MATH 6B PROBLEM SET 7 SOLUTIONS
Problem 1
(+5 if correct) This was made an extra credit problem.
cr(Gn ) 1. If we label the vertices of Gn 1 through n clockwise around the cycle
from Cn , contract n-1 and n and n/2 and n/2+1 to get Gn1 as a minor. Repeati
Ma/CS 6b: Problem Set 1
Due noon, Thursday, January 15
1. The girth of a graph is the length of the minimum cycle in it. For a positive integer
k 3:
(a) Prove that any graph with minimum degree k and girth 4 has at least 2k vertices.
(b) Prove that any gr
Ma/CS 6b: Problem Set 3
Due noon, Thursday, January 29
1. Let G = (V, E) be a k-connected graph, for some k 2, and let |V | 2k. Prove that
G contains a (simple) cycle of length at least 2k (hint: consider the longest cycle in G and
how the additional vert
Ma/CS 6b: Problem Set 6
Due noon, Thursday, February 19
1. Prove that for any positive m, we have R(mK2 , mK2 ) = 3m 1.
2. (NO COLLABORATION) Reprove Question 3 of Assignment 1 by using a probabilistic
proof.
3/2
3. Prove that R(4, t) > c t3/2 for some co
MATH 6B PROBLEM SET 4 SOLUTIONS
Problem 1
Since the graph is 3-regular we get 3|V | = 2|E| |E| = 3|V |/2. Similarly every vertex is part of
exactly 3 faces of length 4, 6 and 8 thus |F | = |V |/4+|V |/6+|V |/8 (for instance if F4 is the number
of faces of
MATH 6B PROBLEM SET 5 SOLUTIONS
Problem 1
Denote by B be the adjacency matrix of the subgraph of G which has all the
blue edges of G (and all its vertices). Similarly, let R be the adjacency matrix of
the subgraph of G which has all the red edges of G.
Le
MATH 6B PROBLEM SET 3 SOLUTIONS
ALEX MUN
Problem 1
Let C be a maximum length cycle of G, and assume for contradiction that
|C| < 2k. There exists a vertex v not on C that is adjacent to a vertex u1 that is on
C. Let u2 be a neighbor of u1 on C. If there i
Ma/CS 6b: Problem Set 10
Due noon, Thursday, March 19th
1. Let C1 V n be a linear code with check matrix H1 , of dimension k1 , and of distance
d1 , and let C2 V n be a linear code with check matrix H2 , of dimension k2 , and of distance
d2 . Consider the
MATH 6B PROBLEM SET 2 SOLUTIONS
Problem 1
We will construct a vertex cover S of size at most 2c inductively as follows. Start with S empty.
Pick any edge (u, v) E and add both u and v to S. Now delete all of the edges incident to both
u and v to obtain a
Ma/CS 6b: Problem Set 9
Due noon, Thursday, March 12th
1. Consider a graph G = (V, E) with an adjacency matrix A and V = cfw_v1 , . . . , vn . Let
D be a diagonal matrix with Di,i = deg vi , and let B = D A.
(a) (8 points) Prove that the smallest eigenval
MATH 6B PROBLEM SET 1 RUBRIC AND SOLUTIONS
ALEX MUN
Generic Proofwriting Points
(-1) Did not explicitly state a denition that is important to proof. Small penalty
if the intent can be determined unambiguously with a small amount of work (otherwise its a m
Ma/CS 6b: Problem Set 7
Due noon, Thursday, February 26
1. (NO COLLABORATION) Consider the graph Cn for an even n 6. For each vertex
v of this cycle, we add an edge between v and the vertex that is at a distance of n/2 from
v in Cn . This yields a 3-regul
Ma/CS 6b: Problem Set 8
Due noon, Thursday, March 5th
Hlders inequality. Given a1 , . . . , an , b1 , . . . , bn R, and 1 < p, q such that 1/p + 1/q =
o
1, we have
( n
)1/p ( n
)1/q
n
p
q
|ai bi |
ai
bi
.
i=1
i=1
i=1
Notice that this is a generalizatio
Ma/CS 6b: Problem Set 4
Due noon, Thursday, February 5
1. (NO COLLABORATION) Let G = (V, E) be a connected 3-regular plane graph such
that every vertex of V participates in one face of length four, one of length six, and one of
length eight. Find the valu
Ma/CS 6b: Problem Set 2
Due noon, Thursday, January 22
1. We mentioned in class that no ecient algorithm is known for nding a minimum
vertex cover. Given a graph G = (V, E) (which may not be bipartite) with a minimum
vertex cover of size c, describe an al
Ma/CS 6b: Problem Set 5
Due noon, Thursday, February 12
1. Consider a graph G = (V, E), two vertices s, t V , and an integer k > 0. Moreover,
every edge of E is colored either red or blue. Describe an algorithm for nding the number
of (not necessarily sim