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第87行： − ==Properties== − + + + 第97行：
第98行： − 二部图可以用几种不同的方式来表示:+ − − * A graph is bipartite [[if and only if]] it does not contain an [[Cycle (graph theory)|odd cycle]].<ref>{{harvtxt|Asratian|Denley|Häggkvist|1998}}, Theorem 2.1.3, p. 8. Asratian et al. attribute this characterization to a 1916 paper by [[Dénes Kőnig]]. For infinite graphs, this result requires the [[axiom of choice]].</ref> − − * A graph is bipartite if and only if it is 2-colorable, (i.e. its [[chromatic number]] is less than or equal to 2).<ref name="adh98-7"/> − − * The [[Spectral graph theory|spectrum]] of a graph is symmetric if and only if it's a bipartite graph.<ref>{{citation − − | last = Biggs | first = Norman − − | last = Biggs | first = Norman − − | last = Biggs | first = Norman − − | edition = 2nd − − | edition = 2nd − − 2nd − − | isbn = 9780521458979 − − | isbn = 9780521458979 − − 9780521458979 − − | page = 53 − − | page = 53 − − 53 − − | publisher = Cambridge University Press − − | publisher = Cambridge University Press − − 剑桥大学出版社 − − | series = Cambridge Mathematical Library − − | series = Cambridge Mathematical Library − − | series = Cambridge Mathematical Library − − | title = Algebraic Graph Theory − − | title = Algebraic Graph Theory − − | title = 代数图论 − − | url = https://books.google.com/books?id=6TasRmIFOxQC&pg=PA53 − − | url = https://books.google.com/books?id=6TasRmIFOxQC&pg=PA53 − − Https://books.google.com/books?id=6tasrmifoxqc&pg=pa53 − − | year = 1994}}.</ref> − − | year = 1994}}.</ref> − − | year = 1994} . </ref > − − − − ===Kőnig's theorem and perfect graphs=== − − In bipartite graphs, the size of [[minimum vertex cover]] is equal to the size of the [[maximum matching]]; this is [[Kőnig's theorem (graph theory)|Kőnig's theorem]].<ref>{{cite journal − − In bipartite graphs, the size of minimum vertex cover is equal to the size of the maximum matching; this is Kőnig's theorem.<ref>{{cite journal − − 在二部图中，最小顶点覆盖的大小等于最大匹配的大小，这就是 k-nig 定理。{ cite journal − − | author = Kőnig, Dénes − − | author = Kőnig, Dénes − − 作者: k nig，d é nes − − | authorlink = Dénes Kőnig − − | authorlink = Dénes Kőnig − − | authorlink = d é nes k nig − − | title = Gráfok és mátrixok − − | title = Gráfok és mátrixok − − | title = Gráfok és mátrixok − − | journal = Matematikai és Fizikai Lapok − − | journal = Matematikai és Fizikai Lapok − − | journal = Matematikai és Fizikai Lapok − − | volume = 38 − − | volume = 38 − − 38 − − | year = 1931 − − | year = 1931 − − 1931年 − | pages = 116–119}}.</ref><ref>{{citation+ + − | pages = 116–119}}.</ref><ref>{{citation+ + − 116-119} . </ref > < ref > { citation+ + − | last1 = Gross | first1 = Jonathan L. − | last1 = Gross | first1 = Jonathan L. − 1 = Gross | first1 = Jonathan l.+ − | last2 = Yellen | first2 = Jay+ − | last2 = Yellen | first2 = Jay+ − 2 = Yellen | first2 = Jay+ − − | edition = 2nd − − | edition = 2nd − − 2nd − − | isbn = 9781584885054 − − | isbn = 9781584885054 − − 9781584885054 − − | page = 568 − − | page = 568 − − 568 − − | publisher = CRC Press − − | publisher = CRC Press − − | publisher = CRC Press − − | series = Discrete Mathematics And Its Applications − − | series = Discrete Mathematics And Its Applications − − 系列 = 离散数学及其应用 − − | title = Graph Theory and Its Applications − − | title = Graph Theory and Its Applications − − | title = 图论及其应用 − − | url = https://books.google.com/books?id=-7Q_POGh-2cC&pg=PA568 − − | url = https://books.google.com/books?id=-7Q_POGh-2cC&pg=PA568 − − Https://books.google.com/books?id=-7q_pogh-2cc&pg=pa568 − − | year = 2005}}.