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2) What is the maximum number of edges possible for any bipartite graph between A and B?

2) What is the maximum number of edges possible for any bipartite graph between A and B?

Advanced Engineering Mathematics
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]?
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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