6. For many applications of matchings, it makes sense to use bipartite graphs. You might wonder,...

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Advance Math

6. For many applications of matchings, it makes sense to usebipartite graphs. You might wonder, however, whether there is a wayto find matchings in graphs in general.
For which n does the complete graph Kn have a matching?
Prove that if a graph has a matching, then |V||V| is even.
Is the converse true? That is, do all graphs with |V||V| evenhave a matching?
What if we also require the matching condition? Prove ordisprove: If a graph with an even number of vertices satisfies|N(S)|â‰¥|S||N(S)|â‰¥|S| for all SâŠ†V,SâŠ†V,then the graph has amatching.
Please keep straight to the point and short if possible, I givegood ratings on good legible writings and correctness. THANKS!!

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In a simple graph a matching is a set of edges such that no twoedges have a common vertexThe terminology in theentire question is flawed Every nontrivial graph has amatching Any single edge is a matching In the case of perfectmatching we can discuss the said questions1 2 For the proof of See Answer

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