CP^n.     

On potentially <Emphasis Type="Italic">H</Emphasis>-graphic sequences

For given a graph H, a graphic sequence π = (d ₁, d ₂,..., d _n) is said to be potentially H-graphic if there is a realization of π containing H as a subgraph. In this paper, we characterize the potentially (K ₅ − e)-positive graphic sequences and give two simple necessary and sufficient conditions for a positive graphic sequence π to be potentially K ₅-graphic, where K _r is a complete graph on r vertices and K _r-e is a graph obtained from K _r by deleting one edge. Moreover, we also give a simple necessary and sufficient condition for a positive graphic sequence π to be potentially K ₆-graphic. Project supported by National Natural Science Foundation of China (No. 10401010).     

The smallest degree sum that yields potentially Kr,r-graphic sequences

We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(K_r,r,n) such that every n-term graphic sequence π = (d₁,d₂,...,d_n) with term sum σ(π) = d₁ + d₂ + ... + d_n ≥ σ(K_r,r,n) is potentially K_r,r-graphic, where K_r,r is an r × r complete bipartite graph, i.e. π has a realization G containing K_r,r as its subgraph. In this paper, the values σ(K_r,r,n) for even r and n ≥ 4r² - r - 6 and for odd r and n ≥ 4r² + 3r - 8 are determined.     

The smallest degree sum that yields potentially Kr,r-graphic sequences

We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(Kr,r, n) such that every n-term graphic sequence π = (d1, d2,..., dn) with term sum σ(π) = d1 + d2 +…+ dn ≥σ(Kr,r, n) is potentially Kr,r-graphic, where Kr,r is an r × r complete bipartite graph, i.e. πr has a realization G containing Kr,r as its subgraph. In this paper, the values σ(Kr,r,n) for even r and n ≥ 4r2 - r - 6 and for odd r and n ≥ 4r2 + 3r - 8 are determined.     

The Erdös-Jacobson-Lehel conjecture on potentiallyP k-graphic sequence is true

A variation in the classical Turan extrernal problem is studied. A simple graphG of ordern is said to have propertyPk if it contains a clique of sizek+1 as its subgraph. Ann-term nonincreasing nonnegative integer sequence π=(d₁, d₂,⋯, d₂) is said to be graphic if it is the degree sequence of a simple graphG of ordern and such a graphG is referred to as a realization of π. A graphic sequence π is said to be potentiallyP _k-graphic if it has a realizationG having propertyP _k . The problem: determine the smallest positive even number σ(k, n) such that everyn-term graphic sequence π=(d₁, d₂,…, d₂) without zero terms and with degree sum σ(π)=(d ₁+d ₂+ …+d ₂) at least σ(k,n) is potentially P_k-graphic has been proved positive. Project supported by the National Natural Science Foundation of China (Grant No. 19671077) and the Doctoral Program Foundation of National Education Department of China.     

蕴含W6-可图序列

摘要对于给定的图日,如果可图序列π有一个实现包含日作为子图,则称丌是蕴含H-可图的．本文给出了可图序列π蕴含W6-可图的一个充分条件,其中Wτ是τ个顶点的轮图．     

Degree sequences of graphs containing a cycle with prescribed length

Let r ≥ 3, n ≥ r and π = (d ₁, d ₂, ..., d _n ) be a non-increasing sequence of nonnegative integers. If π has a realization G with vertex set V (G) = {v ₁, v ₂, ..., v _n } such that d _G (v _i ) = d _i for i = 1, 2, ..., n and v ₁ v ₂ ... v _r v ₁ is a cycle of length r in G, then π is said to be potentially C _r ″-graphic. In this paper, we give a characterization for π to be potentially C _r ″-graphic. This work was supported by the grant of National Natural Science Foundation of China No. 10861006 and China Scholarship Council.     

蕴含W_6-可图序列

对于给定的图H,如果可图序列π有一个实现包含H作为子图,则称π是蕴含H-可图的.本文给出了可图序列π蕴含W_6-可图的一个充分条件,其中W_r是r个顶点的轮图.     

Graphic sequences with a realization containing intersecting cliques

Let r≥ 1, k≥ 2 and Fm1 ,...,mki;r denote the most general definition of a friendship graph, that is, the graph of Kr+m1 , . . . , Kr+mk meeting in a common r set, where Kr+mi is the complete graph on r + mi vertices. Clearly, | Fm1 ,...,mki;r | = m1+ ··· + mk + r. Let σ(Fm1 ,...,mki;r , n) be the smallest even integer such that every n-term graphic sequence π = (d1, d2, . . . , dn) with term sum σ(π) = d1 + d2 + ··· + dn ≥σ(Fm1 ,...,mki;r,n) has a realization G containing Fm1 ,...,mki;r as a subgraph. In this paper, we determine σ(Fm1 ,...,mki;r,n) for n sufficiently large.     