</ref> An alternative and equivalent form of this theorem is that the size of the [[maximum independent set]] plus the size of the maximum matching is equal to the number of vertices. − − | year = 2005}}.</ref> An alternative and equivalent form of this theorem is that the size of the maximum independent set plus the size of the maximum matching is equal to the number of vertices. − − 2005}}.</ref > 这个定理的另一种等价形式是最大独立集的大小加上最大匹配的大小等于顶点的数目。 − − In any graph without [[isolated vertex|isolated vertices]] the size of the [[minimum edge cover]] plus the size of a maximum matching equals the number of vertices.<ref>{{citation − − In any graph without isolated vertices the size of the minimum edge cover plus the size of a maximum matching equals the number of vertices.<ref>{{citation − − 在任何没有孤立顶点的图中，最小边覆盖的大小加上最大匹配的大小等于顶点的数目。< ref > { citation − − | last1 = Chartrand | first1 = Gary − − | last1 = Chartrand | first1 = Gary − − 1 = Chartrand | first1 = Gary − − | last2 = Zhang | first2 = Ping − − | last2 = Zhang | first2 = Ping − − 2 = Zhang | first2 = Ping − − | isbn = 9780486483689 − − | isbn = 9780486483689 − − 9780486483689 − − | pages = 189–190 − − | pages = 189–190 − − | 页数 = 189-190 − − | publisher = Courier Dover Publications − − | publisher = Courier Dover Publications − − 2012年10月21日 | 出版商 = 多佛出版社 − − | title = A First Course in Graph Theory − − | title = A First Course in Graph Theory − − 图论第一课 − − | url = https://books.google.com/books?id=ocIr0RHyI8oC&pg=PA189 − − | url = https://books.google.com/books?id=ocIr0RHyI8oC&pg=PA189 − − Https://books.google.com/books?id=ocir0rhyi8oc&pg=pa189 − − | year = 2012}}.</ref> Combining this equality with Kőnig's theorem leads to the facts that, in bipartite graphs, the size of the minimum edge cover is equal to the size of the maximum independent set, and the size of the minimum edge cover plus the size of the minimum vertex cover is equal to the number of vertices. − − | year = 2012}}.</ref> Combining this equality with Kőnig's theorem leads to the facts that, in bipartite graphs, the size of the minimum edge cover is equal to the size of the maximum independent set, and the size of the minimum edge cover plus the size of the minimum vertex cover is equal to the number of vertices. − − 2012}}.</ref > 将这个等式与 k nig 定理结合起来得到这样一个结论: 在二部图中，最小边覆盖的大小等于最大独立集的大小，最小边覆盖的大小加上最小顶点覆盖的大小等于顶点的数目。 第331行：
第125行： − 另一类相关结果涉及完美图: 每一个二部图、每一个二部图的补图、每一个二部图的线图以及每一个二部图的线图的补图都是完美的。二部图的完美性是显而易见的(它们的色数是2，它们的最大团大小也是2) ，但二部图的补图的完美性是不那么琐碎的，是对 k nig 定理的另一种重述。这是激发完美图最初定义的结果之一+ − − |volume= 184 |series= Graduate Texts in Mathematics − − |volume= 184 |series= Graduate Texts in Mathematics − − 184 | series = 数学研究生教材 − − |author=Béla Bollobás|publisher =Springer|year= 1998 − − |author=Béla Bollobás|publisher =Springer|year= 1998 − − 1998年 − − |isbn= 9780387984889|url=https://books.google.com/books?id=SbZKSZ-1qrwC&pg=PA165|page=165|author-link= Béla Bollobás }}.</ref> Perfection of the complements of line graphs of perfect graphs is yet another restatement of Kőnig's theorem, and perfection of the line graphs themselves is a restatement of an earlier theorem of Kőnig, that every bipartite graph has an [[edge coloring]] using a number of colors equal to its maximum degree. − − |isbn= 9780387984889|url=https://books.google.com/books?id=SbZKSZ-1qrwC&pg=PA165|page=165|author-link= Béla Bollobás }}.</ref> Perfection of the complements of line graphs of perfect graphs is yet another restatement of Kőnig's theorem, and perfection of the line graphs themselves is a restatement of an earlier theorem of Kőnig, that every bipartite graph has an edge coloring using a number of colors equal to its maximum degree. − − 9780387984889 | url = https://books.google.com/books?id=sbzksz-1qrwc&pg=pa165|page=165|author-link= · 贝拉 · 博洛巴斯}。完全图的线图的补图的完美性是对 k nig 定理的又一次重申，线图本身的完美性是对 k nig 早期定理的重申，即每个二部图都有一个边着色，其颜色数等于它的最大度。 第357行：
