Math 423
Lecture Notes, 2014
Kieka Mynhardt
Fundamental Law of Graph Theory:
Draw a Picture!
Text:
Gary Chartrand, Linda Lesniak, Ping Zhang, Graphs and Digraphs
Fifth Edition, Chapman & Hall/CRC, 2010.
(Referred to below as CLZ.)
1
Introduction to Graphs
Math 423/523 Assignment 4
Due: Thursday, November 6
Part 1. Test yourself
1. (a) How many vertices are there in an r-regular maximal planar graph for each
r 2 f3; 4; 5g?
(b) Draw an example of each type.
Answer
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
Math 423/523 Assignment 1
Due: Thursday, September 24
Part 1. Test yourself
Make sure you can do the problems in Part 1. Dont hand them in for marking, they wont
be marked, nor will I ask anybody to do them on the board. Ill supply solutions so you
can ch
MATH 423
Midterm 2
Time: 80 minutes
Monday, November 17, 2014
Part A. Prove two of the following theorems.
A1. If G is a graph of order n
3 such that (G)
(G), then G is Hamiltonian.
[6]
A2. Every planar graph is 5-colourable. (Dont use the Four Colour The
Math 423/523 Assignment 1
Due: Thursday, September 24
Part 1. Test yourself
Make sure you can do the problems in Part 1. Dont hand them in for marking, they wont
be marked, nor will I ask anybody to do them on the board. Ill supply solutions so you
can ch
Math 423/523 Assignment 6
Due: Thursday, December 3
1. Draw a cubic planar graph without a 1-factor.
[3]
2. Draw a cubic nonplanar graph without a 1-factor.
[3]
3. Draw a cubic graph with bridges, not all on a single path, that has a 1-factor.
[3]
4. Show
Math 423/523 Assignment 2
Due: Monday, 5 October
Part 1. Test yourself
b
j
c
u
h
a
v
g
f
G
1. (a) Determine the radius and the diameter of the graph G above.
(b) Determine Cen(G) and Per(G).
(c) Determine the smallest number of edges that need to be delet
Math 423/523 Assignment 3
Due: Monday, October 26
Part 1. Test yourself
1. Use Whitneys Theorem to show that the graph below is 4-connected.
2. For which values of s and t is Ks;t Hamiltonian? Traceable?
3. Show that if G is a graph of order n such that (
Math 423/523 Assignment 4
Due: Thursday, November 12
Part 1. Test yourself
1. (a) How many vertices are there in an r-regular maximal planar graph for each
r 2 f3; 4; 5g?
(b) Draw an example of each type.
2. Draw a bipartite graph that contains a subdivis
MATH 4/523
Midterm 2
Time: 80 minutes
Monday, November 16, 2015
Part 1: Answer three of Questions 1 4.
(If you answer all four Ill only mark the rst three.)
1. Use Mengers Theorem to prove Whitneys characterisation of k-connected graphs:
A nontrivial grap
Math 423/523 Assignment 5
Due: Thursday, November 26
Part 1. Homework to be done on the board
(Possible bonus marks for people who have had two turns already.)
1. Give an example of a planar 4-chromatic graph that contains triangles but no K4 .
2. Give an
Math 423/523 Assignment 2
Due: Monday, 5 October
Part 1. Test yourself
b
j
c
u
h
a
v
g
f
G
1. (a) Determine the radius and the diameter of the graph G above.
(b) Determine Cen(G) and Per(G).
(c) Determine the smallest number of edges that need to be delet
MATH 4/523
Midterm 1
Time: 80 minutes
Thursday, October 8, 2015
Part 1: Answer three of Questions 1 4.
1. Prove that every graph of order and size
j 2k
4
contains a triangle.
[5]
Solution
j 2k
If = 3 and 4 = 2, then = 3 and the statement is true. If = 4
Math 423/523 Assignment 3
Due: Monday, October 26
Part 1. Test yourself
1. Use Whitneys Theorem to show that the graph below is 4-connected.
Answer
See the internally disjoint paths below (and there should be one more, up to isomorphism, but youll get the
Math 423
Classroom Notes, Chapter 8
8
Vertex Colourings
8.1
The Chromatic Number of a Graph
Denitions:
colour class
(vertex) colouring, -colouring, -colourable, chromatic number, -chromatic,
A vertex colouring, or simply a colouring, of a graph is an ass
Math 423
Classroom Notes, Chapter 11
11
11.1
Edge Colourings
Chromatic Index and Vizings Theorem
Denitions: edge colouring, -edge colouring, edge chromatic number, chromatic index,
-edge chromatic
A (proper) edge colouring of a graph is an assignment of
MATH 423
Midterm 1
Time: 80 minutes
Thursday, October 9, 2014
1. Draw three pairwise non-isomorphic connected cubic graphs of order 8. Explain
carefully why your graphs are non-isomorphic.
