Basic Combinatorics
Math 40210, Section 01 Fall 2012
Homework 5 Solutions
1.5.2 1: n = 24 and 2q = v deg(v ) = 24 3 = 72, so q = 36, meaning that in any
planar representation we must have r = 2 + q n
Basic Combinatorics
Math 40210, Section 01 Spring 2012
Reading for Final exam
Here is a detailed list of all the section of the book that we covered this semester, and that will
be examined in the nal
Basic Combinatorics
Math 40210, Section 01 Spring 2012
Reading for Midterm exam
Here is a detailed list of all the section of the book that we covered so far, and that will be
examined in the midterm
Math 40210: Basic Combinatorics, Spring 2012
Final Exam
Friday May 11
Instructor: David Galvin
Name:
This exam contains 8 problems on 9 pages (including the front cover).
Calculators may be used (but
Math 40210: Basic Combinatorics, Spring 2012
Midterm Exam, Wednesday March 7
Solutions and comments
1. (a) Dene the diameter diam(G) and the radius rad(G) of a graph G. (3 points)
Solution: Its easies
BASIC COMBINATORICS (MATH 40210) SEC 01, SPRING 2012, QUIZ 6
SOLUTIONS
100 identical tickets for rides are distributed among 40 children at a carnival.
(1) In how many ways can this be done (just cons
BASIC COMBINATORICS (MATH 40210) SEC 01, SPRING 2012, QUIZ 5
SOLUTIONS
Dene a sequence recursively as follows: g0 = 1, g1 = 2, gn = gn1 + 2gn2 for n 2.
(1) Use induction on n to show that for all n, g
BASIC COMBINATORICS (MATH 40210) SEC 01, SPRING 2012, QUIZ 4
SOLUTIONS
(1) I take out two Aces from a standard deck of 52 cards. How many ways are there to select three more cards from
the remaining 5
BASIC COMBINATORICS (MATH 40210) SEC 01, SPRING 2012, QUIZ 3
SOLUTIONS
(1) State Eulers formula connecting the number of vertices, edges and regions in a planar representation of a
connected planar gr
BASIC COMBINATORICS (MATH 40210) SEC 01, SPRING 2012, QUIZ 3 DO-OVER
NAME:
(1) The girth of a graph is the length of the shortest cycle in the graph. Show that if a connected planar graph has
n vertic
BASIC COMBINATORICS (MATH 40210) SEC 01, SPRING 2012, QUIZ 2
SOLUTIONS
(1) Write down the Pr fer code (Pr fer sequence) of the following tree:
u
u
1
2
3
4
5
6
7
8
Solution: (2, 3, 6, 2, 3, 7)
(2) Draw
Basic Combinatorics
Math 40210, Section 01 Fall 2012
Homework 7 Solutions
2.1 1a: 53 choice for initial character, 63 for all the rest, so (53)(63)(63)(63)(63) in
total.
2.1 1b: 53 with one characte
Basic Combinatorics
Math 40210, Section 01 Fall 2012
Homework 8 Solutions
n
1.8.1 1: Kn has n edges, each one of which can be given one of two colors; so Kn has 2( 2 )
2
2-edge-colorings.
1.8.1 3: L
Basic Combinatorics
Math 40210, Section 01 Fall 2012
Homework 6 Solutions
1.7.1 1: It does not have a perfect matching. A perfect matching is one which saturates
all vertices, and so in particular mu
Basic Combinatorics (Math 40210) Sec 01, Fall 2012, Quiz 5
Solutions
November 30, 2012
Dene a sequence recursively as follows: g0 = 1, g1 = 2, gn = gn1 + 2gn2 for n 2.
1. Use induction on n to show th
Basic Combinatorics (Math 40210) Sec 01, Fall 2012, Quiz 4
Solutions
October 31, 2012
This question is about the binomial coecient identity
n
k
=
n n1
.
k k1
valid for n k 1.
1. Prove the identity usi
Basic Combinatorics (Math 40210) Sec 01, Fall 2012, Quiz 3
Solutions
October 5, 2012
1. Dene (carefully) (G), the chromatic number of a graph G.
Solution: The chromatic number of G, (G), is the smalle
Math 40210: Basic Combinatorics, Fall 2012
Midterm Exam, Monday, October 8
Solutions
1. (a) (5 pts.) Apply Kruskals algorithm to nd a minimal weight spanning tree in the following graph, and compute t
Basic Combinatorics (Math 40210) Sec 01, Fall 2012, Quiz 1
Solutions
1. For the graph G drawn below, write down an a-d walk that is not a path.
a
b
c
f
e
d
Solution: any walk along edges of the graph
Basic Combinatorics
Math 40210, Section 01 Fall 2012
Homework 1 Solutions
1.1.1 1: See gure 1 of the gures page for one possible representation.
1.1.1 4: Represent the picture as graph with four ver
Basic Combinatorics
Math 40210, Section 01 Fall 2012
Homework 2 Solutions
1.2.1 8d: See solutions to homework 1 for this one
1.2.1 11a: See solutions to homework 1 for this one
1.2.2 1b: In row 1 o
Basic Combinatorics
Math 40210, Section 01 Fall 2012
Homework 3 Solutions
1.3.3 1: We prove this by induction on the number of edges. Since G is connected, it
has at least n 1 edges, so this is our b
Basic Combinatorics
Math 40210, Section 01 Fall 2012
Homework 4 Solutions
1.4.2 2: One possible implementation:
Start with abcgf jiea
From edge cd build, using previously unmarked edges: cdhlponmin
BASIC COMBINATORICS (MATH 40210) SEC 01, SPRING 2012, QUIZ 1
SOLUTIONS
All parts below relate to the following graph G on vertex set cfw_a, b, c, d, e, f .
a
b
c
f
e
d
(1) Write down an a-d walk in G
Basic Combinatorics
Math 40210, Section 01 Spring 2012
Homework 10 not being collected for grading
General information: I encourage you to talk with your colleagues about homework problems,
but to get
Basic Combinatorics
Math 40210, Section 01 Fall 2012
Homework 8 Solutions
n
1.8.1 1: Kn has n edges, each one of which can be given one of two colors; so Kn has 2( 2 )
2
2-edge-colorings.
1.8.1 3: L
Basic Combinatorics
Math 40210, Section 01 Fall 2012
Homework 7 Solutions
2.1 1a: 53 choice for initial character, 63 for all the rest, so (53)(63)(63)(63)(63) in
total.
2.1 1b: 53 with one characte
Basic Combinatorics
Math 40210, Section 01 Fall 2012
Homework 6 Solutions
1.7.1 1: It does not have a perfect matching. A perfect matching is one which saturates
all vertices, and so in particular mu