第133行： − 根据完美图问题，完美图有一个类似于二部图的禁止图: 一个图是二部图的当且仅当它作为一个子图没有奇圈，并且一个图是完美的当且仅当它没有奇圈或它作为诱导子图的补图。二部图，二部图的线图，以及它们的补图在完美图问题证明中的5个基本类型的完美图中的4个。< ref > { citation+ − − | last1 = Chudnovsky | first1 = Maria | author1-link = Maria Chudnovsky − | last1 = Chudnovsky | first1 = Maria | author1-link = Maria Chudnovsky − 1 = Chudnovsky | first1 = Maria | author1-link = Maria Chudnovsky − | last2 = Robertson | first2 = Neil | author2-link = Neil Robertson (mathematician)+ − − | last2 = Robertson | first2 = Neil | author2-link = Neil Robertson (mathematician) − − | last2 = Robertson | first2 = Neil | author2-link = Neil Robertson (数学家) − − | last3 = Seymour | first3 = Paul | author3-link = Paul Seymour (mathematician) − − | last3 = Seymour | first3 = Paul | author3-link = Paul Seymour (mathematician) − − 3 = Seymour | first3 = Paul | author3-link = Paul Seymour (数学家) − − | last4 = Thomas | first4 = Robin | author4-link = Robin Thomas (mathematician) − − | last4 = Thomas | first4 = Robin | author4-link = Robin Thomas (mathematician) − − | last4 = Thomas | first4 = Robin | author4-link = Robin Thomas (数学家) − − | doi = 10.4007/annals.2006.164.51 − − | doi = 10.4007/annals.2006.164.51 − − 2006.164.51 − − | issue = 1 − − | issue = 1 − − 1 − − | journal = [[Annals of Mathematics]] − − | journal = Annals of Mathematics − − 2012年3月24日 | 日志 = 数学纪事 − − | pages = 51–229 − − | pages = 51–229 − − | 页数 = 51-229 − − | title = The strong perfect graph theorem − − | title = The strong perfect graph theorem − − 纽约完美图问题 − − | volume = 164 − − | volume = 164 − − 164 − − | year = 2006| citeseerx = 10.1.1.111.7265| arxiv = math/0212070}}.</ref> − − | year = 2006| citeseerx = 10.1.1.111.7265| arxiv = math/0212070}}.</ref> − − 2006 | citeseerx = 10.1.1.111.7265 | arxiv = math/0212070}} . </ref > − − − − 第433行：
第143行： − 对于一个顶点，相邻顶点的数目称为顶点的度数，表示为 < math > deg (v) </math > 。+ − − The ''degree sum formula'' for a bipartite graph states that − − The degree sum formula for a bipartite graph states that − − 二部图的度和公式表明: − − :<math>\sum_{v \in V} \deg(v) = \sum_{u \in U} \deg(u) = |E|\, .</math> − − <math>\sum_{v \in V} \deg(v) = \sum_{u \in U} \deg(u) = |E|\, .</math> − − <math>\sum_{v \in V} \deg(v) = \sum_{u \in U} \deg(u) = |E|\, .</math> − 第453行：
第150行： − 二部图的度序列是一对列表，每一个列表包含两部分的度。例如，完全二分图 k < sub > 3,5 </sub > 具有度序列 < math > (5,5,5) ，(3,3,3,3,3,3) </math > 。同构二部图具有相同的度序列。然而，度序列通常不能唯一地识别二部图，在某些情况下，不同构的二部图可能具有相同的度序列。+ 第461行：
第158行： − 二部实现问题是寻找度序列为两个给定自然数列表的简单二部图的问题。(尾随零可能被忽略，因为它们通过向有向图添加适当数量的孤立顶点来实现。)+
无编辑摘要
超立方体图，局部立方体和中位数图均为二分图。判别方法是将这些图中的顶点用位向量（二进制位组成的向量）进行标记，然后对比其中两个顶点的位向量，发现当且仅当位向量中只有一个位元是不同的时候，该两个顶点相邻。另外判定该图的二分性可以通过观察每个顶点的位向量，奇数位向量和偶数位向量分别为该图的二分顶点子集。树图和方图都是中位数图，而所有中位数图都是局部立方体。
超立方体图，局部立方体和中位数图均为二分图。判别方法是将这些图中的顶点用位向量（二进制位组成的向量）进行标记，然后对比其中两个顶点的位向量，发现当且仅当位向量中只有一个位元是不同的时候，该两个顶点相邻。另外判定该图的二分性可以通过观察每个顶点的位向量，奇数位向量和偶数位向量分别为该图的二分顶点子集。树图和方图都是中位数图，而所有中位数图都是局部立方体。
===Characterization===
== Properties 属性 ==
=== Characterization 表征 ===
Bipartite graphs may be characterized in several different ways:
Bipartite graphs may be characterized in several different ways:
Bipartite graphs may be characterized in several different ways:
Bipartite graphs may be characterized in several different ways:
二分图可以用以下几种不同的方式来描述其特征：
* A graph is bipartite [[if and only if]] it does not contain an [[Cycle (graph theory)|odd cycle]].<ref>{{harvtxt|Asratian|Denley|Häggkvist|1998}}, Theorem 2.1.3, p. 8. Asratian et al. attribute this characterization to a 1916 paper by [[Dénes Kőnig]]. For infinite graphs, this result requires the [[axiom of choice]].