[6]
Answer
Consider the three graphs below. (There are other examp
Math 423/523 Assignment 3
Due: Monday, October 27
Part 1. Test yourself
1. Show that if is a graph of order such that () , then is non-Hamiltonian.
2
Answer
Let be a maximum independent set, let = () , and apply Theorem 3.16.
It works because | |.
2. Draw
Math 423/523 Assignment 5
Due: Thursday, November 20
Part 1. Test yourself
1. For each 2, give an example of a -regular -chromatic graph.
Answer
Take two copies of and join them with independent edges (no two edges share
an end-vertex). Call this graph .
Math 423/523 Assignment 4
Due: Thursday, November 6
Part 1. Test yourself
1. (a) How many vertices are there in an r-regular maximal planar graph for each
r 2 f3; 4; 5g?
(b) Draw an example of each type.
Answer
(a) Use Theorem 6.10.
If G is a 3-regular ma
MATH 423
Midterm 3
Time: 80 minutes
Monday, December 1, 2014. You may attempt all questions.
1. The Y and Y transforms of a graph are dened as follows (see sketch).
u
u
v
x
w
w
x
Y
If contains a vertex of degree three whose neighbours form an independent
MATH 423
Midterm 2
Time: 80 minutes
Monday, November 17, 2014
Part A. Prove two of the following theorems.
A1. If G is a graph of order n
3 such that (G)
(G), then G is Hamiltonian.
[6]
A2. Every planar graph is 5-colourable. (Don use the Four Colour Theo
Math 423
Classroom Notes, Chapter 10
10
10.1
Matchings and Factorizations
Matchings and Independence in Graphs
Denitions: independent set of edges, matching, maximum matching, edge independence number, matching number, matched/unmatched vertex, perfect ma
Math 423/523 Assignment 2
Due: Monday, 29 September
Part 1. Test yourself
1. (a) Determine the radius and the diameter of the graph above.
(b) Determine Cen() and Per().
(c) Determine the smallest number of edges that need to be deleted from in order
to o
Chapter 2
Structure and Symmetry of Graphs
2
Trees and Connectivity
2.1
Nonseparable Graphs
Cut-vertices
Denition:
cut-vertex
A vertex is a cut-vertex of a graph if the number of components of is
greater than the number of components of ,
i.e., in symbo
Math 423
Classroom Notes, Chapter 4
4
Digraphs
4.1
Strong Digraphs
Denitions: digraph, arcs, adjacent to, adjacent from, indegree, outdegree, in-neighbourhood, out-neighbourhood
A directed graph ( ), or digraph for short, consists of a nite nonempty set
Math 423
Classroom Notes, Chapter 3
3.1
Eulerian Graphs
C
River Pregel
D
A
B
Figure 1: The bridges of Knigsberg, East Prussia, 18th Century
The Knigsberg Bridge Problem
In the 18th Century the people in the city of Knigsberg, East Prussia, used to go wand
Math 423/523 Assignment 1
Due: Thursday, September 18
Part 1. Test yourself
Make sure you can do the problems in Part 1. Dont hand them in for marking, they wont
be marked, nor will I ask anybody to do them on the board. Ill supply solutions so you
can ch
Math 423
Classroom Notes, Chapter 6
6
Planar Graphs
6.1
The Euler Identity
Denitions:
polyhedron, faces, Platonic solids
A polyhedron is a 3-dimensional object whose boundary consists of polygonal plane
surfaces, called the faces of the polyhedron. The b
Math 423/523 Assignment 6
Due: Monday, December 1
1. (a) Determine whether the graph below is Hamiltonian or not.
(b) Determine 0 ().
[2]
[2]
u
v
Answer
(a) The contrapositive of Theorem 3.16 is quicker to use in this case than Grinbergs
Theorem. Note tha