21-301 Combinatorics
Homework 5
Due: Monday, October 26
1. Prove that if u, v are the only vertices of odd degree in a graph G, then there is a path
from u to v in G.
Solution: We have to show that u, v are in the same component of G. But is they are
in d
21-301 Combinatorics
Homework 3
Due: Monday, September 28
1. Suppose that in the Tower of Hanoi problem there are n sets of k rings of the same
size. For example you there could be two rings of size 1, two rings of size 2 and 2 rings
of size 3, here n = 3
21-301 Combinatorics
Homework 3
Due: Monday, October 5
1. Suppose that you are asked to multiply a collection of m m matrices to form the
product A1 A2 An+1 . Let C0 = 1 and let Cn be the number of ways to do this. For
example C2 = 2. We can compute (A1 A
21-301 Combinatorics
Homework 9
Due: Monday, November 23
1. Consider the following take-away game: There is a pile of n chips. A move consists
of removing 3k chips for some k 1. Compute the Sprague-Grundy numbers g(n) for
n 0.
Solution: After looking at t
21-301 Combinatorics
Homework 6
Due: Monday, November 2
1. Let (G) be the chromatic number of graph G = (V, E). Let (G), (G) denote the
size of the largest independent set of G, clique of G respectively.
Show that
|V |
, (G) .
(G) max
(G)
Show further tha
21-301 Combinatorics
Homework 8
Due: Monday, November 16
1. Let A be an intersecting family of subsets of [n] such that A A implies k |A|
n/2. Show that
n1
.
|A|
i1
i=k
Solution: Let Ai = cfw_A A : |A| = i. Then Ai is an intersecting family and
so by th
21-301 Combinatorics
Homework 10
Due: Wednesday, December 2
1. How many ways are there of k-coloring the squares of the above cross if the group acting
is e0 , e1 , e2 , e3 where ej is rotation by 2j/4. Assume that instead of 13 squares there
are 4n + 1.
21-301 Combinatorics
Homework 7
Due: Friday, November 6
1. Use the pigeon-hole principle to show that for every integer k 1 there exists a power
of 3 that ends with 000 0001 (k 0s).
Solution: If we consider the innite sequence u = 3 mod 10k+1 for = 1, 2,