Session 6: Applied Mathematics Verticillate Structures in Multi-Component Bose-Einstein Condensates

Shu-Ming Chang*
Department of Mathematics
National Tsing Hua University
Hsinchu, Taiwan
m863254@am.nthu.edu.tw

Chang-Shou Lin
Department of Mathematics
National Chung-Cheng University
Chia-Yi, Taiwan
cslin@math.ccu.edu.tw

Tai-Chia Lin
Department of Mathematics
National Taiwan University
Taipei, Taiwan
tclin@math.ntu.edu.tw

Wen-Wei Lin and Shih-Feng Shieh
National Tsing Hua University
Hsinchu, Taiwan
wwlin@am.nthu.edu.tw and sfshieh@am.nthu.edu.tw

Abstract : In this paper we propose two iterative methods, a Jacobi-type iteration (JI) and a Gauss-Seidel-type iteration (GSI), for the computation of energy states of the time-independent vector Gross-Pitaevskii equation (VGPE) which describes a multi-component Bose-Einstein condensate (BEC). A discretization of the VGPE leads to a nonlinear algebraic eigenvalue problem (NAEP). We prove that the GSI method converges locally and linearly to a solution of the NAEP if and only if the associated minimized energy functional problem has a strictly local minimum. The GSI method can thus be used to compute ground states and positive bound states, as well as the corresponding energies of a multi-component BEC. Numerical experience shows that the GSI converges much faster than JI and converges globally within 10 to 20 steps and discovers a new phenomenon called verticillate multiplying, i.e., the generation of multiple verticillate structures.




On the Relations between the Permanental and Characteristic Polynomials of Coronoid Systems

Rong Si Chen
Department of Statistics
Fuzhou University
Fuzhou, Fujian, 350002, P.R.China
chenrongsi@fzu.edu.cn

Abstract : A coronoid system is a benzenoid system with a hole, i.e., a non-hexagonal internal face. Several relations between the coefficients of the permanental and characteristic polynomials of benzenoid systems were recently established by Gutman et al.. In this note we investigate relations between the coefficients of the permanental and characteristic polynomials of coronoid systems.




The Integer Discrete Transform and Its Application in Image Processing

LiZhi Cheng
Department of Mathematics and System Science
National University of Defense Technology
Changsha, Hunan, 410073 P.R. China
clzcheng@vip.sina.com

Abstract : This paper summarizes our recently works related to the construction of integer discrete transforms that includes integer DCTs, DSTs, DWTs, DHT and integer wavelets transforms with parameters, discuss the application of these integer transforms in image processing such as in digital watermarking, information hiding, information disgusising, image super-resolution, and finally, we use the combination of wavelet and multi-grid method to solve ill-conditioned systems.




Poncelet Property Arising from Numerical Range

Mao-Ting Chien
Department of Mathematics
Soochow University
Taipei, Taiwan 11102
mtchien@scu.edu.tw

Abstract : Let C and D be two ellipses on the plane with C inside D. A well known Poncelet's closure theorem asserts that if there is an n-polygon inscribed in D and circumscribed about C, then for any point l on D there is a unique such inscribed n-polygon having l as one of its vertices. The inside ellipse C is called a Poncelet curve.

The numerical range of a matrix A Mn is defined as the set W(A) = { x*  A  x : x Cn, |x| = 1 }. Let A Mn be a completely non-unitary contraction. It is known that the boundary curve of W(A) is a Poncelet curve. We study the Poncelet property arising from the numerical range.




Modeling Intervention Measures and Severity-dependent Public Response during Severe Acute Respiratory Syndrome Outbreak

Sze-Bi Hsu
Department of Mathematics
National Tsing Hua University
Hsinchu, Taiwan

Ying-Hen Hsieh*
Department of Applied Mathematics
National Chung Hsing University
Taichung, Taiwan
hsieh@amath.nchu.edu.tw

