蔡少棠院士傅京孙院士的师兄我们图论的大师Seifollah Louis Hakimi的论文清单：
Hakimi, S. L. Graphs with two
kinds of elements. J. Franklin Inst. 270 (1960), 451--467. MR0119813；
Hakimi, S. L. On trees of a
graph and their generation. J. Franklin Inst. 272 (1961), 347--359.
Hakimi, S. L. On the
realizability of a set of trees. IRE Trans. CT-8 (1961), 11--17.
Hakimi, S. L. On
realizability of a set of integers as degrees of the vertices of a linear graph.
I. J. Soc. Indust. Appl. Math. 10 (1962), 496--506.（Havel–Hakimi算法是图论中用于判定非负整数序列能否构成简单图度序列的递归算法；Kleitman, D. J.; Wang, D. L. Algorithms for
constructing graphs and digraphs with given valences and factors. Discrete
Math. 6 (1973), 79--88.）
Hakimi, S. L. Simultaneous
flows through a communication network. IRE Trans. CT-9 (1962), 169--175.
Hakimi, S. L. On realizability
of a set of integers as degrees of the vertices of a linear graph. II.
Uniqueness. J. Soc. Indust. Appl. Math. 11 (1963), 135--147.
Hakimi, S. L. Optimum
distribution of switching centers in a communication network and some related
graph theoretic problems. Operations Res. 13 (1965), 462--475.
S. L. Hakimi,他的师弟S. S. Yau丘錫生合作Distance matrix
of a graph and its realizability. Quart. Appl. Math. 22 (1965), 305--317.
Hakimi, S. L. On the degrees
of the vertices of a directed graph. J. Franklin Inst. 279 (1965),
290--308.
他的师兄Wataru Mayeda（写了一本很厚的图论书）, Seifollah Louis Hakimi,
Wai-kai Chen（陈惠开）, Narsingh Deo合作Generation of
complete trees. IEEE Trans. Circuit Theory CT-15 (1968), 101--105.（Narsingh Deo是他的博士并博文是“哈密顿树”并是他的6个有博士的人之一有22个博士、，其他5个是Howard Frank有3个博士、Jon Gustav Bredeson仅有1个博士、Kyung-Yong Chwa仅有1个博士、Ching-Chung Kuo仅有1个博士、Shachindra Maheshwari仅有1个博士。这哈密顿树图博士写了一本世界图论名著，并在计算机科学的许多领域都做出重要贡献如并行算法“Parallel algorithms
and architectures report of a workshop”“Data
Structures for Parallel Computation on Shared-Memory Machines”、“Parallel
Algorithms and Implementations”）。
Hakimi, Seifollah Louis; Jon G. Bredeson合作, Graph theoretic
error-correcting codes. IEEE Trans. Inform. Theory IT-14 (1968), 584--591.
Frank, H.; Hakimi, S. L. Parametric
analysis of statistical communication nets. Quart. Appl. Math. 26 (1968),
249--263.
Manherz, R. K.; Hakimi, S.
L. The
generalized inverse in network analysis and quadratic error-minimization
problems. IEEE Trans. Circuit Theory CT-16 (1969), 559--562.
Frank, H.; Hakimi, S. L. Parametric
synthesis of statistical communication nets. Quart. Appl. Math. 27 (1969),
105--120.
L. S. Bobrow,
S. L. Hakimi, Graph
theoretic prefix codes and their synchronizing properties. Information and
Control 15 (1969), 70--94.（1969年7篇，这里附3篇，但很奇怪1970年没有）；
Hakimi, S. L. Steiner's
problem in graphs and its implications. Networks 1 (1971/72), 113--133.
A. N. Patrinos,
S. L. Hakimi, The
distance matrix of a graph and its tree realization. Quart. Appl. Math. 30
(1972/73), 255--269.
S. L. Hakim, S. N.Maheshwari合作Optimum
locations of centers in networks. Operations Res. 20 (1972), 967--973.
A. T. Amin, S.
L. Hakimi, Upper
bounds on the order of a clique of a graph. SIAM J. Appl. Math. 22 (1972),
569--573.
Hakimi, S. L.; Amin, A. T. On the design of
reliable networks. Networks 3 (1973), 241--260.
