How can we show that a maximum matching in $G$ has $|V_1|$ edges

Asked Dec 12 '20 at 19:43

Active Dec 12 '20 at 20:05

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A bipartite graph $G = (V_1, V_2, E)$, with $V_1$ and $V_2$ being the two partitions of the set of vertices, is $d$-regular provided every vertex in $G$ has degree exactly $d$.

Show that a maximum matching in $G$ has $|V_1|$ edges. I'm thinking to use standard max-flow-min-cut theorem.

edited Dec 12 '20 at 20:05

Martin Sleziak

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asked Dec 12 '20 at 19:43

tien lee

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Does this answer your question? Prove that a $k$-regular bipartite graph has a perfect matching – ArsenBerk Dec 12 '20 at 19:47
@ArsenBerk I didn't understand that answer. Can you please elaborate based on this question – tien lee Dec 12 '20 at 19:49
Do you know Hall's Theorem? Without using it, this might be hard to prove. If you know Hall's Theorem, I can try to explain the answer given there. – ArsenBerk Dec 12 '20 at 19:55
I'm thinking to use max flow min cut theorem because Halls theorem was not explained to me – tien lee Dec 12 '20 at 20:10

How can we show that a maximum matching in $G$ has $|V_1|$ edges

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