Editing Bipartite graph (section)

===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>{{citation
  | author = Kőnig, Dénes
  | author-link = Dénes Kőnig
  | title = Gráfok és mátrixok
  | journal = Matematikai és Fizikai Lapok
  | volume = 38
  | year = 1931
  | pages = 116–119}}.</ref><ref>{{citation
 | last1 = Gross | first1 = Jonathan L.
 | last2 = Yellen | first2 = Jay
 | edition = 2nd
 | isbn = 9781584885054
 | page = 568
 | publisher = CRC Press
 | series = Discrete Mathematics And Its Applications
 | title = Graph Theory and Its Applications
 | url = 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. 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
 | last1 = Chartrand | first1 = Gary | author1-link = Gary Chartrand
 | last2 = Zhang | first2 = Ping | author2-link = Ping Zhang (graph theorist)
 | isbn = 9780486483689
 | pages = 189–190
 | publisher = Courier Dover Publications
 | title = A First Course in Graph Theory
 | url = 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.

Another class of related results concerns [[perfect graph]]s: every bipartite graph, the [[complement (graph theory)|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 [[complement (graph theory)|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
|volume= 184 |series= Graduate Texts in Mathematics
|author=Béla Bollobás|publisher	=Springer|year= 1998
|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.

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 (graph theory)|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
 | last1 = Chudnovsky | first1 = Maria | author1-link = Maria Chudnovsky
 | last2 = Robertson | first2 = Neil | author2-link = Neil Robertson (mathematician)
 | last3 = Seymour | first3 = Paul | author3-link = Paul Seymour (mathematician)
 | last4 = Thomas | first4 = Robin | author4-link = Robin Thomas (mathematician)
 | doi = 10.4007/annals.2006.164.51
 | issue = 1
 | journal = [[Annals of Mathematics]]
 | pages = 51–229
 | title = The strong perfect graph theorem
 | volume = 164
 | year = 2006| citeseerx = 10.1.1.111.7265| arxiv = math/0212070| s2cid = 119151552 }}.</ref> It follows that any subgraph of a bipartite graph is also bipartite because it cannot gain an odd cycle.<ref>{{citation|url=http://www.sfu.ca/~mdevos/notes/graph/345_matchings.pdf|title=Matchings|publisher=Simon Fraser University|first=Matt|last=DeVos|work=Lecture notes: Introduction to Graph Theory, Math 345}}</ref>