A. T. Amin, S.
L. Hakimi, Graphs
with given connectivity and independence number or networks with given measures
of vulnerability and survivability. IEEE Trans. Circuit Theory CT-20
(1973), 2--10.
E. F. Schmeichel, S. L. Hakimi,. Pancyclic graphs
and a conjecture of Bondy and Chvátal. J. Combinatorial Theory Ser. B 17
(1974), 22--34.
Hakimi, S. L. On the existence
of graphs with prescribed degrees and connectivity. SIAM J. Appl. Math. 26
(1974), 154--164.
Hakimi, S. L.; Amin, A. T. Characterization
of connection assignment of diagnosable systems. IEEE Trans. Comput. C-23
(1974), 86--88.（关键词有graph models，其后1975年没有论文）
Maheshwari, Shachindra N.;
Hakimi, S. Louis. On models for
diagnosable systems and probabilistic fault diagnosis. IEEE Trans. Comput.
C-25 (1976), no. 3, 228--236.（虽是接着前一篇的做，但与图论更疏远一些，算是第一篇非图论的论文）
Patrinos, A. N.; Hakimi, S.
L. Relations
between graphs and integer-pair sequences. Discrete Math. 15 (1976), no. 4,
347--358.
Schmeichel, E. F.; Hakimi,
S. L. On the
existence of a traceable graph with prescribed vertex degrees. Ars Combin.
4 (1977), 69--80.
Schmeichel, E. F.; Hakimi,
S. L. On
planar graphical degree sequences. SIAM J. Appl. Math. 32 (1977), no. 3,
598--609.
Hakimi, S. L.; Schmeichel,
E. F. On the
connectivity of maximal planar graphs. J. Graph Theory 2 (1978), no. 4,
307--314.
Hakimi, S. L.; Schmeichel,
E. F. On
$p$-centers in networks. Transportation Sci. 12 (1978), no. 1, 1--15.
Hakimi, S. Louis;
Schmeichel, Edward F. Graphs and their
degree sequences: a survey. Theory and applications of graphs (Proc.
Internat. Conf., Western Mich. Univ., Kalamazoo, Mich., 1976), pp. 225--235, Lecture
Notes in Math., Vol. 642, Springer, Berlin-New York, 1978.（1978年4篇，仅列3篇）
S. L. Hakimi, E. F.
Schmeichel, C.Thomassen合作On the number of
Hamiltonian cycles in a maximal planar graph. J. Graph Theory 3 (1979), no.
4, 365--370.（Carsten Thomassen是1976年才博士毕业的哈密顿图世界大师，他的导师的导师是Omar Wing周昌教授是文革后第一个来美籍哈密顿图大师赖虹建教授的母校的）
哈佛大学王浩的博士Shimon Even（写了2本名著图论算法和组合数学算法）的博士O. Kariv,
S. L. Hakimi,.An
algorithmic approach to network location problems. II. The $p$-medians. SIAM
J. Appl. Math. 37 (1979), no. 3, 539--560.
Kariv, O.; Hakimi, S. L. An algorithmic
approach to network location problems. I. The $p$-centers. SIAM J. Appl.
Math. 37 (1979), no. 3, 513--538.
（1978年5篇，仅列3篇）
Hakimi, S. L.; Schmeichel,
E. F. The
number of triangles in a triangulation of a set of points in the plane.
Elem. Math. 35 (1980), no. 6, 137--142.
Kyung-Yong Chwa, S.
L. Hakimi. Schemes
for fault-tolerant computing: a comparison of modularly redundant and $t$-diagnosable
systems. Inform. and Control 49 (1981), no. 3, 212--238.
Nakajima,
K.; Leung, J. Y.-T.; Hakimi, S. L. Optimal two
processor scheduling of tree precedence constrained tasks with two execution
times. Performance Evaluation 1 (1981), no. 4, 320--330 (1982).
S. C.Ntafos,
S. L. Hakimi. On
the complexity of some coding problems. IEEE Trans. Inform. Theory 27
(1981), no. 6, 794--796. （1981年5篇，仅列3篇）
Nakajima, Kazuo; Hakimi, S.
Louis.
Complexity results for scheduling tasks with discrete starting times. J.