Abstract : The Severe Acute Respiratory Syndrome (SARS) epidemic of November 2002 to July 2003 came with much fanfare and left swiftly, resulting in more than 8000 probable cases worldwide and 774 casualties. Many had believe that the simple age-old system of quarantine those individuals suspected of being infected had been instrumental in quick containment of the infections. In this work we propose a differential equations model which includes quarantine and other intenvention measures implemented by the health authority, include those to prevent nosocomial infections and decrease frequency of contacts among general public. We also consider the possible behavior change by the general populance to avoid infection in response to the severity of the outbreak in general, and to these measures in particular. Full analysis is given for the model without quarantine. For the general model with quarantine, basic reproduction number is derived and partial analysis is provided. We will show that introducing quarantine measures in the model could produce bistability in the system, thus changing the basic dynamics of the system. We also give examples where bistable steady states (one disease-free and the another endemic) exist. The results indicate that the measures will be effective in reducing infections only if the quarantined/isolated SARS patients can successfully reduce their contact rate and/or transmission probabilities for each contact. Hence diligent adherence to the quarantine order by the quarantined individuals is essential to any intevention measures. Moreover, the effectiveness of quarantine for infectious diseases like SARS, for which no infection is being prevented during the quarantine period, can only be indirect and therefore must be combined with other intervention measures in order to fully contain the outbreaks.




The Inverse Problems of Optimization

Siming Huang
Institute of Policy and Management
Chinese Academy of Sciences
Beijing 100080, China
simhua@mail.casipm.ac.cn

Abstract : We will summerize the results of inverse problems of optimization, both invserse problems of combinatorial optimization and inverse problems of mathematical programming problems. We will discuss its implications in computational complexity theory and more deep implications.




A Computational Study of the Two Layered Model for Blood Flow Through a Circular Tube

Sanjeev Kumar
Department of Mathematics
Institute of Basic Science, Khandari
Dr. B.R. Ambedkar University, Agra.(India) 282002
sanjeevibs@yahoo.co.in

Abstract : Hemodialysis is a very important phenomenon during the study of the blood, because it removes metabolic wastes uric acid etc. from the blood. The most of the wastes removed during the process are contained in plasma. So to analyze the hemodialysis, one must first understand the flow behavior of the blood in the tubes, therefore, we are considering a two-layered model of the blood flow through a circular tube hemodialyser. Since the viscosity of the plasma layer is less than that of a blood, hence this plasma layer near the wall acts as a natural lubricant for the blood flow. In dialysis, we have flows which have peripheral plasma layer with less red cells. Such flows are called two-layered. Since red blood cells contain an iron compound, so the blood may be considered as a magnetic fluid. It has been already studied that a magnetic field can push red cells away from the hemodialyser membrane. We solved numerically this problem using the finite difference techniques and obtained the result for blood velocity for the different values of pressure force.




A Continuation BSOR-Lanczos-Galerkin Method for Solving Multi-Component Bose-Einstein Condensate

Shu-Ming Chang
Department of Mathematics
National Tsing Hua University
Hsinchu, 300, Taiwan
m863254@am.nthu.edu.tw

Yuen-Cheng Kuo*
Department of Mathematics
National Center for Theoretical Sciences
Hsinchu, 300, Taiwan
m883207@am.nthu.edu.tw

Wen-Wei Lin
Department of Mathematics
National Tsing Hua University
Hsinchu, 300, Taiwan
wwlin@am.nthu.edu.tw

Abstract : We develop a continuation BSOR-Lanczos-Galerkin method for the computation of positive bound states of time-independent, coupled Gross-Pitaevskii equations (CGPEs) which describe a multi-component Bose-Einstein condensate (BEC). A discretization of the CGPEs leads to a nonlinear algebraic eigenvalue problem (NAEP). The solution curve with respect to some parameter of the NAEP is then followed by the proposed method. For a single-component BEC, we prove that there exists a unique global minimizer (the ground state) which is represented by an ordinary differential equation with the initial value. For a multi-component BEC, we prove that m identical ground/bound states will bifurcate into m different ground/bound states at a finite repulsive inter-component scattering length. Numerical results show that various positive bound states of a two/three-component BEC are solved efficiently and reliably by the continuation BSOR-Lanczos-Galerkin method.




An Extremal Problem on Potentially Kp,1,1-graphic Sequencesf

Chunhui Lai
Department of Mathematics
Zhangzhou Teachers College, Zhangzhou
Fujian 363000, P. R. of China
zjlaichu@public.zzptt.fj.cn

Abstract : A sequence S is potentially Kp,1,1 graphical if it has a realization containing a Kp,1,1 as a subgraph, where Kp,1,1 is a complete 3-partite graph. Let s(Kp,1,1, n) denote the smallest degree sum such that every n-term graphical sequence S with s(S) s(Kp,1,1, n) is potentially Kp,1,1 graphical. In this paper, we prove that s(Kp,1,1, n) (p+1)(n-1)+1 for p and n are all even; s(Kp,1,1, n) (p+1)(n-1)+2 for p is even but n is odd; s(Kp,1,1, n) (p+1)(n-1)+2 for p is odd. We conjectured that equality holds for n 2p+4. We proved that this conjecture is true for p = 3, n 7.

f Project Supported by NNSF of China(10271105), NSF of Fujian, Science and Technology Project of Fujian, Fujian Provincial Training Foundation for ``Bai-Quan-Wan Talents Engineering" , Project of Fujian Education Department and Project of Zhangzhou Teachers College.




