Question

6. Suppose, in the example taught in class of arranging dance partners at a party with eight girls and eight boys, that each girl likes exactly four boys, and that each boy is liked by exactly four girls. Without knowing anything else about the graph, show that a complete matching is possible. (Hint: Use Hall's theorem.)

Answer 1

Evey regulan bpasetide graph mateking perfect Lemma has a Proof: Let G be k-regulos bipartike araph with bpartition (A,B). Let XSA and Qat t be the humbes of edges with one end in X. eveay vedex in x has Since k, it llows vertex in Jagees klxl=t. Simila ess than or that t is degaee N(X) has So kN(X). It ollous that xis at most N(x)). Thus, by Halls theotem, equal to uivelent Smatehing OY matching coveig A, there is a every maximum natching covers A maximum covers B Smilaby evey Hence paoved girls, B-boys A = given problem, k-4 and Al 1Bl=3 the above lemma thee exfsts By complete a waicking

Please upvote the answer.