Foundations of Data Science

1. For what value of p do cycles first appear?
2. For what value of p do isolated vertices disappear and the graphs become connected?
Exercise 4.28 Consider G(n, p) with p = 1
3n .
1. Use the second moment method to show that with high probability there exists a
simple path of length 10.
2. Argue that on the other hand, it is unlikely there exists any cycle of length 10.
Exercise 4.29 Complete the second moment argument of Theorem 4.13 to show that for
p = d , d > 1, G(n, p) almost surely has a cycle.
n
Hint: If two cycles share one or more edges, then the union of the two cycles is at least
one greater than the union of the vertices.
Exercise 4.30 Draw a tree with 10 vertices and label each vertex with a unique integer
from 1 to 10. Construct the Prüfer sequence (Appendix 12.9.6) for the tree. Given the
Prüfer sequence, recreate the tree.
Exercise 4.31 Construct the tree corresponding to the following Prüfer sequences (Appendix
12.9.6)
1. 113663 (1,2),(1,3),(1,4),(3,5),(3,6),(6,7), and (6,8)
2. 552833226.
Exercise 4.32 What is the expected number of isolated vertices in G(n, p) for p = 1 ln n
? 2 n
Exercise 4.33 Theorem 4.17 shows that for some c > 0 and p = c ln n/n, G(n, p) has
diameter O (ln n). Tighten the argument to pin down as low a value as possible for c.
Exercise 4.34 What is diameter of G(n,p) for various values of p?
Exercise 4.35
1. List five increasing properties of G (n, p).
2. List five non increasing properties .
Exercise 4.36 Consider generating the edges of a random graph by flipping two coins,
one with probability p 1 of heads and the other with probability p 2 of heads. Add the edge
to the graph if either coin comes down heads. What is the value of p for the generated
G(n, p) graph?
p
Exercise 4.37 In the proof of Theorem 4.19, we proved for p 0 (n) such that lim 0 (n)
= 0
n→∞ p(n)
that G(n, p 0 ) almost surely did not have property Q. Give the symmetric argument that
p(n)
for any p 1 (n) such that lim = 0, G(n, p
n→∞ p 1 (n) 1) almost surely has property Q.
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