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New definitions are in bold and key topics covered are in a bulleted list.
This schedule is approximate and subject to change!
Weeks 1 & 2 — Introduction (4 classes)
1/31: (Notes pages 0-15) 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
- Syllabus discussion.
- What is a graph?
- How to describe a graph.
- Degree sequence of a graph.
- Theorem 1.1.2.
2/2: (Notes pages 16-24) path graph Pn, cycle Cn, complete graph Kn, bipartite graph, complete bipartite graph Km,n, wheel graph Wn, star graph Stn, cube graph, 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
- Proof of Theorem 1.1.2.
- A dictionary of graphs.
- Schlegel diagrams of Platonic solids.
- When are two graphs the same?
- Larger graphs from smaller graphs.
2/7: (Notes pages 25-32) subgraph, induced subgraph, proper subgraph, path in G from a to b, connected graph, connected component, tree, forest, cut vertex, bridge
- Smaller graphs from larger graphs.
- Groupwork on definitions.
- Connected graphs and connected components
- Theorem 1.3.1
- 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.
- Remark: If you remove an edge from a cycle, the remaining graph is connected.
- Lemma B. Let G be a connected graph. Let C be a cycle contained in G. Let
e be any edge in C. Then the graph G\e is connected.
2/9: tree, forest, cut vertex, bridge
- Trees and forests.
- Theorems 1.3.2, 1.3.3, and 1.3.5.
- Theorems 2.4.1 and 3.2.1
- Cut vertex and bridges
Weeks 3 & 4 — Coloring graphs (2 classes)
2/14: (Notes pages 33-45) (proper) coloring, chromatic number,
- Proper colorings, chromatic number.
- Groupwork: the chromatic number of Gr.
2/16: Discussion of Homework 2
2/21: No class; Presidents' Day
2/23: critical graph, bipartite, edge coloring, proper edge coloring, edge chromatic number, snark, turning trick
- Lemma C. If H is a subgraph of G, and chi(H)=k, then chi(G)>=k.
- Cliques and their relation to chromatic number.
- Critical graphs, Thms 2.1.1-2.1.2.
- Critical graphs, Thms 2.1.3-2.1.4.
- Bipartite graphs, Thm 2.1.6.
- edge coloring, proper edge coloring, edge chromatic number.
- Thms 2.2.1-2.2.2
- Snarks
- Turning trick.
- Line graph of a graph
2/28: (Notes pages 46-58) Decomposition, Hamiltonian cycle, (closed) knight's tour
- Decomposition of a graph into smaller graphs.
- A snark has no Hamiltonian cycle
- Hamiltonian cycle decomposition of Kn.
- Knight's tours. (Here is the article Knight's Tours on a Torus.)
3/1: (Notes pages 59-70) 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
- Vocabulary of pseudographs.
- Eulerian circuits.
- Thm 3.1.2. Even degrees implies Eulerian.
- Eulerian Trails.
- de Bruijn sequences.
3/28: (Notes pages 71-82).
- Planar graphs
- Euler's Formula.
- Maximal planar graphs.
- dual graph, self-dual graph
- Maps, normal maps.
- Four Color Theorem (not proved).
- History of the four color theorem.
3/30: (Notes pages 83-92).
- Six Color Theorem (proved).
- Five Color Theorem (proved).
- Kempe Chains argument
- Modifications of graphs.
- Kuratowski's Theorem.
4/4:
- Five Color Theorem (proved).
- Kempe Chains argument
- Modifications of graphs.
- Kuratowski's Theorem.
4/6: Discussion of Homework 6.
4/11: (Notes pages 93-100).
- What is an algorithm?
- Matchings
- Maximum Matchings
- Hungarian Algorithm
4/13: Peer review day
4/27: (Notes pages 101-109).
- Stable Marriages
- Gale-Shapely Algorithm
5/2: (Notes pages 110-122).
- Networks, Flows
- Max flow/Min Cut
- Ford-Fulkerson Algorithm
5/4: (Notes pages 123-129).
- Transshipments
- Dynamic Networks
5/9: (Notes pages 130-137).
- Weighted graphs
- Minimum-weight spanning tree
- Kruskal's algorithm
- The traveling salesman problem
- Finding a good TSP-tour
5/11: Homework 8 discussion.
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