Algorithms 3 (1982), no. 4, 344--361.
Nakajima, K.; Hakimi, S. L.;
Lenstra,
J. K. Complexity
results for scheduling tasks in fixed intervals on two types of machines.
SIAM J. Comput. 11 (1982), no. 3, 512--520.
Hakimi, S. L.; Schmeichel,
E. F. Bounds
on the number of cycles of length three in a planar graph. Israel J. Math.
41 (1982), no. 1-2, 161--180.
Kreutzer, S. E.; Hakimi, S.
L. Adaptive
fault identification in two new diagnostic models. Twenty-first annual Allerton
conference on communication, control, and computing (Monticello, Ill.,
1983), 353--362, Univ. Illinois, Urbana, IL, 1983.
斯坦福大学教授Nimrod Megiddo,
Eitan Zemel, S. L.
Hakimi.. The
maximum coverage location problem. SIAM J. Algebraic Discrete Methods 4
(1983), no. 2, 253--261.
Hakimi, S. Louis. On locating new
facilities in a competitive environment. European J. Oper. Res. 12 (1983),
no. 1, 29--35.
Hakimi, S. Louis;
Schmeichel, Edward F. An adaptive
algorithm for system level diagnosis. J. Algorithms 5 (1984), no. 4,
526--530.
Hakimi, S. Louis; Nakajima,
Kazuo. On
adaptive system diagnosis. IEEE Trans. Comput. 33 (1984), no. 3, 234--240.
Abdol-Hossein. Esfahanian,
S. L. Hakimi. On
computing the connectivities of graphs and digraphs. Networks 14 (1984),
no. 2, 355--366.
Hakimi, S. Louis. Further results
on a generalization of edge-coloring. Graph theory with applications to
algorithms and computer science (Kalamazoo, Mich., 1984), 371--389,
Wiley-Intersci. Publ., Wiley, New York, 1985.
Abdol-Hossein. Esfahanian,
S. L. Hakimi.. Fault-tolerant
routing in de Bruijn communication networks. IEEE Trans. Comput. 34 (1985),
no. 9, 777--788.
Hakimi, S. Louis; Kariv,
Oded. A
generalization of edge-coloring in graphs. J. Graph Theory 10 (1986), no.
2, 139--154.
Kreutzer, S. E.; Hakimi, S.
L. System-level
diagnosis: analysis of two new models. Inform. Sci. 40 (1986), no. 2,
117--130.
Hyeong-Ah Choi, S.
L. Hakimi. Data
transfers in networks with transceivers. Networks 17 (1987), no. 4, 393--421.
Choi, Hyeong-Ah; Hakimi, S.
Louis. Scheduling
file transfers for trees and odd cycles. SIAM J. Comput. 16 (1987), no. 1,
162--168.
Hakimi, S. L.; Schmeichel,
E. F. A note
on the vertex arboricity of a graph. SIAM J. Discrete Math. 2 (1989), no.
1, 64--67.
Hakimi, S. L.; Schmeichel,
E.; Weinstein, J. Partitioning
planar graphs into independent sets and forests. Proceedings of the
Twenty-first Southeastern Conference on Combinatorics, Graph Theory, and
Computing (Boca Raton, FL, 1990). Congr. Numer. 78 (1990), 109--118.
Bauer, D.; Hakimi, S. L.;
Schmeichel, E. Recognizing
tough graphs is NP-hard. Discrete Appl. Math. 28 (1990), no. 3, 191--195.
Tai Ching Tuan, S.
L. Hakimi. River
routing with a small number of jogs. SIAM J. Discrete Math. 3 (1990), no.
4, 585--597. （1990年6篇，仅列3篇）
Martine Labbé, S.
L. Hakimi. Market
and locational equilibrium for two competitors. Oper. Res. 39 (1991), no.
5, 749--756.
Hakimi, S. Louis; Labbé, Martine;
Schmeichel,
Edward. The Voronoi partition of a network and its implications in location
theory. ORSA J. Comput. 4 (1992), no. 4, 412--417.
Bagchi, Anindo; Hakimi, S.
Louis. Data
transfers in broadcast networks. IEEE Trans. Comput. 41 (1992), no. 7,
842--847.
