Math 412: Introduction to Graph Theory (Spring 2018)

Mikhail Lavrov


Location

Grading

The course grade will be calculated as follows, out of 660 points total:

You can check your grades via Moodle.

Your grade total will be converted to a letter grade according to the following scale:

A+ ≥ 620 B+ ≥ 490 C+ ≥ 360 D+ ≥ 250
A ≥ 570 B ≥ 440 C ≥ 320 D ≥ 210
A- ≥ 530 B- ≥ 400 C- ≥ 280 D- ≥ 180

It is possible to take the course for 4 credits rather than 3, at the cost of extra homework questions and more difficult exams. If you want to do this, you need to register at the math office in Altgeld Hall soon after the start of the course.

Homework

There will be 10 homework assignments, to be turned in at the beginning of class the day they are due. Of these homework assignments, the top 8 homework scores will determine your grade. Each assignment is graded out of 20 points, for a total of 160. The only extra credit given will be a 1-point bonus for typesetting your homework (for example, but not necessarily, in LaTeX); graph diagrams can still be hand-drawn.

If you cannot attend class, you can submit a scanned copy or a photo of your homework by e-mail before class begins. Please do not do this if you can attend class.

If the homework assignment is received after class on the due date, but before the next class, it will be accepted as late, for a 2-point penalty. Homework will not be accepted after the next class for any reason.

Exams

There will be three evening midterm exams: Wednesday 2/14 and Wednesday 3/14 from 7pm to 8:30pm in Talbot 103, and Wednesday 4/18 from 7pm to 8:30pm in Engineering Hall Room 106B8. Correspondingly, three lectures will be canceled, not necessarily in the same week as the midterm exams; these are also marked in the syllabus below.

The final exam will be given on Monday, May 7, 1:30-4:30pm, in 245 Altgeld Hall (our usual classroom).

Detailed syllabus

The course follows chapters 1-7 of the textbook Introduction to Graph Theory by D. B. West. The syllabus below will initially describe a tentative plan for what parts of the textbook will be covered when. As the semester progresses, I will update the syllabus with information about what actually happened in class, adjustments to my plans for the future, and links to homework assignments.

Date Chapter Details Homework
Wed January 17 Chapter 1
Fundamentals
Section 1.1: What is a graph?
Fri January 19 Section 1.1: Isomorphic graphs
Mon January 22 Section 1.2: Paths, cycles, and trails
Wed January 24 Section 1.2: Eulerian circuits
Fri January 26 Section 1.3: Vertex degrees HW 1 due
Mon January 29 Section 1.3: Graphic sequences
Wed January 31 Section 1.4: Directed graphs
Fri February 2 Section 1.4: Tournaments HW 2 due
Mon February 5Chapter 2
Trees
Section 2.1: Trees and forests
Wed February 7 Section 2.1: More on distance
Fri February 9 Section 2.2: Spanning trees HW 3 due
Mon February 12 Section 2.2: Counting trees
Wed February 14 Section 2.3: Optimization Exam: 7pm in Talbot 103 (Topics)
Fri February 16 No class
Mon February 19 Chapter 3
Matchings
Section 3.1: Matchings; Berge's theorem
Wed February 21 Section 3.1: Hall's theorem
Fri February 23 Section 3.1: Vertex and edge covers HW 4 due
Mon February 26 Section 3.2: Stable matchings
Wed February 28 Section 3.3: Tutte's theorem
Fri March 2 Section 3.3: The Berge–Tutte formula HW 5 due
Mon March 5 Section 3.3: Petersen's theorem
Wed March 7 Chapter 4
Connectivity
Section 4.1: Initial definitions
Fri March 9 Section 4.2: Whitney's theorem HW 6 due
Mon March 12 Section 4.2: Menger's theorem
Wed March 14 Section 4.2: Fan lemma Exam: 7pm in Talbot 103 (Topics)
Fri March 16 No class
Mon March 19Spring break: no class
Wed March 21
Fri March 23
Mon March 26Chapter 4
Network Flows
Section 4.3: Network flows
Wed March 28 Section 4.3: The min-cut max-flow theorem Infinite Loops
Fri March 30 Section 4.3: Corollaries of min-cut max-flow HW 7 due
Mon April 2Chapter 6
Planar Graphs
Section 6.1: Planar graphs; Euler's theorem
Wed April 4 Section 6.1: Special classes of planar graphs
Fri April 6 Section 6.2: Conditions for planarity HW 8 due
Mon April 9 Section 6.2: Conflict graphs
Wed April 11 Section 6.3: Map coloring
Fri April 13 No class
Mon April 16 Chapter 5
Coloring
Section 5.1: Basic results HW 9 due
Wed April 18 Section 5.1: Brooks's theorem Exam: 7pm in Eng. Hall 106B8 (Topics)
Fri April 20 Section 5.2: The Mycielskian
Mon April 23 Section 5.2: Color-critical graphs
Wed April 25 Section 5.3: The chromatic polynomial
Fri April 27 Chapter 7
Cycles
Section 7.1: Edge chromatic number HW 10 due. Vizing's refs: [1], [2], [3], [4]
Mon April 30 Section 7.2: Hamiltonian cycles; tough graphs
Wed May 2 Section 7.2: Hamiltonian cycles; Dirac's theorem Not HW 11
Mon May 7 Final Exam 1:30-4:30pm in 245 Altgeld Hall (Topics)

Last updated May 3, 2018. Mikhail Lavrov <mlavrov@illinois.edu>