Let A and B each be sets of N labeled vertices, and consider bipartite graphs b 1) How many possible ways are there to match or pair vertices between A ar 2) What is the maximum number of edges possible for any bipartite graph b 3) Show by example that there is a bipartite graph between A and B with N² Questions 2.2 and 2.3 indicate that even if the bipartite graph is almost full of still no have a perfect matching. However, we can show that perfect matching less edge-heavy bipartite graphs
Let A and B each be sets of N labeled vertices, and consider bipartite graphs b 1) How many possible ways are there to match or pair vertices between A ar 2) What is the maximum number of edges possible for any bipartite graph b 3) Show by example that there is a bipartite graph between A and B with N² Questions 2.2 and 2.3 indicate that even if the bipartite graph is almost full of still no have a perfect matching. However, we can show that perfect matching less edge-heavy bipartite graphs
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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please provide solution for Q6
![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%2F830a2972-402c-4d09-8911-3454714bdf99%2Fk68j7_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]?
Transcribed Image Text:Consider taking |E| = aN, i.e., the total number of edges is proportional to the number of vertices. This is a
relatively sparse number of edges, given the total number of edges that can exist between A and B.
6) Show that taking |E| = 3N, the expected number of matchings goes to 0 as N → o.
7) Show that taking |E| = 4N, the expected number of matchings goes to infinity as N → .
We may conclude that constructing random bipartite graphs in this way, perfect matchings become exceedingly
common once the number of edges crosses some threshold between 3N and 4N, but are relatively uncommon prior
to that.
The following approximations may be useful in 2.6 and 2.7:
(*)
1
V2TN V a –
-1 ( (a – 1)a-1
1
- (eN)N
V2T Ne
(1)
()"
N! z V27N
Bonus: Show that if |E| = 3N, the probability that the generated graph contains even a single perfect matching goes
to 0 as N → o.
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