Graph Theory, Spring 2014
Topics, Notes, and Homework
For each date below, you will find the day's new definitions, the homework assignment that is due that day, any lecture notes for downloading, and the key topics that are covered that day.
This schedule is approximate and subject to change!
Introduction (4 classes)
Monday, January 27
New definitions: graph, vertex, edge, finite graph, multiple edges, loop, simple graph, adjacent, neighbors, incident, endpoint, degree, degree sum, isolated vertex, leaf, end vertex, degree sequence, graphic, Havel-Hakimi algorithm
- Notes from Section 1.1 (Notes pages 0–14) '
- Syllabus discussion.
- What is a graph?
- How to describe a graph.
- Degree sequence of a graph.
- Theorem 1.1.2.
Wednesday, January 29
Homeworks #0 and #1 are due on Wednesday, January 29.
(.tex file)
New definitions: path graph Pn, cycle Cn, complete graph Kn, bipartite graph, complete bipartite graph Km,n, wheel graph Wn, star graph Stn, cube graph
- Proof of Theorem 1.1.2.
- Notes from Section 1.2 (Notes pages 15–23) '
- A dictionary of graphs.
- Schlegel diagrams of Platonic solids.
- When are two graphs the same?
Monday, February 3
Homework #2 is due on Monday, February 3.
(.tex file)
New definitions: Petersen graph, Grotzsch graph, Platonic solid, Schlegel diagram, Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron, equal graphs, isomorphic graphs, disjoint union, union, graph complement, self-complementary graph, subgraph, induced subgraph, proper subgraph
- Larger graphs from smaller graphs.
- Smaller graphs from larger graphs.
- Groupwork on definitions.
Wednesday, February 5
Homework #3 is due on Wednesday, February 5.
(.tex file)
New definitions: path in G from a to b, connected graph, tree, forest, bridge
- Notes from Section 1.3. (Notes pages 24–31) '
- Connected graphs
- Lemma A. If there is a path from a to b in G and a path from b to c in G, then there is a path from a to c in G.
- Lemma B. Let G be a connected graph. Suppose that G contains a cycle C and e is an edge of C. The graph H=G \ e is connected.
- Theorems 1.3.1, 1.3.2, 1.3.3, and 1.3.5.
- Trees and forests.
- Theorems 2.4.1 and 3.2.1
Monday, February 10
Homework #4 is to be prepared for presentation on Monday, February 10.
(.tex file)
- Discussion Day
Coloring (2 classes)
Wednesday, February 19
There is no homework due today.
New definitions: (vertex) coloring, proper coloring, clique, clique number, critical graph,
- Theorems 2.4.1 and 3.2.1
- Notes from Sections 2.1 and 2.2 (Notes pages 32–44) '
- (Vertex) coloring, proper coloring
- Chromatic number
- Lemma C. If H is a subgraph of G, then χ(H)≤χ(G).
Thursday, February 20
Homework #5 is due on Thursday, February 20.
(.tex file)
New definitions: bipartite graph, edge coloring, edge chromatic number, snark, turning trick
- Critical graphs
- Bipartite graphs
- Edge coloring
- Vizing's Theorem
- Snarks
- Edge chromatic number of complete graphs
Cycles and Circuits (2 classes)
Monday, February 24
Homework #6 is due on Monday, February 24.
(.tex file)
New definitions: Spanning subgraph, decomposition, perfect matching, Hamiltonian path, Hamiltonian cycle
- Notes from Section 2.3. (Notes pages 45–51) '
- Knight's tours. (Notes pages 52–57) '
- Spanning subgraphs
- Decomposition
- Perfect matchings, Perfect matching decompositions
- Hamiltonian cycle, Hamiltonian cycle decompositions
- History of Knight's Tours: http://faculty.olin.edu/~sadams/DM/ktpaper.pdf
- Knight's Tours on a Torus: http://www.mimuw.edu.pl/~rytter/TEACHING/ALCOMB/knight_torus.pdf
Wednesday, February 26
Homework #7 is due on Wednesday, February 26.
(.tex file)
New definitions: (closed) knight's tour, walk, trail, path, open, closed, circuit, cycle, loop, Eulerian circuit, Eulerian Trail, Eulerian circuit, Eulerian Trail, sequence, binary sequence, de Bruijn sequence, de Bruijn graph
- Eulerian graphs and de Bruijn cycles (Notes pages 58–69) '
- Vocabulary of pseudographs.
- Eulerian circuits.
- Thm 3.1.2. Even degrees implies Eulerian.
- Eulerian Trails.
- de Bruijn sequences.
Monday, March 3
Homework #8 is to be prepared for presentation on Monday, March 3.
(.tex file)
- Discussion Day
Graph Statistics (2 classes)
Wednesday, March 5
Project topic due today.
Homework #9 is due on Wednesday, March 5. (.tex file) It only contains definitions.
