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第41行： − + − + − − | last1 = Wasserman | first1 = Stanley − − | last1 = Wasserman | first1 = Stanley − − 1 = Wasserman | first1 = Stanley − − | author-link1= Stanley Wasserman − − | author-link1= Stanley Wasserman − − 1 = Stanley Wasserman − − | last2 = Faust | first2 = Katherine − − | last2 = Faust | first2 = Katherine − − | last2 = Faust | first2 = Katherine − − | isbn = 9780521387071 − − | isbn = 9780521387071 − − 9780521387071 − − | pages = 299–302 − − | pages = 299–302 − − | 页数 = 299-302 − − | publisher = Cambridge University Press − − | publisher = Cambridge University Press − − 剑桥大学出版社 − − | series = Structural Analysis in the Social Sciences − − | series = Structural Analysis in the Social Sciences − − 系列 = 社会科学中的结构分析 − − | title = Social Network Analysis: Methods and Applications − − | title = Social Network Analysis: Methods and Applications − − 社会网络分析: 方法和应用 − − | url = https://books.google.com/books?id=CAm2DpIqRUIC&pg=PA299 − − | url = https://books.google.com/books?id=CAm2DpIqRUIC&pg=PA299 − − Https://books.google.com/books?id=cam2dpiqruic&pg=pa299 − − | volume = 8 − − | volume = 8 − − 8 − − | year = 1994}}.</ref> − − | year = 1994}}.</ref> − − | year = 1994} . </ref > 第147行：
第81行： − + − − * The [[complete bipartite graph]] on ''m'' and ''n'' vertices, denoted by ''K<sub>n,m</sub>'' is the bipartite graph <math>G = (U, V, E)</math>, where ''U'' and ''V'' are disjoint sets of size ''m'' and ''n'', respectively, and ''E'' connects every vertex in ''U'' with all vertices in ''V''. It follows that ''K<sub>m,n</sub>'' has ''mn'' edges.<ref>{{harvtxt|Asratian|Denley|Häggkvist|1998}}, p. 11.</ref> Closely related to the complete bipartite graphs are the [[crown graph]]s, formed from complete bipartite graphs by removing the edges of a [[perfect matching]].<ref>{{citation − − | last1 = Archdeacon | first1 = D. | author1-link = Dan Archdeacon − − | last1 = Archdeacon | first1 = D. | author1-link = Dan Archdeacon − − 1 = Archdeacon | first1 = d | author1-link = Dan Archdeacon − − | last2 = Debowsky | first2 = M. − − | last2 = Debowsky | first2 = M. − − 2 = Debowsky | first2 = m. − − | last3 = Dinitz | first3 = J. | author3-link = Jeff Dinitz − − | last3 = Dinitz | first3 = J. | author3-link = Jeff Dinitz − − 3 = j.| author3-link = Jeff Dinitz − − | last4 = Gavlas | first4 = H. − − | last4 = Gavlas | first4 = H. − − 4 = Gavlas | first4 = h. − − | doi = 10.1016/j.disc.2003.11.021 − − | doi = 10.1016/j.disc.2003.11.021 − − | doi = 10.1016/j.disc. 2003.11.021 − − | issue = 1–3 − − | issue = 1–3 − − 第一季，第三集 − − | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]] − − | journal = Discrete Mathematics − − 2012年3月24日 | 日志 = 离散数学 − − | pages = 37–43 − − | pages = 37–43 − − | 页数 = 37-43 − − | title = Cycle systems in the complete bipartite graph minus a one-factor − − | title = Cycle systems in the complete bipartite graph minus a one-factor − − | title = 完全二分图的循环系统减去一个因子 − − | volume = 284 − − | volume = 284 − − 284 − − | year = 2004| doi-access = free − − | year = 2004| doi-access = free − − 2004 | doi-access = free − − }}.</ref> − − }}.</ref> − − } . </ref > + − −
→Examples
== Examples ==
== Examples ==
When modelling [[heterogeneous relation|relations]] between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an ''affiliation network'', a type of bipartite graph used in [[social network analysis]].<ref>{{citation
When modelling [[heterogeneous relation|relations]] between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an ''affiliation network'', a type of bipartite graph used in [[social network analysis]].