关于蕴含Ar,s—可图序列的注记

设G＝（V（G）,E（G）是n阶简单图,其顶点集V（G）＝{v1,…,vr,vr 1,…,vr s,…,vn},n={d1,…,dr 1,…,dr s,…,dn}是G的度序列,且vi的度为dio称G具有性质Ar,s,如果{v1,…,vr,vr 1,…,vr x}的导出子图是完全二部图Kr,s,且{v1,…,vr}和{vr 1,…,vr s}是Kr,s顶点集的二部划分,序列π={d1,…,dr,dr 1,…,dr s,…,dn}称为是蕴含Ar,s－可图的序列判别准则。     

A Rao-type characterization for a sequence to have a realization containing a split graph

For a given graph H, a non-increasing sequence π=(d₁,d₂,…,d_n) of nonnegative integers is said to be potentially H-graphic if there exists a realization of π containing H as a subgraph. The split graph on r+s vertices is denoted by S_r,s. In this paper, we give a Rao-type characterization for π to be potentially S_r,s-graphic. A simplification of this characterization is also presented.     

Conditions for -graphic sequences to be potentially -graphic

An r-graph is a loopless undirected graph in which no two vertices are joined by more than r edges. An r-complete graph on m+1 vertices, denoted by , is an r-graph on m+1 vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π=(d₁,d₂,…,d_n) of nonnegative integers is said to be r-graphic if it is realizable by an r-graph on n vertices. An r-graphic sequence π is said to be potentially -graphic if it has a realization containing as a subgraph. In this paper, some conditions for r-graphic sequences to be potentially -graphic are given. These are generalizations from 1-graphs to r-graphs of four theorems due to Rao [A.R. Rao, The clique number of a graph with given degree sequence, in: A.R. Rao (Ed.), Proc. Symposium on Graph Theory, in: I.S.I. Lecture Notes Series, vol. 4, MacMillan and Co. India Ltd., (1979), 251–267; A.R. Rao, An Erdös-Gallai type result on the clique number of a realization of a degree sequence (unpublished)] and Kézdy and Lehel [A.E. Kézdy, J. Lehel, Degree sequences of graphs with prescribed clique size, in: Y. Alavi et al., (Eds.), in: Combinatorics, Graph Theory, and Algorithms, vol. 2, New Issues Press, Kalamazoo Michigan, 1999, 535–544].     

有一个实现包含Kr +1的可图序列

设K _{r +1}是一个r +1个顶点的完全图. 一个可图序列π =(d₁, d₂,…, d_n)称为是蕴含K _r+1 -可图的, 如果π有一个实现包含 K _{r +1}作为子图. 该文进一步研究了蕴含K _{r+1 -}可图序列的一些新的条件, 证明了这些条件包含文献[14,10,11]中的一些主要结果和当n≥5r/2 +1时,σ(K _r+1, n)之值(此值在文献[2]中被猜测, 在文献[6,7,8,3]中被证实). 此外, 确定了所有满足n≥5, d₅≥4 且不蕴含K₅ -可图序列π=(d₁, d₂,…, d_n)的集合.     

An extremal problem on potentially Kr,s-graphic sequences

We consider a variation of a classical Turán-type extremal problem (F. Chung, R. Graham, Erd s on Graphs: His Legacy of Unsolved Problems, AK Peters Ltd., Wellesley, 1998, Chapter 3) as follows: Determine the smallest even integer σ(K_r,s,n) such that every n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂++d_nσ(K_r,s,n) is potentially K_r,s-graphic, where K_r,s is a r×s complete bipartite graph, i.e., π has a realization G containing K_r,s as its subgraph. In this paper, we first give sufficient conditions for a graphic sequence being potentially K_r,s-graphic, and then we determine σ(K_r,r,n) for r=3,4.     

整数偶序列为蕴含强连通的Beineke－Harary判准

一个由ｎ个非负整数有序对构造的序列是有向可图的,如果它是某个有向图的度序列．一个有向可图序列是蕴含强连通的,如果它是某个强连通有向图的度序列．Ｂｅｉｎｅｋｅ和Ｈａｒａｒｙ给出了一个有向可图序列为蕴含强连通的判准．Ｂｅｉｎｅｋｅ－Ｈａｒａｒｙ判准的充分性证明是“相当长”的（见［１］）．本文的目的是给出Ｂｅｉｎｅｋｅ－Ｈａｒａｒｙ判准的充分性的一个简短证明．