New Fixed Point Theorems and Almost Periodic Type Solutions for Delay Integral Equation

Yicheng Liu and Zhixiang Li*
Department of Mathematics and System Science,
Science School, National University of Defence Technology
Changsha, 410073, P.R. China
liuyc2001@hotmail.com

Abstract : Motivated by generalizing the principle of contraction mapping in complete metric space, Boyd and Wong[2] and Burton[5] introduced a class of contraction mapping respectively, and showed two useful fixed point theorems. In this paper, we introduce the concept of property L and obtain several fixed point results which were generalization of Boyd and Wong's [2] and Burton's [5]. As an application, some existence results of positive almost periodic solution are obtained, which extend the corresponding results in [6], [8].

Key words:  Fixed point theorem, Almost periodic solution, Delay integral equation.

2000 Mathematics Subject Classification:  34K14, 47H10




The Approximate Solution of A Class of Heat Conduction Equations with Phase Change and Its Applications

Feng Ling
Department of Computer Science
Zhaoqing University
Zhaoqing, Guangdong, 526061, P. R. China
lingf@zqu.edu.cn

Abstract : This study discusses the solution of the following equations:

T(x,t) =





T1(x,t),
if 0 < x < X1,
T2(x,t),
if X1 < x < X1+X2,
T3(x,t),
if X1+X2 < x <

where

2 T1
x2
= 1
a1
T1
x
,  0 < x < X1
(1)

T1(0,t) = A0sinwt,        x = 0, t > 0
(2)

2 T2
x2
= 1
a2
T2
x
,   X1 < x < X1+X2
(3)

2 T3
x2
= 1
a3
T3
x
, X1+X2 < x <
(4)

T1(x,0) = T2(x,0) = T3(x,0) = 0,        x > 0,     t = 0
(5)

T1(X1,t) = T2(X1,t),        x = X1    
(6)

T2(X1+X2,t) = T2( X1+X2,t),        x = X1+X2
(7)

K1 T1
x
= K2 T2
x
,        x = X1
(8)

K2 T1
x
= K3 T2
x
,    x = X1+X2
(9)


lim
x  
T3(x,t) = 0
(10)

The temperature, T(x,t), as a function of depth and time, is found from a knowledge of the periodically varying surface temperature and the thermal parameters of a stratified medium. The applications of this approximate method to some problems in the geo-science also are discussed.




Ky Fan Section Theorem and its Applications in Topological Ordered Spaces

Qun Luo
Department of Mathematics
Zhaoqing University
Zhaoqing, Guangdong 526061, P.R.China
luoqun@zqu.edu.cn

Abstract : In this paper, in topological ordered spaces, we obtain Ky Fan's section Theorem and discuss its applications.

Ky Fan's section Theorem  Let X be a nonempty L convex subset of a topological semilattice with path-connected intervals M, and C X ×X:

  1. for any y X, the set {x X : (x, y) \not C } is either L-convex or empty.

  2. the mapping x {y X : (x, y) C } is transfer closed valued.

  3. for any x X, (x, x) C.

  4. there exists x0 X such that the set cl { y X : (x0, y) C } is compact.

Then there exists y* X such that X ×{y* } C.




Uncertainty Analysis for Parallel Car-Crash Simulation Resultsf

Li-Quan Mei
School of Science
Xi'an Jiaotong University
Xi'an, 710049, P.R.China
lqmei@mail.xjtu.edu.cn

Abstract : Small changes in parameters, load cases or model specifications for crash simulation may result in huge changes in the results, characterizing the crash behavior of an automotive design. For a BMW model differences between the position of a node in two simulation runs of up to 10 cm were observed, just as a result of round-off differences in the case of parallel computing. The paper shows that numerical properties of the simulation codes as well as bifurcations in the crash behavior in the certain parts of the design are reasons for scatter of simulation results. The tool DIFF-CRASH compares simulation results and uses data mining technology to cluster those nodes of the car model, which show similar scatter among the simulation runs and then trace back to a certain part to remove the uncertain behavior. DIFF-CRASH is the only activity using data mining technology for crash simulation stability analysis.
Keywords: Uncertainty analysis, Crash Simulation, Data Mining, Clustering.