* 当且仅当它不包含奇数环的时候，该图为二分图。
* A graph is bipartite if and only if it is 2-colorable, (i.e. its [[chromatic number]] is less than or equal to 2).
* 当且仅当图是2色系图2-colorable（即，其色数小于或等于2，详见“图着色”）时，该图为二分图。
* The [[Spectral graph theory|spectrum]] of a graph is symmetric if and only if it's a bipartite graph.
* 当且仅当它是二分图时，图的频谱spectrum是对称的。
=== Kőnig's theorem and perfect graphs 柯尼希定理和完美图 ===
In bipartite graphs, the size of minimum vertex cover is equal to the size of the maximum matching; this is Kőnig's theorem.[16][17] An alternative and equivalent form of this theorem is that the size of the maximum independent set plus the size of the maximum matching is equal to the number of vertices. In any graph without isolated vertices the size of the minimum edge cover plus the size of a maximum matching equals the number of vertices.[18] Combining this equality with Kőnig's theorem leads to the facts that, in bipartite graphs, the size of the minimum edge cover is equal to the size of the maximum independent set, and the size of the minimum edge cover plus the size of the minimum vertex cover is equal to the number of vertices.
In bipartite graphs, the size of minimum vertex cover is equal to the size of the maximum matching; this is Kőnig's theorem.[16][17] An alternative and equivalent form of this theorem is that the size of the maximum independent set plus the size of the maximum matching is equal to the number of vertices. In any graph without isolated vertices the size of the minimum edge cover plus the size of a maximum matching equals the number of vertices.[18] Combining this equality with Kőnig's theorem leads to the facts that, in bipartite graphs, the size of the minimum edge cover is equal to the size of the maximum independent set, and the size of the minimum edge cover plus the size of the minimum vertex cover is equal to the number of vertices.
在二分图中，最小顶点覆盖Minimum vertex cover（顶点数）等于最大匹配数Maximum matching(边数)；这就是柯尼希定理Kőnig's theorem。该定理的另一种等效形式是，最大独立集Maximum independent set（顶点数）加上最大匹配数等于顶点的数量。在任何没有孤立顶点的图中，最小边覆盖Minimum edge cover（边数）加上最大匹配数等于顶点数。于是，将以上等式与柯尼希定理相结合，得出以下事实：在二分图中，最小边覆盖数值等于最大独立集数值，最小边覆盖数值加上最小顶点覆盖数值等于顶点数。
Another class of related results concerns perfect graphs: every bipartite graph, the complement of every bipartite graph, the line graph of every bipartite graph, and the complement of the line graph of every bipartite graph, are all perfect. Perfection of bipartite graphs is easy to see (their chromatic number is two and their maximum clique size is also two) but perfection of the complements of bipartite graphs is less trivial, and is another restatement of Kőnig's theorem. This was one of the results that motivated the initial definition of perfect graphs.<ref>{{citation|title=Modern Graph Theory
Another class of related results concerns perfect graphs: every bipartite graph, the complement of every bipartite graph, the line graph of every bipartite graph, and the complement of the line graph of every bipartite graph, are all perfect. Perfection of bipartite graphs is easy to see (their chromatic number is two and their maximum clique size is also two) but perfection of the complements of bipartite graphs is less trivial, and is another restatement of Kőnig's theorem. This was one of the results that motivated the initial definition of perfect graphs.<ref>{{citation|title=Modern Graph Theory
另一类相关特性与完美图有关：每个二分图，以及其补图、其线图、其线图的补图均为完美图。其实关于二分图的完美性很容易就可以看出来（他们的色数为2，最大的团数同样为2），但是二分图的补图完美性判定就更简单些，可以看作是柯尼希定理的另一种陈述。其实在定义完美图这一概念的初期，它是作为其中之一的论证结果而存在的。同时完美图线图的补图完美性也是作为柯尼希定理的另一种陈述，线图的完美性陈述了比较早期的柯尼希定理，即每一个二分图的着色边，其色数都等于其最大节点度数。
