查看“︁立方图”︁的源代码

{{expand language|1=en|page=Cubic graph|time=2019-04-23T07:48:12+00:00}}

[[File:Petersen1_tiny.svg|右|缩略图|[[佩特森圖|彼得森图]]是立方图]]
[[File:Biclique_K_3_3.svg|右|缩略图|180x180像素|[[完全二分图]] <math>K_{3,3}</math>是立方二分图]]

在[[图论]]中，若一个[[图 (数学)|图]]的每个[[顶点 (图论)|顶点]][[度 (图论)|度数]]均为三，则称其为'''立方图'''（Cubic graph）、3-'''[[正則圖|正则图]]'''或'''三次图'''。

[[佩特森圖|彼得森图]]、[[三間小屋問題|汤玛森图]]等都是立方图。

== 对称性 ==
1932年，{{link-en|Ronald M. Foster|R. M. Foster}}首先寻找立方{{link-en|对称图|Symmetric graph}}的例子，并收集为{{link-en|Foster census|Foster census}}。<ref name="Ref_Foster">{{Citation|first1=R. M.|last1=Foster|title=Geometrical Circuits of Electrical Networks|journal=[[Transactions of the American Institute of Electrical Engineers]]|volume=51|pages=309–317|year=1932|doi=10.1109/T-AIEE.1932.5056068|issue=2}}.</ref>许多著名的图都是立方对称图，如[[汤玛森图]]、[[彼得森图]]等。[[威廉·湯瑪斯·圖特]]用满足下列性质的最大整数''s''来对立方对称图进行分类：图的[[自同构群]]在其所有长度为''s''的路径（其中不能有重复的边）组成的集合上作用是传递的。他证明了''s''最大只能取5，也即''s''的可能值是1到5。<ref>{{Citation
 | doi = 10.4153/CJM-1959-057-2
 | last = Tutte
 | first = W. T.
 | journal = Can. J. Math.
 | pages = 621–624
 | title = On the symmetry of cubic graphs
 | url = http://cms.math.ca/cjm/v11/p621
 | volume = 11
 | year = 1959
 | accessdate = 2019-05-10
 | archive-date = 2011-07-16
 | archive-url = https://web.archive.org/web/20110716145555/http://cms.math.ca/cjm/v11/p621
 }}.</ref>

==图着色与独立集==
根据[[布鲁克定理]]，除了''K''<sub>4</sub>以外的任何连通立方图都可以用至多三种颜色染色。也即，这样的连通立方图至少存在一个包含n/3个顶点的[[独立集]]，其中n是该图的顶点数。

根据[[Vizing定理]]，任一立方图的{{link-en|边色数|Edge chromatic number}}只能为三或四。3-边着色又称Tait-着色，Tait-着色方式将边集分割为三个[[匹配_(图论)|完美匹配]]。根据{{tsl|en|Kőnig%27s_theorem_(graph_theory)|Kőnig's_theorem}}每个二分立方图都有一个Tait-着色。

==哈密顿回路==
关于立方图是否具有[[哈密顿回路]]有许多研究。1880年，{{link-en|P.G. Tait|Peter Tait (physicist)}}猜想任一立方多面体图都有哈密顿回路。1946年，[[威廉·湯瑪斯·圖特]]提出了{{link-en|Tait猜想|Tait's conjecture}}的反例，有46个点的{{link-en|图特图|Tutte graph}}。1971年，图特猜想所有的二分立方图都有哈密顿回路。然而Joseph Horton提出了图特猜想的反例，有96个点的{{link-en|Horton图|Horton graph}}。<ref>Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 240, 1976.</ref>在这之后，{{link-en|Mark Ellingham|Mark Ellingham}}又提出了两个反例：{{link-en|Ellingham–Horton图|Ellingham–Horton graph}}。<ref>Ellingham, M. N. "Non-Hamiltonian 3-Connected Cubic Partite Graphs."Research Report No. 28, Dept. of Math., Univ. Melbourne, Melbourne, 1981.</ref><ref>{{Citation|last1=Ellingham|first1=M. N.|last2=Horton|first2=J. D.|title= Non-Hamiltonian 3-connected cubic bipartite graphs|journal=[[Journal of Combinatorial Theory]]|series=Series B| volume=34| pages=350–353| year=1983| doi=10.1016/0095-8956(83)90046-1|issue=3|doi-access=free}}.</ref>{{link-en|Barnette猜想|Barnette's conjecture}}（目前仍是猜想）将Tait猜想与图特猜想结合起来，称任一二分立方多面体图都有哈密顿回路。当一个立方图有哈密顿回路时，可以使用{{link-en|LCF表示法|LCF notation}}简洁地表示。

如果从所有<math>n</math>阶立方图中随机选取一个，那么它有相当大概率有哈密顿回路：当<math>n</math>趋近于无穷时，这个概率趋近于1。<ref>{{citation
 | last1 = Robinson | first1 = R.W.
 | last2 = Wormald | first2 = N.C.
 | doi = 10.1002/rsa.3240050209
 | issue = 2
 | journal = Random Structures and Algorithms
 | pages = 363–374
 | title = Almost all regular graphs are Hamiltonian
 | volume = 5
 | year = 1994}}.</ref>

{{link-en|David Eppstein|David Eppstein}}猜想任一<math>n</math>阶立方图最多有<math>2^{n/3}</math>（约等于<math>1.260^n</math>）条不同的哈密顿回路，且给出了极限情况下的例子。<ref>{{citation
 | last = Eppstein
 | first = David
 | authorlink = David Eppstein
 | arxiv = cs.DS/0302030
 | issue = 1
 | journal = [[Journal of Graph Algorithms and Applications]]
 | pages = 61–81
 | title = The traveling salesman problem for cubic graphs
 | url = http://jgaa.info/accepted/2007/Eppstein2007.11.1.pdf
 | volume = 11
 | year = 2007
 | doi = 10.7155/jgaa.00137
 | accessdate = 2020-12-27
 | archive-date = 2021-02-24
 | archive-url = https://web.archive.org/web/20210224121929/http://jgaa.info/accepted/2007/Eppstein2007.11.1.pdf
 }}.</ref>目前为止，得到证明的最佳估计为<math> O({1.276}^n)</math>。<ref>{{citation
 | last = Gebauer
 | first = H.
 | contribution = On the number of Hamilton cycles in bounded degree graphs
 | title = Proc. 4th Workshop on Analytic Algorithmics and Combinatorics (ANALCO '08)
 | url = https://epubs.siam.org/doi/pdf/10.1137/1.9781611972986.8
 | year = 2008
 }}.</ref>

==另见==
* {{link-en|一些简单的立方图|Table of simple cubic graphs}}

== 参考文献 ==
{{reflist}}

== 外部連結 ==
*{{cite web|url=http://mapleta.maths.uwa.edu.au/~gordon/remote/foster/|first1=Gordon|last1=Royle|title=Cubic symmetric graphs (The Foster Census)|deadurl=yes|archiveurl=https://web.archive.org/web/20111023234733/http://mapleta.maths.uwa.edu.au/~gordon/remote/foster/|archivedate=2011-10-23|df=}}
*{{mathworld|urlname=BicubicGraph|title=Bicubic Graph}}
*{{mathworld|urlname=CubicGraph|title=Cubic Graph}}

[[Category:图族]]