Math 145: Combinatorics
Homework 7 Solutions
13.3.3 Let G be a connected graph such that all vertices but one have degree at most d (one vertex may have
degree larger than d). Prove that G is (d + 1)-colorable.
Solution: Choose arbitrarily a color for the
Math 145:
Homework 2
This problem set is due Wednesday, January 22.
Do problems 2.5.5, 3.3.2, and 3.8.10, in addition to the following:
2.5.5 There is a class of 40 girls. There are 18 girls who like to play chess, and 23 who like to play soccer.
Several
Math 145: Combinatorics
Homework 6
This problem set is due Wednesday, February 19.
13.3.1 Prove that these graphs are not 3-colorable:
A
B
F
C
(a)
E
D
G
Solution: Suppose that a 3-coloring of this graph exists. The nodes marked B and C must be different
c
Math 145: Combinatorics
Homework 3 Solutions
2.5.7. We select 38 even positive integers, all less than 1000. Prove that there will be two of them whose
difference is at most 26.
Solution: Each even positive integer less than 1000 belongs to one of the int
Math 145: Combinatorics
Homework 4 Solutions
1.8.27. Alice has 10 balls (all different). First, she splits them into two piles; then she picks one of the piles
with at least two elements, and splits it into two; she repeats this until each pile has only o
Math 145: Combinatorics
Homework 5 Solutions
7.3.5 Does there exist a graph whose vertices have degree:
(a) 0, 2, 2, 2, 4, 4, 6?
Solution: No. Suppose that such a graph exists. Then it has 7 vertices and one has degree 6, and
so that vertex is adjacent to
Math 145: Combinatorics
Homework 8 Solutions
7.3.2 When does a connected graph contain two walks such that every edge is used by exactly one of them,
exactly once?
Solution: Each walk must enter and exit each vertex, except for the rst and last vertices o
Math 145: Combinatorics
Homework 9
This problem set is due Wednesday, March 12.
Do problems *7.3.3, 10.4.10, and 10.4.15 in the book, in addition to the following problems. (I am giving
7.3.3 a star even with the answer given in the back of the book.)
7.3
Math 145: Combinatorics
Solutions to the nal
1. How many words are there that are anagrams of the letters of UCDAVIS, plus one other wildcard
letter which can be any of the 26 letters of the alphabet?
Solution: From Nicole Geiss:
2. A bishop on a chessboa
Math 145: Combinatorics
Solutions to the Second Midterm
1. Which of the following sequences are possible as the degree sequence of a graph?
(a)
(b)
(c)
4, 4, 1, 1, 1, 1, 1, 1, 1
4, 4, 1, 1, 1, 1, 1, 1, 0
4, 4, 1, 1, 1, 1, 0, 0, 0
Solution: From Ramona Kho
Math 145: Combinatorics
Solutions to the First Midterm
1. In how many ways can you divide 15 students into 5 study groups with 3 students each? (The
students have 15 different names, but the study groups are not numbered or anything like
that.)
Solution:
Math 145: Combinatorics
Homework 10 Solutions
10.4.3 Follow how the algorithm for constructing a perfect matching works on this graph:
Solution: The greedy matching (from top to bottom) is indicated by the red edges. Starting from one
of the unmatched nod
MATH 145: HOMEWORK 1 SOLUTIONS
1.6.2
a) In how many ways can 8 people play chess [where two ways are the same if the tables are interchanged,
or if the players at a table switch places]?
Solution 1: There are 8! ways to assign 8 seats to 8 people. In doin