Bagchi, A.; Schmeichel, E.
F.; Hakimi, S. L. Sequential
information dissemination by packets. Networks 22 (1992), no. 4, 317--333.
Hakimi, S. L.; Schmeichel,
E. F. Gossiping
in radio networks. Ars Combin. 35 (1993), A, 155--160.
Ueno, S.; Bagchi, A.;
Hakimi, S. L.; Schmeichel, E. F. On minimum
fault-tolerant networks. SIAM J. Discrete Math. 6 (1993), no. 4, 565--574.
Hakimi, S. L.; Schmeichel,
E. F.; Labbé, Martine. On locating
path- or tree-shaped facilities on networks. Networks 23 (1993), no. 6,
543--555.
Labbé, Martine; Schmeichel,
E. F.; Hakimi, S. Louis. Approximation
algorithms for the capacitated plant allocation problem. Oper. Res. Lett.
15 (1994), no. 3, 115--126
Corneil, Derek G.; Masuyama,
Shigeru; Hakimi, S. Louis. Edge-disjoint
packings of graphs. Discrete Appl. Math. 50 (1994), no. 2, 135--148.
Bagchi, A.; Schmeichel, E.
F.; Hakimi, S. L. Parallel
information dissemination by packets. SIAM J. Comput. 23 (1994), no. 2,
355--372.
Hakimi, S. L.; Mitchem, J.;
Schmeichel, E. F. Degree-bounded
coloring of graphs: variations on a theme by Brooks. J. Graph Theory 20
(1995), no. 2, 177--194.
Moh, Teng-Sheng; Chang,
Tsu-Shuan; Hakimi, S. Louis. Globally
optimal floorplanning for a layout problem. IEEE Trans. Circuits Systems I
Fund. Theory Appl. 43 (1996), no. 9, 713--720.
Labbé, M.; Schmeichel, E.
F.; Hakimi, S. L. Errata and
comments on: "Approximation algorithms for the capacitated plant
allocation problem'' [Oper. Res. Lett. 15 (1994), no. 3, 115–126; MR1288700
(95f:90047)]. Oper. Res. Lett. 18 (1996), no. 4, 205.
Hakimi, S. L.; Mitchem, J.;
Schmeichel, E. Star arboricity
of graphs. Discrete Math. 149 (1996), no. 1-3, 93--98. （1996年4篇，仅列3篇）
Hakimi, S. Louis;
Schmeichel, Edward F. Locating
replicas of a database on a network. Networks 30 (1997), no. 1, 31--36.
Hakimi, S. Louis;
Schmeichel, Edward F.; Young, Neal E. Orienting
graphs to optimize reachability. Inform. Process. Lett. 63 (1997), no. 5,
229--235.
Ge, Zhengyu; Hakimi, S.
Louis. Disjoint
rooted spanning trees with small depths in Debruijn and Kautz graphs. SIAM
J. Comput. 26 (1997), no. 1, 79--92. （1997年4篇，仅列3篇）
Hakimi, S. Louis; Mitchem,
John; Schmeichel, Edward. Short proofs of
theorems of Nash-Williams and Tutte. Ars Combin. 50 (1998), 257--266.
Hakimi, S. L.; Labbé,
Martine; Schmeichel, E. F. Locations on
time-varying networks. Centrality concepts in network location. Networks 34
(1999), no. 4, 250--257.
Hakimi, S. Louis;
Schmeichel, Edward F. Improved bounds
for the chromatic index of graphs and multigraphs. J. Graph Theory 32
(1999), no. 4, 311--326.
Griggs, Jerrold R.;
Woltermann, Michael; Borchers, B.; Callan, D.; Chapman, R. J.; Chilcott, J.;
Cull, P.; Ferguson, A.; Martin, K.; Gagola, G. G.; Golomb, S. W.; Guilford, J.;
Hakimi, S. L.; Schmeichel, E.; Haugland, J. K.; Jepson, C. H.; Johnsonbaugh,
R.; Kundgen, A.; Levandosky, S.; Lewis, J. T.; Lindsey, J. H., II; Lossers, O.
