Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
3) Show by example that there is a bipartite graph between A and B with N² – N edges, and no perfect matching.

3) Show by example that there is a bipartite graph between A and B with N² – N edges, and no perfect matching.

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
See similar textbooks
icon
Related questions
Question

please provide complete handwritten solution for Q3

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]?
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS