Math 4690/6690, Spring 2010
Daniel Krashen
Course Syllabus
Exam schedule:
- Exam 1
Thursday, February 25 - Exam 2
Date to be determined (during class time) - Final Exam
Tuesday, May 4
12:00 - 3:00 pm
- Practice Sheet for Exam 1 (pdf)
Homework Assignments
- Due Thursday, January 21
- Textbook problems 1.1/1, 2, 6, 7, 11
- Prove that the composition of two bijetive functions is bijective (use the definition of
functions and bijectivity which were given in class)
- Due Thursday, January 28
- Textbook problems 1.2/1, 3, 4
- Prove that if G is a graph and v, w are verticies of G
such that there exists a walk from v to w then there exists a path from v
to w.
- Prove that if G is a graph with no cycles, and e is an
edge of G then the graph obtained by deleting the edge e from G cannot
be connected.
- (6690 Question) For a vertex v in a graph G, let X be
the collection of subgraphs of G which contain v and are connected. Let H
be the graph obtained as the union of the graphs (verticies and edges) in
X. Show that a vertex w is in H if and only if there exists a path from v
to w.
- (6690 Question) Suppose G is a graph with no cycles.
Construct a new graph G' as follows : the vertices of G' consist of the paths
of length 2 in G, and we say that two of these are conected by an edge in
G' if the paths share some common edge. Prove that G' contains no
cycles, or exhibit a counterexample to this claim.
- Due Thursday, February 4
- Textbook problems 1.3/2, 4, 8, 9, 11
- Show that if G is a connected graph, and H and H' are connected subgraphs whose union is G
then H and H' must share a common vertex.
- Due Thursday, February 11
- Textbook problems 1.3/5, 6, 12
- Show that if G is a graph such that there are at most 3
distinct paths between any two points, and exactly m cycles, show that if you
add an edge to G, the resulting graph has no more than m+3 cycles.
- Suppose G is a connected graph with exactly one cycle C,
and that C has 5 edges (a cycle of length 5). How many different spanning trees
are contained in G?
- (6690) Textbook problems 1.3/10, 13
- Due Thursday, February 18
- Textbook problems 2.1/1, 6, 9, 12
- No justification is needed for problem 9.
- Show that if G=Kn is the complete graph on n vertices, then its
chromatic number is n. Show further, that if we remove any edge from G, the
chromatic number of the resulting graph is n-1.
- Due Thursday, March 25
- 2.2/3 (explain reasoning), 4 --- don't forget Vizing's Theorem (as we
stated it in class!)
- 2.3/4, 7 (Hand in only the cases Q3, O, D)
- 2.4/1, 4
- 3.1/7
- Due Thursday, April 1
- Do the problems in the lecture notes on
equivalence relations
- 4.1/1, 2
Lecture Outlines (PDF format):
Homework Assignments
- Due Thursday, January 21
- Textbook problems 1.1/1, 2, 6, 7, 11
- Prove that the composition of two bijetive functions is bijective (use the definition of functions and bijectivity which were given in class)
- Due Thursday, January 28
- Textbook problems 1.2/1, 3, 4
- Prove that if G is a graph and v, w are verticies of G such that there exists a walk from v to w then there exists a path from v to w.
- Prove that if G is a graph with no cycles, and e is an edge of G then the graph obtained by deleting the edge e from G cannot be connected.
- (6690 Question) For a vertex v in a graph G, let X be the collection of subgraphs of G which contain v and are connected. Let H be the graph obtained as the union of the graphs (verticies and edges) in X. Show that a vertex w is in H if and only if there exists a path from v to w.
- (6690 Question) Suppose G is a graph with no cycles. Construct a new graph G' as follows : the vertices of G' consist of the paths of length 2 in G, and we say that two of these are conected by an edge in G' if the paths share some common edge. Prove that G' contains no cycles, or exhibit a counterexample to this claim.
- Due Thursday, February 4
- Textbook problems 1.3/2, 4, 8, 9, 11
- Show that if G is a connected graph, and H and H' are connected subgraphs whose union is G then H and H' must share a common vertex.
- Due Thursday, February 11
- Textbook problems 1.3/5, 6, 12
- Show that if G is a graph such that there are at most 3 distinct paths between any two points, and exactly m cycles, show that if you add an edge to G, the resulting graph has no more than m+3 cycles.
- Suppose G is a connected graph with exactly one cycle C, and that C has 5 edges (a cycle of length 5). How many different spanning trees are contained in G?
- (6690) Textbook problems 1.3/10, 13
- Due Thursday, February 18
- Textbook problems 2.1/1, 6, 9, 12
- No justification is needed for problem 9.
- Show that if G=Kn is the complete graph on n vertices, then its chromatic number is n. Show further, that if we remove any edge from G, the chromatic number of the resulting graph is n-1.
- Due Thursday, March 25
- 2.2/3 (explain reasoning), 4 --- don't forget Vizing's Theorem (as we stated it in class!)
- 2.3/4, 7 (Hand in only the cases Q3, O, D)
- 2.4/1, 4
- 3.1/7
- Due Thursday, April 1
- Do the problems in the lecture notes on equivalence relations
- 4.1/1, 2