P.; Martin, R.; Morris, R.; Myerson, G.; Nieto, J H.; Rykken, E.; Sorensen, J.;
Schlosberg, J.; Seaman, W.; Ward, J. T.; Zeuge, Z.; GCHQ Problems
Group; WMC Problems GROUP. Problems and Solutions: Solutions: Tiling
Rectangles with Trominoes: 10641. Amer. Math. Monthly 107 (2000), no. 2, 179.
Coffman, William C.; Hakimi,
S. Louis; Schmeichel, Edward. Bounds for the
chromatic number of graphs with partial information. Discrete Math. 263
(2003), no. 1-3, 47--59.
Hakimi, S. Louis;
Schmeichel, Edward. Improved bounds
for the chromatic number of a graph. J. Graph Theory 47 (2004), no. 3,
217--225.
Bauer, D.; Hakimi, S. L.;
Kahl, N.; Schmeichel, E. Best monotone
degree bounds for various graph parameters. Proceedings of the Thirty-Ninth
Southeastern International Conference on Combinatorics, Graph Theory and
Computing. Congr. Numer. 192 (2008), 75--83.
Bauer, D.; Hakimi, S. L.;
Kahl, N.; Schmeichel, E. Sufficient
degree conditions for $k$-edge-connectedness of a graph. Networks 54
(2009), no. 2, 95--98.
其后是同性不同名的人的论文：
Hakimi,
Said; Zertiti, Abderrahim. Radial positive
solutions for a nonpositone problem in a ball. Electron. J. Differential
Equations 2009, No. 44, 6 pp.
Hakimi, Said. Nonexistence of radial positive solutions for a
nonpositone system in an annulus. Electron. J. Differential Equations 2011, No.
152, 7 pp.
跟着做他的工作主要有：
Lantz, Benjamin. On the
Havel-Hakimi Residue of Degree Sequences and Its Relation to the Independence
Number. Thesis (Ph.D.)–University of Rhode Island. ProQuest LLC, Ann Arbor,
MI, 2020. 84 pp.
Barrus, Michael D.; Molnar, Grant. Graphs with the
strong Havel-Hakimi property. Graphs Combin. 32 (2016), no. 5, 1689--1697.
Brualdi, Richard A.; Fritscher, Eliseu. Tournaments
associated with multigraphs and a theorem of Hakimi. Discrete Math. 338
(2015), no. 2, 229--235.
Yin, Jian-Hua. A Havel-Hakimi
type procedure and a sufficient condition for a sequence to be potentially
$S_{r,s}$-graphic. Czechoslovak Math. J. 62(137) (2012), no. 3, 863--867.
Barrus, Michael D. Havel-Hakimi
residues of unigraphs. Inform. Process. Lett. 112 (2012), no. 1-2, 44--48.
Xie, Gao Gang; Zhang, Da Fang; Wang, Zhong Sheng; Miu, Ying Hua. Diagnosis
algorithms based on the Chaw & Hakimi model. (Chinese) Hunan Daxue
Xuebao 25 (1998), no. 6, 96--101.
Griggs, Jerrold R.; Kleitman, Daniel J. Independence
and the Havel-Hakimi residue. Graph theory and applications (Hakone, 1990).
Discrete Math. 127 (1994), no. 1-3, 209--212.
Mao, Jing Zhong. On E. F.
Schmeichel-S. L. Hakimi conjecture about planar graphical degree sequences.
Kexue Tongbao (English Ed.) 32 (1987), no. 3, 145--146. MR0898119
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Mao, Jing Zhong. The conjecture
of E. F. Schmeichel and S. L. Hakimi concerning degree sequences of planar
graphs. (Chinese) Kexue Tongbao (Chinese) 30 (1985), no. 24, 1852--1853.
Böttger, G.; Harders, H. Note on a problem
by S. L. Hakimi concerning planar graphs without parallel elements. J. Soc.
Indust. Appl. Math. 12 (1964), 838--839.
这些论文主要跟着做Havel–Hakimi算法-它是图论中用于判定非负整数序列能否构成简单图度序列的递归算法；提出于Hakimi, S. L. On
realizability of a set of integers as degrees of the vertices of a linear graph.
I. J. Soc. Indust. Appl. Math. 10 (1962), 496--506.论文-其中建立了算法基础框架；（ Kleitman, D. J.; Wang, D. L. Algorithms for
constructing graphs and digraphs with given valences and factors. Discrete
Math. 6 (1973), 79--88. 提出广义版本）
Michael Alexander Harrison的博士论文是“Combinatorial
Problems in Boolean Algebras and Applications to the Theory of Switching”(他的妻子是同系教授Susan Lois Graham) ，他的博士Ivan M. Havel在美国数学评论中有59篇其中标题中含“图”的12图论论文：
Berrachedi,
Abdelhafid; Havel, Ivan; Mulder, Henry Martyn. Spherical and clockwise spherical
graphs. Czechoslovak Math. J. 53(128) (2003), no. 2, 295--309.
Havel, Ivan;
Zelinka, Bohdan. On 2-periodic graphs of a certain graph operator. Discuss.
Math. Graph Theory 21 (2001), no. 1, 13--30.
Ivan M. Havel在1976年发表2篇分别在离散数学杂志和人工智能杂志发表：Geller, Matthew M.;
Harrison, Michael A.; Havel, Ivan M. Normal forms of deterministic grammars.
Discrete Math. 16 (1976), no. 4, 313--321.和
Štepankova, Olga; Havel, Ivan M. A logical theory of robot problem
solving. Artificial Intelligence 7 (1976), no. 2, 129--161.
哈密顿图的泛圈性和偶泛圈性
赵克文
（海南热带海洋学院，数学与信息学科研究所，海南三亚，572022）
摘要：1988年张社民在国内的论文和1993年再在国际组合数学顶刊J. Combin. Theory发表的论文中都得到2个结果：(1) 若G是n 阶哈密顿图，如果存在G的一点x并对任何和x不相邻的点y都有d (x)+d(y)≥n，则G是泛圈图或Kn/2, n/2。(2) 若G=(X, Y;
E) 是n阶哈密顿二分图，|X|
=|Y|=n>3.，如果存在X的一点x并对Y中每一点y都有d(x)+d(y)≥n+1，则G是偶泛圈图。也就是张社民考虑了所有和x不相邻的点的情况，但这随着n 趋于无穷就更复杂。现在本论文得出3个点的情形，并对2个点的情形提出泛圈性猜想。
关键词：哈密顿图；泛圈性。
中图分类号：O157.5
本文的定义、概念、记号等参考众所熟悉的邦迪和默蒂的名著《图论及其应用》，并得到下面结果：
定理1：图Cn={1, 2，…n, 1}一个n阶哈密顿圈，若存在一点x（比如x=1）d(x)>n/2，并2和n-1相邻，则G是泛圈图。
证明：因2和n-1相邻则有长为n-1的不含点1的圈Cn-1，并因d(1)>n/2，则对任何m£n-1必存在Cm，即必存在Cn上连续点数m-1个的路的2端点和1都相邻（否则和1不相邻的点数就不会少相邻的点数，就有d(1)£ n/2，矛盾）。从而证明G是泛圈图。
定理2：图Cn={1, 2，…n, 1}一个n阶哈密顿圈，若存在一点x（比如x=1并2和n-1不相邻）d(1)>n/2并d(2)+d(n-1) >n，则G是泛圈图（这是只考虑哈密顿圈上相连接的3点就行；而张社民校长的是n趋于无穷大也要考虑无穷大）；
证明：因d(2)+d(n-1) >n，所以存在某点m和n-1相邻并m+1和2相邻（否则将有d(2)+d(n-1) £n，矛盾）。从而转化为定理1的情形。
定理3：图Cn={1, 2，…n, 1}一个n阶哈密顿圈，若存在一点x（可认为x=1），d(x)>n/2，则当m£n/2时，必存在某2点y、w（y-w=m）都和点1都相邻，从而存在长为3至n/2的圈。
证明：其证明显然，因为若不存在这样的2点则d(x)£n/2，矛盾。
定理4： 图Cn是一个n阶哈密顿圈，若存在Cn上相连（或相距2）的2点x，y有d(x)+d(y) >n（d(x)+d(y) >n+1）并x，y的邻边不交叉（即x，y的邻边各全在它俩的一边），则易知G是泛圈图；
证明：不妨认为d(x) ³d(y)，由d(x)+d(y) >n，得d(x) ³ n/2且显然x和它这边的d(x) ³ n/2的点都相邻，从而容易构造出不同长度全部的n-2的圈，即G是泛圈图。