Homework #9 is due on Wednesday, March 5. (.tex file) It only contains definitions.
New definitions: connected component, cut vertex, cut set, disconnecting set, connectivity, edge connectivity, maximum, maximal, girth, distance, diameter, independent set, independence number, vertex cover, vertex cover number
- Graph Statistics (Notes pages 70–76) '
- Connectivity and Edge connectivity
- Girth, distance, and diameter
- Cliques, Independent sets, Vertex covers
Monday, March 10
Homework #10 is due on Monday, March 10.
(.tex file)
- Graph Statistics worksheet
Exam 1 Information
- The first exam of the semester will take place on Monday, March 17.
- The exam covers all material covered through March 10.
- Here are more details about the first exam.
- My students often ask for an example of the style of exam that I am liable to give. I am including my exam from last year. The topics covered by the exam are the same topics, but you should expect your exam to be very different because there are many ways for me to ask questions that test your knowledge on these topics. Click here for last year's exam.
Wednesday, March 12
- Question and Answer Day
Monday, March 17
- Exam 1
Planarity (4 classes)
Wednesday, March 19
Before class, compare and constrast the definitions of plane drawing and planar graph.
New definitions: drawing, simple curve, plane drawing, plane graph, planar graph, region, face, outside face, maximal planar, dual graph
- Notes from Sections 8.1 and 8.2 (Notes pages 77–88) '
- Planar graphs.
- Euler's Formula.
- Maximal planar graphs.
Monday, March 24
Project outline due Monday.
Homework #11 is due on Monday, March 24.
(.tex file)
New definitions: dual graph, map, normal map, Kempe chain
- Notes from Sections 8.2, 8.3, and 9.1 (Notes pages 89–98) '
- dual graph, self-dual graph
- Maps, normal maps.
- Four Color Theorem (not proved).
- History of the four color theorem.
- Six Color Theorem (proved)
Wednesday, March 26
Homework #12 is due on Wednesday, March 26.
(.tex file)
New definitions: deletion, contraction, minor, subdivision
- Five Color Theorem (proved)
- Kempe Chains argument
- Modifications of graphs.
- Kuratowski's Theorem.
Monday, March 31
Continue Wikipedia work.
New definitions: crossing number, thickness, genus of a graph, torus
- Notes from Sections 9.1, 9.2, and 10.3 (Notes pages 99–105) '
- Statistics of nonplanarity.
- Crossing number of a graph
- Thickness of a graph
- Genus of a graph
Graph Algorithms (6 classes)
Wednesday, April 2
Draft started in Wikipedia.
New definitions: algorithm, correctness, matching, Hungarian algorithm, M-alternating path, M-augmenting path
- Notes from Section 7.2 and more (Notes pages 106–113) '
- Algorithms.
- Maximal, maximum, perfect matchings.
- Hungarian algorithm.
- Correctness of the Hungarian algorithm.
Monday, April 7
Homework #13 is to be prepared for presentation on Monday, April 7.
(.tex file)
- Discussion Day
Wednesday, April 9
- (no class)
— Spring Break —
Wednesday, April 23
Homework #14 is due on Wednesday, April 23.
(.tex file)
New definitions: stable matching
- Notes about stable matchings (Notes pages 114–122) '
- Stable matchings
- The play
- Proof of correctness
- Proof of male optimality
Monday, April 28
Complete Wikipedia Project.
New definitions: directed edges, network, flow, cut, max flow, min cut, augment a flow
- Notes about network flow (Notes pages 123–135) '
- Network
- Flow in a network, MAX FLOW
- Edge cuts, MIN CUT
Wednesday, April 30
New definitions: Ford-Fulkerson algorithm, companion graph, transshipment, dynamic network
- Ford-Fulkerson algorithm and examples
- Notes about transshipment (Notes pages 136–142) '
- Transshipment
- Dynamic Network
Monday, May 5
Homework #15 is due on Monday, May 5.
(.tex file)
New definitions: weighted graph, Kruskal's Algorithm, Hamiltonian Cycle, Traveling Salesman Tour
- Notes from Section 7.1 and TSP (Notes pages 143–150) '
- Minimum Weight Spanning Trees
- Traveling Salesman Problem
Exam 2 Information
- The second exam of the semester will take place on Monday, May 12.
- The exam covers all material covered since the first exam.
- Here are more details about the second exam.
- My students often ask for an example of the style of exam that I am liable to give. I am including my exam from last year. The topics covered by the exam are the same topics, but you should expect your exam to be very different because there are many ways for me to ask questions that test your knowledge on these topics. Click here for last year's exam. (Recognize that material from Questions 1b and 2 is not covered on this exam.)
Wednesday, May 7
- Question and Answer Day
Monday, May 12
- Exam 2
Wednesday, May 14
- Presentations
Wednesday, May 21, 4–6pm (Final Exam Period)
- Presentations
