MATH 443 ASSIGNMENT 1
Jean-Sbastien Basque-Girouard
e
University of British-Columbia
September 11, 2013
1. (a) Proof. Let
d(v) = 2|E|,
vV
as we have seen in class. Since (G) d(v) for all v V , we have that
|V | (G) =
(G)
vV
d(v) = 2|E|.
vV
Thus
(G)
2|E|
MATH 443: GRAPH THEORY
September 2013
This outline has two pages
SCHEDULE: Section 101, 9:30-11:00 TTh in MATH 102
INSTRUCTOR: Richard Anstee
Office: Math Annex 1114, phone 604-822-6105
email [email protected]
Home: phone 604-325-8877
OFFICE HOURS: tenta
MATH 443: Special DeBruijn Sequence that yields a Card Trick
This example was shown to me by Ron Graham. It is a special deBruijn sequence of 32 bits (to
be thought of as a cyclic sequence) that can be generated in an easy way.
000010101110110001111100110
MATH 443
Problems #1
Due Tuesday September 17.
These problems are for classroom presentations and some fraction of them may be final exam
questions.
1. Use the notations (G) and (G) for the minimum and maximum degrees.
a) Show that (G)
2e(G)
n(G)
(G).
b
MATH 443
Problems #2
Due Tuesday October 1.
These problems are for classroom presentations and some fraction of them may be final exam
questions.
5. Let G be a graph with (G) 2, Show that G contains a cycle of length at least (G) + 1.
6. Let G be a graph
MATH 443
Assignment #3
Due Tuesday November 5.
1. Let G be a simple 3-regular graph. Recall that P4 refers to the path of three edges (and 4
vertices).
a) Prove that if G has a (edge) decomposition into P4 s then G has a perfect matching.
b) Prove that if
MATH 443
Problems #4
Due Tuesday October 22.
These problems are for classroom presentations and some fraction of them may be final exam
questions.
17. Let A be n n the adjacency matrix of a graph G on n vertices where A = (aij ) and aij = 1 if
ij E and ai
MATH 443
Problems #5
Due Thursday Nov 14.
These problems are for classroom presentations and some fraction of them may be final exam
questions.
30. Let G be a planar graph with girth(G) = k. Show that |E(G)|
k(|V (G)|2)
k2
31. Let x1 , x2 , . . . , xn be
MATH 443
Problems #3
Due Tuesday October 15.
These problems are for classroom presentations and some fraction of them may be final exam
questions.
11. Assign integer weights to the edges of Kn . Prove that on every cycle the total weight is even
if and on
MATH 443
Problems #7
Due Thursday Nov 21.
These problems are for classroom presentations and some fraction of them may be final exam
questions.
35. Let L(G) denote the line graph of G. Let G be a simple graph with (L(G) = 2. Show that
G is a vertex disjoi
MATH 443
Problems #5
Due Thursday Nov 7.
These problems are for classroom presentations and some fraction of them may be final exam
questions.
23. Let G be a cubic simple connected planar map with 3 faces of size 4, s faces of size 6 and t
faces of size 1
MATH 443
Various Notations
In this course, a graph G consists of a nite set V (G) = V of vertices and a nite set E(G) = E
of edges such that each edge e = uv has associated with it two endpoints u, v V which need not
be distinct. A loop is an edge with bo
MATH 443
Midterm
Thursday October 24, 2013.
For all questions, explanations are required. Aim for clarity and precision. If you cant solve a
problem, interesting progress to a solution may get partial credit.
1. Let G be a connected simple graph with (G)
MATH 443
Assignment #1
Due Tuesday September 24.
1. A claw is a 4 vertex graph of 3 edges with one vertex joined to each of the other three.
Let G be a simple graph with all vertices of degree 3 i.e. G is cubic. Prove that G has a
decomposition into claws
MATH 443 PROBLEM SET 3
Jean-Sbastien Basque-Girouard
e
University of British-Columbia
December 15, 2013
11. Proof. Assume that on every cycle, the total weight is even. Then the number of odd edges
in any cycle is even. We get a bipartition of the graph b
MATH 443 PROBLEM SET 4
Jean-Sbastien Basque-Girouard
e
University of British-Columbia
October 18, 2013
17. Proof. We can use induction. The statement is clearly true for the base case: the adjacency
matrix encodes the paths of length 1. Assume that each e
MATH 443 PROBLEM SET 2
Jean-Sbastien Basque-Girouard
e
University of British-Columbia
October 1, 2013
5. Proof. Let G be a graph such that (G) 2. Let x0 , . . . , xk be the longest path in G. Then
the neighbours of xk are in the path, otherwise we could m
MATH 443 PROBLEM SET 7
Jean-Sbastien Basque-Girouard
e
University of British-Columbia
November 25, 2013
35. Proof.
(L(G) = 2 (G) = 2 d(v) 2 v.
36.
37. Proof. Suppose that t > r + s. Then the graph does not have a Hamiltonian cycle, since the
vertices of a
MATH 443
Assignment #4
Due Tuesday November 26.
1. a) Let G be a graph with even degrees. Show that G can be decomposed into even length
closed trails and a set of vertex disjoint odd cycles.
We can begin by decomposing E(G) into cycles. An even cycle in
MATH 443
Assignment #4
Due Thursday November 28.
1. a) Let G be a graph with even degrees. Show that G can be decomposed into even length
closed trails and a set of vertex disjoint odd cycles.
b) Let G be a graph on n vertices with the property that every
MATH 443
Assignment #3
Due Tuesday November 5.
1. Let G be a simple 3-regular graph. Recall that P4 refers to the path of three edges (and 4
vertices).
a) Prove that if G has a (edge) decomposition into P4 s then G has a perfect matching.
I asked this rst
MATH 443
Assignment #1
Due Tuesday September 24.
1. A claw is a 4 vertex graph of 3 edges with one vertex joined to each of the other three.
Let G be a simple graph with all vertices of degree 3 i.e. G is cubic. Prove that G has a
decomposition into claws
MATH 443
Assignment #2
Due Tuesday October 7.
1. Let D = (N, A) be a directed graph with capacities on arcs denoted u(e). A circulation is a
ow x = (x(e) : e A) with
x(e) v N
x(e) =
e : t(e)=v
e : h(e)=v
0 x(e) u(e) e A
A ow cycle is a ow y given by a dir
MATH 443
Assignment #2
Due Tuesday October 7.
1. Let D = (N, A) be a directed graph with capacities on arcs denoted c(e). A circulation is a
ow x = (x(e) : e A) with
x(e) v N
x(e) =
e : t(e)=v
e : h(e)=v
0 x(e) c(e) e A
Show that x can be written as a sum
MATH 443
Topics
Tree descriptions.
A graph with all degrees even has a cycle
Euler Tours
Halls Theorem
A regular bipartite has a perfect matching
Maximum Matching equal minimum vertex cover in a bipartite graph.
Tuttes Matching theorem.
Max Flow Min Cut T