f This project was funded by the German Ministry for Education and Research(BMB+F)




Uncertainty Principle for KdV

Mikhail Kovalyov
Department of Mathematics
University of Alberta
Edmonton, AB, T6G 2G1, Canada
mkovalyo@ualberta.ca

Abstract : The Heisenbeg uncertainty principle was obtained by Heisenberg and Bohr alomst a century ago in the context of quantum mechanics, its mathematical formulation based on Fourier transform followed a few years later. We show that a rather large class of nonlineaar waves obtained as solutions of KdV satisfies a remarkably simmilar uncertainty principle




A Robust Algorithm for Reconstructing Surfaces with Boundaries even Possibly Non-orientable

Xiao-Yuan Qian
Department of Applied Mathematics
Dalian University of Technology
Dalian, China 116024
xyqian@vip.sina.com

Abstract : In this paper we propose an algorithm which is applied to reconstruct surfaces from scattered point data. Given a set of unorganized points sampled from a surface with boundary, even possibly non-orientable, the algorithm is applied to automatically reconstruct a topologically correct surface in the form of a piecewise-linear model.




On Bandwidth of Refinement of Caterpillar Graphs

Yung-Ling Lai and Ding-Tai Ron*
Department of Computer Science and Information Engineering
National Chiayi University
Chaiyi, Taiwan
yllai@mail.ncyu.edu.tw, jacky@mail.csie.ncyu.edu.tw

Abstract : In some festival or big party, when there were a huge amount of users using mobile phone, it happens very often that either it is too busy in the GSM Network so one can not dial out or easy to get disconnection while talking to another mobile user. Setting some base stations to enhance the mobile communication in the busy communication region is a reasonable solution. In the GSM Network the communication between two nonadjacent stations will transited through a shortest path. When the stations on the path is busy, the signal may consider go through another path which is longer than the original path. Then there is dilation in the network of the transition. A GSM network can be represented by a graph G = (V,E) where the vertices indicate the base stations and the edges denote the adjacency between base stations. Then, the bandwidth of G represents the dilation of the network and adding an additional vertex v on edge (u,v) , which is known as an elementary refinement operation to graph G, is the same as adding a base station between two adjacent base stations. In this paper, we present some results by applying the elementary refinement operation to the edges on the caterpillar graphs.




Bifurcation Analysis for Energy States of a Two Component Bose-Einstein Condensate

Yuen-Cheng Kuo
Department of Mathematics,
National Tsing Hua University
Hsinchu, 300, Taiwan
d883207@oz.nthu.edu.tw

Wen-Wei Lin
Department of Mathematics
National Tsing Hua University
Hsinchu, 300, Taiwan
wwlin@am.nthu.edu.tw

Shih-Feng Shieh*
Department of Mathematics
National Tsing Hua University
Hsinchu, 300, Taiwan.
sfshieh@am.nthu.edu.tw

Abstract : In this paper, we prove that the solution curve of the ground/positive bound states of a two-component Bose-Einstein condensate has a supercritical pitchfork bifurcation at some finite values of the inter-component scattering length. The ground state solutions will bifurcate into two symmetric solutions with respect to some suitable axis on the symmetric domain, when two components of BEC has equal intra-component scattering lengths. Furthermore, we show that the ground/positive bound states repel each other and form segregated nodal domain when the repulsive scattering length goes to infinity.




The Method of Fundamental Solutions for a Three-Dimensional Inverse Problem in Linear Elasticity

Jun Wang, Fangyu Sun*, Bangti Jin
Department of Mathematics
Zhejiang University
Hangzhou 310027, P.R. China
fysun@sun.zju.edu.cn

Abstract : In this paper, we consider the application of the method of fundamental solutions for the numerical solution of a three-dimensional inverse problem in linear elasticity, i.e., Cauchy problems associated with Navier equations. Regularization methods are the most efficient and powerful methods for ill-posed problems, and two regularization methods, i.e., Tikhonov regularization method and truncated singular value decomposition, are employed to solve the system of Linear equations arising from the method of fundamental solutions, with the regularization parameter determined by the L-curve method. Numerical experiments indicate that the method proposed can yield stable, accurate solutions for the inverse problem, and both the two regularization methods have high accuracy.