According to the strong perfect graph theorem, the perfect graphs have a forbidden graph characterization resembling that of bipartite graphs: a graph is bipartite if and only if it has no odd cycle as a subgraph, and a graph is perfect if and only if it has no odd cycle or its complement as an induced subgraph. The bipartite graphs, line graphs of bipartite graphs, and their complements form four out of the five basic classes of perfect graphs used in the proof of the strong perfect graph theorem.<ref>{{citation
According to the strong perfect graph theorem, the perfect graphs have a forbidden graph characterization resembling that of bipartite graphs: a graph is bipartite if and only if it has no odd cycle as a subgraph, and a graph is perfect if and only if it has no odd cycle or its complement as an induced subgraph. The bipartite graphs, line graphs of bipartite graphs, and their complements form four out of the five basic classes of perfect graphs used in the proof of the strong perfect graph theorem.<ref>{{citation
根据强完美图定理，完美图具有类似于二分图的禁止图特征Forbidden graph characterization：当且仅当它没有奇环的子图时，图才是二分的；当且仅当它没有奇环或其补图作为导出子图时，图才是完美的。二分图，其线图以及补图构成了完美图五种基本类别中的四个，被用于证明了强完美图定理。
=== Degree 度 ===
===Degree===
For a vertex, the number of adjacent vertices is called the [[degree (graph theory)|degree]] of the vertex and is denoted <math>\deg(v)</math>.
For a vertex, the number of adjacent vertices is called the [[degree (graph theory)|degree]] of the vertex and is denoted <math>\deg(v)</math>.
For a vertex, the number of adjacent vertices is called the degree of the vertex and is denoted <math>\deg(v)</math>.
For a vertex, the number of adjacent vertices is called the degree of the vertex and is denoted <math>\deg(v)</math>.
对于一个顶点，其相邻顶点的数量称为该顶点的度数，并表示为deg（v）。二分图的总度数和公式为：
The degree sequence of a bipartite graph is the pair of lists each containing the degrees of the two parts <math>U</math> and <math>V</math>. For example, the complete bipartite graph K<sub>3,5</sub> has degree sequence <math>(5,5,5),(3,3,3,3,3)</math>. Isomorphic bipartite graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a bipartite graph; in some cases, non-isomorphic bipartite graphs may have the same degree sequence.
The degree sequence of a bipartite graph is the pair of lists each containing the degrees of the two parts <math>U</math> and <math>V</math>. For example, the complete bipartite graph K<sub>3,5</sub> has degree sequence <math>(5,5,5),(3,3,3,3,3)</math>. Isomorphic bipartite graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a bipartite graph; in some cases, non-isomorphic bipartite graphs may have the same degree sequence.
一个二分图的度数序列是一对列表，每个列表包含两个部分U和V的度数。例如，完全二分图K3,5具有度数序列（5,5,5），（3,3,3,3,3）。其同构二分图具有相同的度数序列。但是，度数序列通常不是定义二分图的唯一方法；在某些情况下，非同构二分图可能具有相同的度数序列。
The bipartite realization problem is the problem of finding a simple bipartite graph with the degree sequence being two given lists of natural numbers. (Trailing zeros may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the digraph.)
The bipartite realization problem is the problem of finding a simple bipartite graph with the degree sequence being two given lists of natural numbers. (Trailing zeros may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the digraph.)
二分实现问题bipartite realization problem是通过已知的两组自然数序列作为度数序列，来查找简单的二分图的判定问题（数理逻辑）。（尾随零可以忽略，因为通过向图添加适当数量的孤立顶点可以轻松实现尾随零。）