When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.<ref>{{citation
When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.<ref>{{citation
当建立两类不同对象之间的模型关系时，二部图往往自然而然地出现。例如，一个足球运动员和俱乐部的图，如果一个运动员为一个俱乐部效力，那么该运动员和俱乐部之间就有一条边，这是一个隶属关系网络的自然例子，这是一种用于社会网络分析的二部图。< ref > { citation
当建立两类不同对象之间的模型关系时，二部图往往自然而然地出现。例如，一个足球运动员和俱乐部的图，如果一个运动员为一个俱乐部效力，那么该运动员和俱乐部之间就有一条边，这是一个隶属关系网络的自然例子，这是一种用于社会网络分析的二部图。
* [[Cycle graph]]s with an even number of vertices are bipartite.<ref name="s12"/>
* [[Cycle graph]]s with an even number of vertices are bipartite.<ref name="s12"/>
* Every [[planar graph]] whose [[Glossary of graph theory#Genus|faces]] all have even length is bipartite.<ref>{{citation|first=Alexander|last=Soifer|authorlink=Alexander Soifer|year=2008|title=The Mathematical Coloring Book|title-link= The Mathematical Coloring Book |publisher=Springer-Verlag|isbn=978-0-387-74640-1|pages=136–137}}. This result has sometimes been called the "two color theorem"; Soifer credits it to a famous 1879 paper of [[Alfred Kempe]] containing a false proof of the [[four color theorem]].</ref> Special cases of this are [[grid graph]]s and [[squaregraph]]s, in which every inner face consists of 4 edges and every inner vertex has four or more neighbors.<ref>{{citation|title=Combinatorics and geometry of finite and infinite squaregraphs|first1=H.-J.|last1=Bandelt|first2=V.|last2=Chepoi|first3=D.|last3=Eppstein|author3-link=David Eppstein|arxiv=0905.4537|journal=[[SIAM Journal on Discrete Mathematics]]|volume=24|issue=4|pages=1399–1440|year=2010|doi=10.1137/090760301}}.</ref>
* Every [[planar graph]] whose [[Glossary of graph theory#Genus|faces]] all have even length is bipartite.<ref>{{citation|first=Alexander|last=Soifer|authorlink=Alexander Soifer|year=2008|title=The Mathematical Coloring Book|title-link= The Mathematical Coloring Book |publisher=Springer-Verlag|isbn=978-0-387-74640-1|pages=136–137}}. This result has sometimes been called the "two color theorem"; Soifer credits it to a famous 1879 paper of [[Alfred Kempe]] containing a false proof of the [[four color theorem]].</ref> Special cases of this are [[grid graph]]s and [[squaregraph]]s, in which every inner face consists of 4 edges and every inner vertex has four or more neighbors.
* The [[complete bipartite graph]] on ''m'' and ''n'' vertices, denoted by ''K<sub>n,m</sub>'' is the bipartite graph <math>G = (U, V, E)</math>, where ''U'' and ''V'' are disjoint sets of size ''m'' and ''n'', respectively, and ''E'' connects every vertex in ''U'' with all vertices in ''V''. It follows that ''K<sub>m,n</sub>'' has ''mn'' edges.<ref>{{harvtxt|Asratian|Denley|Häggkvist|1998}}, p. 11.
* [[Hypercube graph]]s, [[partial cube]]s, and [[median graph]]s are bipartite. In these graphs, the vertices may be labeled by [[bitvector]]s, in such a way that two vertices are adjacent if and only if the corresponding bitvectors differ in a single position. A bipartition may be formed by separating the vertices whose bitvectors have an even number of ones from the vertices with an odd number of ones. Trees and squaregraphs form examples of median graphs, and every median graph is a partial cube.<ref>{{citation|first=Sergei|last=Ovchinnikov|title=Graphs and Cubes|series=Universitext|publisher=Springer|year=2011}}. See especially Chapter 5, "Partial Cubes", pp. 127–181.</ref>
* [[Hypercube graph]]s, [[partial cube]]s, and [[median graph]]s are bipartite. In these graphs, the vertices may be labeled by [[bitvector]]s, in such a way that two vertices are adjacent if and only if the corresponding bitvectors differ in a single position. A bipartition may be formed by separating the vertices whose bitvectors have an even number of ones from the vertices with an odd number of ones. Trees and squaregraphs form examples of median graphs, and every median graph is a partial cube.<ref>{{citation|first=Sergei|last=Ovchinnikov|title=Graphs and Cubes|series=Universitext|publisher=Springer|year=2011}}. See especially Chapter 5, "Partial Cubes", pp. 127–181.</ref>
==Properties==
==Properties==