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 = 2 + 36 24 = 14.
1.5.2 4: If G is a tree, then 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 exam. The list is broken into two sections: before the
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 exam.
Introduction to Chapter 1
Section 1.1.1
Sectio
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 will not be needed.).
Show all your work on the paper p
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 easiest to dene these terms by rst saying that the eccentrici
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 considering the number of tickets each child gets), if each
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, gn 2fn , where fn is the nth Fibonacci number (dened by
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 50, in such a way that the ve cards together form a full
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 graph.
Solution: V E + F = 2, where V is number of vertic
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 vertices, m edges and girth g , then
2g
g
n
.
g2
g2
Hint: Loo
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 the tree whose Pr fer code is (2, 2, 2, 2) (use the te
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 character, 53 63 with two characters, so (53) + (53)(63) +
(53)
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: Let : E (Kk ) cfw_r, b be a 2-colouring of the edges of
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 must saturate the vertex at the center. Suppose, wlog,
th
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 that for all n, gn 2fn , where fn is the nth Fibonacci nu
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 using the algebraic representation of
n
k
.
Solution:
n
k
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 smallest integer k such there exists a
coloring of the vertic
Basic Combinatorics (Math 40210) Sec 01, Fall 2012, Quiz 2
Solutions
September 20, 2012
1. Draw the tree on vertex set cfw_1, 2, 3, 4, 5, 6, 7 whose Prfer code is (3, 2, 3, 2, 1).
u
Solution (3 pts): 4 and 6 are leaves joined to 3; 3 is joined to 2, which
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 the weight of the tree you nd.
1
3
1
1
1
3
2
2
3
2
2
Sol
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 that starts at a, ends at d, and repeats an edge and/or
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 vertices (one for each of the four land
masses) and seven
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 of the adjacency matrix of C2k , there are 1s in positio
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 base case. The base case is trivial: G is a tree so
it i
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: cdhlponminjkghc
Patch rst two together: abcdhlponminjkghcgf jiea
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 that is not a path.
Solution: Lots of possible answers;
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 the most out of this ungraded homework assignment, you
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: Let : E (Kk ) cfw_r, b be a 2-colouring of the edges of
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 character, 53 63 with two characters, so (53) + (53)(63) +
(53)
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 must saturate the vertex at the center. Suppose, wlog,
th