Math 2.0

Combinatorics Theorem

Hall's theorem for bipartite graphs

Builds on NeighbourhoodBipartite graphMatching

Statement

Let GG be a finite bipartite graph with parts XX and YY. For W⊆XW \subseteq X, let N(W)N(W) be the set of vertices of YY that are adjacent to some vertex of WW. Then GG has a matching that covers every vertex of XX if and only if ∣N(W)∣≥∣W∣|N(W)| \ge |W| for every W⊆XW \subseteq X.

Remarks

If ∣X∣=∣Y∣|X| = |Y|, such a matching is a perfect matching. Thus the condition then characterizes the perfect matchings. A consequence: a finite regular bipartite graph with at least one edge has a perfect matching. The theorem is the graph form of Hall's marriage theorem: let XX index the sets and let YY be their union.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Bipartite graph, definition, no evidence levelBipartite graphdefinitionGraph, definition, no evidence levelGraphdefinitionMatching, definition, no evidence levelMatchingdefinitionNeighbourhood, definition, no evidence levelNeighbourhooddefinitionHall's theorem for bipartite graphs, statement, no evidence levelHall's theorem for bi…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Bipartite graph definition
  2. Graph definition
  3. Matching definition
  4. Neighbourhood definition
  5. Hall's theorem for bipartite graphs statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Hall's_marriage_theorem, Graph theoretic formulation
About this page
Tag0X2GJPNPFN
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Kindstatement, theorem
WritingW0 stub
MSC05C70
StatusThis copy of the site has no build data. Run m2 status 0X2GJPNPFN.
UsesNeighbourhood, Bipartite graph, Matching

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