Let A and B each be sets of N labeled vertices, and consider bipartite graphs between A and B. 1) How many possible ways are there to match or pair vertices between A and B?
Let A and B each be sets of N labeled vertices, and consider bipartite graphs between A and B. 1) How many possible ways are there to match or pair vertices between A and B?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let A and B each be sets of N labeled vertices, and consider bipartite graphs between A and B.
1) How many possible ways are there to match or pair vertices between A and B?
2) What is the maximum number of edges possible for any bipartite graph between A and B?
3) Show by example that there is a bipartite graph between A and B with N² – N edges, and no perfect matching.
Questions 2.2 and 2.3 indicate that even if the bipartite graph is almost full of all the edges it might have, it may
still no have a perfect matching. However, we can show that perfect matchings are relatively common with much
less edge-heavy bipartite graphs.
4) Starting with no edges between A and B, if N edges are added between A and B uniformly at random, what
is the probability that those N edges form a perfect matching?
5) Starting with no edges between A and B, if |E| many edges are add
between A and B uniformly at random,
what is the expected number of perfect matchings in the resulting graph? Hint: if S is a set of edges in a
potential perfect matching, let Xs =1 if all the edges in S are added to the graph, and Xs = 0 if any of them
are missing. What is E[Xs]?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a229896-97dd-4061-9c05-fb264594ea86%2F5b44ef2e-28d7-4dbd-af60-31515e50e651%2Fvepjp0k_processed.png&w=3840&q=75)
Transcribed Image Text:Let A and B each be sets of N labeled vertices, and consider bipartite graphs between A and B.
1) How many possible ways are there to match or pair vertices between A and B?
2) What is the maximum number of edges possible for any bipartite graph between A and B?
3) Show by example that there is a bipartite graph between A and B with N² – N edges, and no perfect matching.
Questions 2.2 and 2.3 indicate that even if the bipartite graph is almost full of all the edges it might have, it may
still no have a perfect matching. However, we can show that perfect matchings are relatively common with much
less edge-heavy bipartite graphs.
4) Starting with no edges between A and B, if N edges are added between A and B uniformly at random, what
is the probability that those N edges form a perfect matching?
5) Starting with no edges between A and B, if |E| many edges are add
between A and B uniformly at random,
what is the expected number of perfect matchings in the resulting graph? Hint: if S is a set of edges in a
potential perfect matching, let Xs =1 if all the edges in S are added to the graph, and Xs = 0 if any of them
are missing. What is E[Xs]?
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