Matchings without Hall's theorem

In practice matchings are found not by following the proof of Hall's theorem but by starting with some matching and improving it by finding augmenting paths.  Given a matching in a bipartite graph on vertex classes and , an augmenting path is a path from to such that ever other edge of is an edge of .  Replace by produces a matching with .

Theorem.  Let be a spanning subgraph of .  If (i) or (ii) is -regular, then has a perfect matching.

Proof. Let be a maximal matching in with .

(i) Choose  , .  We have and .  Since there is an such that is adjacent to and is adjacent to .  Then is an augmenting path.

(ii) Without loss of generality, is connected.  Form the directed graph on by taking the directed edge  ( ) whenever is an edge of .  Add directed edges arbitrarily to to obtain a -regular digraph , which might contain multiple edges; since is connected we have to add at least one directed edge.  The edge set of decomposes into directed cycles.  Choose a cycle containing at least one new edge of , and let be a maximal sub-path of containing only edges of .  Let , be the start- and endpoints of respectively.  Then we can choose and , whence  is an augmenting path, where is the result of "pulling back" from to , replacing each visit to a in by use of the edge of .

 

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