Key words: Method of fundamental solutions, Cauchy problem, Regularization, Linear elasticity, Inverse problem.




Numerical Approximation for Optimal Control of Damped Nonlinear Klein-Gordon Equations

Quan-Fang Wang
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing 100080, P. R. China
wang@amss.ac.cn

Abstract : Recent development of control theory cause the rapidly research of numerical approximation for controlled systems. This work is to address numerical approximation of optimal control for nonlinear damped klein-Gordon equations. We implement three processes to achieve prospective purpose.

(a). Present theoretical results on optimal control based upon variational method thanks to J. L. Lions.

(b). Construct a semi-discrete algorithm to solve nonlinear control state system and adjoint system based on finite element approach.

(c). Implement numerical approximation to get optimal control and optimal state by minimizing the quadratic cost functions.

Furthermore, error estimation and necessary optimality condition are verified the effectiveness of our scheme. As our future work, the proposed approximation can be applied to a wide class nonlinear control systems arising in real world.




On Characterizations of Rough Approximation Operators

Wei-Zhi Wu
Information College
Zhejiang Ocean University
Zhoushan, Zhejiang, 316004, P. R. China
wuwz@zjou.net.cn

Abstract : The theory of rough sets, proposed by Pawlak in 1982, is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. Using the concepts of lower and upper approximations in rough set theory, knowledge hidden in information systems may be unravelled and expressed in the form of decision rules. The basic operators in rough set theory are approximations. There are at least two approaches for the development of the rough approximation operators, the constructive and axiomatic approaches. This paper discusses a general framework for the study of rough approximation operators within which both constructive and axiomatic approaches are used. In the constructive approach, various pairs of lower and upper rough approximation operators are defined. Basic properties of rough approximation operators are investigated. The connections between binary relations and rough approximation operators are further established. In the axiomatic approach, operator-oriented characterizations of rough approximation operators are proposed, that is, approximation operators are defined by axioms. Different axiom sets of upper and lower set-theoretic operators guarantee the existence of different types of binary crisp or fuzzy relations which produce the same operators.

Keywords: Approximation operators; Binary relations; Fuzzy sets; Fuzzy rough sets; Rough fuzzy sets; Rough sets.




Online Learning Based on SVM

Xiaowei Yang*
Department of Applied Mathematics
South China University of Technology
Guangzhou 510640, P. R. China
xwyang@scut.edu.cn

Zhifeng Hao
Department of Applied Mathematics
South China University of Technology
Guangzhou 510640, P. R. China
mazfhao@scut.edu.cn

Shu Yu
College of Computer Science and Engineering
South China University of Technology
Guangzhou 510640, P. R. China
Yu-shu@126.com

Abstract : Support vector machine (SVM) is a powerful new tool for data classification and function estimation. The training problem in SVM is equivalent to solve a linearly constrained convex quadratic programming (QP) problem with a number of variables equal to the one of data points. This optimization problem is known to be challenging when the number of data points exceeds a few thousands. In order to speed up learning and apply SVM into the practice, the algorithms for online learning based on SVM have been researched. In this paper, these algorithms are reviewed in detail and the future work is also given.




Error Estimate On The Loop Subdivision Surfaces

Xiao-Ming Zeng
Department of Mathematics
Xiamen University
Xiamen, 361005, China
xmzeng@jingxian.xmu.edu.cn

Abstract : Loop subdivision surfaces is a very popular subdivision scheme for triangular meshes. It is known that Loop subdivision scheme is an approximating subdivision scheme. Using Loop subdivision rules any triangular mesh can be refined. In the limit of an infinite number of subdivisions a smooth surface is obtained. Then, how well is the rate of convergence of control meshes? The aim of this paper is to give the theoretical analysis to this problem. we estimate the rate of convergence of control meshes of Loop surfaces by means of the first order difference of control points. We prove that the control meshes converge to the limit surface at an exponential rate. Our estimate is the optimum. In addition, as an application of our estimate we give a computational formula of subdivision depth for Loop subdivision surfaces by means of the bound of Kobbelt-Daubert-Seidel.