推论4： 图Cn是一个n阶哈密顿圈，若存在Cn上的2点x，y有d(x)+d(y) >n并x，y的邻边不交叉（即x，y的邻边各全在它俩的一边），则易知G是泛圈图；
证明：由条件可得出，除至多2个点外的每一个点都至少和x，y之一相邻，从而类似定理4容易易构造出不同长度全部的n-2的圈，即G是泛圈图。
由定理4，已知x，y的邻边不交叉时的情况；那么若x，y的邻边有交叉时的情况又如何呢？（如若存在2点x，y有d(x)+d(y) >n，甚至仅³n，打个比方如若d(x)边和d(y)边都相交，则d(x)条边和d(y)条边与Cn上的路就围城构成2d(x)´d(y)个不同的圈（> 2d(x)´(n- d(x))个圈或³2 d(x)´(n- d(x))个圈），比x，y的邻边都不交叉的是泛圈图的定理4的不同圈多一倍，那么G更应是泛圈图吗？这是比张社民教授的定理好得多的结果（知x，y的邻边有交叉时则这2边和Cn上的路就围城构成2个不同圈（若存在2点x，y有d(x)+d(y) >n（甚至仅³n），则d(x)条边和d(y)条边与Cn上的路就围城构成d(x)´d(y)个不同的圈（> d(x)´(n- d(x))个圈或³ d(x)´(n- d(x))个圈）。
泛圈图中长为3和n的圈是基本的，弱只缺至多某一圈长不是3和n的圈而含其它各长度的圈的图称为弱泛圈图。基于定理4，下面2点度和情形的猜想虽没有十足的理由猜想是泛圈图，但猜想是弱泛圈图是有理由的（因存在C3、C4、Cn-1、Cn的一般通常至少都是弱泛圈图）且也是一个巨大的进步，就此我们提出它：
猜想：图Cn是一个n阶哈密顿圈，若存在Cn上的2点x，y有d(x)+d(y) >n并x，y的邻边有交叉，则G是弱泛圈图。
总之，应先证明弱泛圈图，才进一步尝试证明泛圈图。
关于二分哈密顿图，就不难得出相应于上面定理结论的结果，这里就略去。
致谢：作者感谢导师柳柏濂教授的指教，感谢师弟周波教授的帮助！
参考文献
1、张社民，pancyclism and
bipancyclism of hamiltonian graphs,数学研究与评论1988年第02期,314。
2.、Shemin Zhang, Pancyclism and bipancyclism of
Hamiltonian graphs. J. Combin. Theory Ser. B 60 (1994), 2,
159—168。
3、J.A.邦迪，U.S.R.默蒂，《图论及其应用》，科学出版社，1984年
Pancyclism and
bipancyclism of Hamiltonian graphs
Kewen Zhao
(Hainan
Tropical Ocean University, Institute of Mathematics and Information Science,
Sanya, Hainan, 572022)
Abstract: In 1988, Zhang Shemin published
papers in domestic magazines and again in 1993 in the international top journal
J. Combin. Theory, where he obtained the following results: (1) If G is a hamiltonian graph of order n and if there exists a vertex xÎV(G) such that d(x)+d(y)≥n for each y not adjacent to x,
then G is either pancyclic or the
complete bipartite graph Kn/2,
n/2. (2) Let G =(X,
Y; E) be a hamiltonian bipartite graph with |X| =|Y| =n>3. If there exists a vertex xÎX such
that d(x)+d(y)≥n+1 for
each yÎY not
adjacent to x, then G is bipancyclic. Thus, Shemin Zhang
considers all points that are not adjacent to x, which becomes more complex as n tends to infinity. In this paper, we
derive the case of 3 points and propose a pancyclism conjecture for the case of
2 points.
Key words: Hamiltonian graphs; Pancyclism.
作者赵克文的通信地址：海南三亚市海南热带海洋学院数学与信息学科研究所，邮编572022；电子信箱：kwzhao2006@163.com