Introduction (4 classes)

1/28: 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.

2/4: path graph P_n, cycle C_n, complete graph K_n, bipartite graph, complete bipartite graph K_m,n, wheel graph W_n, star graph St_n, 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?

2/6: 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.

2/11: 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

2/13:

• Discussion of Homework 2

Coloring (3 classes)

2/20: (vertex) coloring, proper coloring,

• 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).

2/25: critical graph, 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

2/27:

• Notes from Section 2.3. (Notes pages 45–51) 
• Spanning subgraphs
• Decomposition
• Perfect matchings, Perfect matching decompositions
• Hamiltonian cycle, Hamiltonian cycle decompositions

3/4:

• Question and Answer Day

3/6:

• Exam 2

3/11:   (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

• Knight's tours. (Notes pages 52–57) 
• Eulerian graphs and de Bruijn cycles (Notes pages 58–69) 
• Vocabulary of pseudographs.
• Eulerian circuits.
• Thm 3.1.2. Even degrees implies Eulerian.
• (Here is the article Knight's Tours on a Torus.)

3/13:   sequence, binary sequence, de Bruijn sequence, de Bruijn graph

• (no new notes)
• Eulerian Trails.
• de Bruijn sequences.
• (Here is the article Knight's Tours on a Torus.)

3/18: 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 70–81) 
• 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.

3/20:

• Discussion of Homework 4

3/25-4/1:

• SPRING BREAK!

4/3: dual graph, map, normal map, kempe chain,

• Notes from Sections 8.2, 8.3, and 9.1 (Notes pages 82–91) 
• Six Color Theorem (proved).
• Five Color Theorem (proved).
• Five Color Theorem (proved).
• Kempe Chains argument

4/8: deletion, contraction, minor, subdivision

• Modifications of graphs.
• Kuratowski's Theorem.

4/10: crossing number, thickness, and genus of a graph, torus,

• Notes from Sections 9.1, 9.2, and 10.3 (Notes pages 92–98) 
• Statistics of nonplanarity.
• Crossing number of a graph
• Thickness of a graph
• Genus of a graph

4/15: algorithm, correctness, matching, perfect matching, Hungarian algorithm, M-alternating path, M-augmenting path

• Notes from Section 7.2 and more (Notes pages 99–107) 
• Algorithms.
• Maximal, maximum, perfect matchings.
• Hungarian algorithm.
• Correctness of the Hungarian algorithm.

4/17: stable matching

• Notes about stable matchings (Notes pages 108–116) 
• Stable matchings
• The play
• Proof of correctness
• Proof of male optimality

4/22:

• Discussion of Homework 7

4/24: directed edges, network, flow, cut, max flow, min cut, augment a flow

• Notes about network flow (Notes pages 117–129) 
• Network
• Flow in a network, MAX FLOW
• Edge cuts, MIN CUT

4/29: Ford-Fulkerson algorithm, companion graph, transshipment, dynamic network

• Ford-Fulkerson algorithm and examples
• Notes about transshipment (Notes pages 130–136) 
• Transshipment
• Dynamic Network

5/1: weighted graph, spanning tree, Kruskal's Algorithm, Hamiltonian Cycle, Traveling Salesman Tour

• Notes from Section 7.1 and TSP (Notes pages 137–144) 
• Minimum Weight Spanning Trees
• Traveling Salesman Problem

5/6:

• TBD

5/8:

• Question and Answer Day

5/13:

• Exam 2

5/15:

• Presentations

Wednesday, 5/22 (4pm-6pm)

• Presentations