Some New Results on Hamiltonian Graphs

Kewen Zhao
Department of Mathematics
Qiongzhou University
Wuzhishan,572200
Hainan, P.R.China
Kewen@bxemail.com

Abstract : Let G be a simple graph of order n 3. We denote by d(x), N(x), respectively, the degree number and the neighborhood of x in G. And we denote by a, d, respectively, the independent number and the minimum degree of G.

In 1952 Dirac [1] obtained the classical result of Hamiltonian:

Theorem A[1]: Let G be a graph of order n 3, if d /2, then G is Hamiltonian.

In 1960 Ore [2] generalized the above result of Dirac as follow:

Theorem B[2]: Let G be a graph of order n 3, if d(x)+d(y) n for each pair of nonadjacent vertices x,y, then G is a Hamiltonian.

In 1984 Fan [3] generalized the above results:

Theorem C[3]: Let G be a 2-connected graph of order n 3, if max {d(x),d(y)} n/2 for each pair vertices x,y of d(x,y) = 2, then G is Hamiltonian.

In 1991 Faudree et al [4] considered neighborhood unions condition and obtained:

Theorem D[4]: G is 2-connected graph of order n 3 and, if |N(x) N(y)| n - d for each pair nonadjacent vertices x,y V(G), then G is Hamiltonian.

Theorem E[4]: Let G be a 2-connected graph of order n 19 , if |N(x) N(y)| (2n+5)/3 for each pair nonadjacent vertices x,y, then G is pancyclic.

In 1994 we [5] improved the Theorem E of Faudree et alas follow:

Theorem F[5]: Let G be a 2-connected graph of order n 18, if |N(x) N(y)| (2n+4)/3 for each pair nonadjacent vertices x,y, then G is pancyclic.(MR 96B:05093 by Faudree RJ).

Now, In this Abstract we show the follow progress on Theorem E-F:

Theorem 1: Let G be a 2-connected graph of order n 10, if |N(x) N(y)| (2n-3)/3 for each pair nonadjacent vertices x,y, then G is pancyclic.

In 1993 Prof. Guantao Chen (Georgia State University)[6] generalized the Fan type condition of Theorem C as follow:

Theorem G[6]: Let G be a 2-connected graph of order n 3, if max{d(x),d(y)} n/2 for each pair vertices x,y with satisfying 1 |N(x)N(y)| a-1, then G is Hamiltonian.

In 1996 Prof. Guantao Chen, Egawa,Liu X and Saitoet al [7] generalized above results of Theorem A-C:

Theorem H[7]: If G is a k-connected (k 2) graph of order n, and max{ d(v),v S } n/2 for every independent set S of order k with exist two distinct vertices x,y in S satisfies d(x,y) = 2, then G is Hamiltonian.

Now, in this Abstract we generalize the above results of Theorem A-C and Theorem G-H as follow:

Theorem 2: If G is a k-connected (k 2) graph of order n, and max{ d(v),v S } n/2 for every independent set S of order k which two distinct vertices x,y in S satisfies 1 |N(x) N(y)| a-1, then G is Hamiltonian.

And we generalize Theorem A-D and Theorem G as follow:

Theorem 3: G is 2-connected graph and, for each pair vertices x,y V(G) with 1 |N(x) N(y)| a-1, max{d(x),d(y)} n/2 or |N(x) N(y)| n-d, then G is Hamiltonian.

Reference

[1] Dirac GA, Some theorems on abstract graphs. Proc. London Math.Soc.2 (1952) 69-81.

[2] Ore.O, Note on Hamiltonian circuits. Amer.Math.Monthly 67(1960) 55.

[3] Fan GH, New sufficient conditions for cycles in graphs. J.Combin.Theory Ser.B 37(1984) 221-227

[4] Faudree RJ, Gould RJ, Jacobson MS and Lesniak L, Neighborhood unions and highly Hamilton graphs. Ars Combinatoria 31( 1991) 139-148.

[5] Kewen Zhao, Neighborhood condition for pancyclic graphs.Proceedings of the Third China-USA international conference on Graph Theory,Combinatorics, Algorisms and Applications. World Scientific Publishing.Co Pte Ltd 1994,233-240.

[6] Chen GT, Hamiltonian graphs involving neighborhood intersections. Discrete Math.112 (1993) 253-258.

[7] Chen GT, Egawa Y,Liu X and Saito A, Essential independent set and Hamiltonian cycles. J.Graph Theory 21(1996) 243-250

[8] Gould RJ, Advances on the Hamiltonian problem-A survey. Graph and Combin. 19(2003) 7-52.


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