Let A and B cach 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 Iess cdge-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 added 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]?
Let A and B cach 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 Iess cdge-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 added 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]?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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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 \( N^2 - N \) 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 added 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 \( X_S = 1 \) if all the edges in \( S \) are added to the graph, and \( X_S = 0 \) if any of them are missing. What is \( E[X_S] \)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78f85931-ed68-43c6-82f9-075f499bfced%2Fa0b4036c-dbf6-4e0a-b6fa-82e8307668bf%2Fspr9tws_processed.jpeg&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 \( N^2 - N \) 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 added 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 \( X_S = 1 \) if all the edges in \( S \) are added to the graph, and \( X_S = 0 \) if any of them are missing. What is \( E[X_S] \)?
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