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For many applications of matchings, it makes sense to use bipartite graphs. You might wonder, however, whether there is a way to find matchings in graphs in general. (a) For which n does the complete graph K, have a matching? N (b) Prove that if a graph has a matching, then |V|is even. (c) Is the converse true? That is, do all graphs with |V| even have a matching?

For many applications of matchings, it makes sense to use bipartite graphs. You might wonder, however, whether there is a way to find matchings in graphs in general. (a) For which n does the complete graph K, have a matching? N (b) Prove that if a graph has a matching, then |V|is even. (c) Is the converse true? That is, do all graphs with |V| even have a matching?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Discrete Maths Oscar Levin 3rd eddition 4.6.6:

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For many applications of matchings, it makes sense to use bipartite graphs. You might wonder, however, whether there is a way to find matchings in graphs in general. 6. (a) For which n does the complete graph K, have a matching? (b) Prove that if a graph has a matching, then |V| is even. (c) Is the converse true? That is, do all graphs with |V| even have a matching? (d) What if we also require the matching condition? Prove or dis- prove: If a graph with an even number of vertices satisfies |N(S)| > |S| for all S C V, then the graph has a matching.
Transcribed Image Text:For many applications of matchings, it makes sense to use bipartite graphs. You might wonder, however, whether there is a way to find matchings in graphs in general. 6. (a) For which n does the complete graph K, have a matching? (b) Prove that if a graph has a matching, then |V| is even. (c) Is the converse true? That is, do all graphs with |V| even have a matching? (d) What if we also require the matching condition? Prove or dis- prove: If a graph with an even number of vertices satisfies |N(S)| > |S| for all S C V, then the graph has a matching.
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