Introduction (3 classes)

2/1: 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.

2/6: 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.

2/8: 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

• A dictionary of graphs.
• Schlegel diagrams of Platonic solids.
• When are two graphs the same?
• Larger graphs from smaller graphs.
• Smaller graphs from larger graphs.
• Groupwork on definitions.

Graph Statistics (3 classes)

2/15: path in G from a to b, connected graph, disconnected graph, connected component, cut vertex, cut set

• Groupwork on definitions.
• Notes from Section 1.3 and additional concepts in connectivity.   (Notes pages 24–34) 
• Connected graphs and connected components
• Vertex and edge connectivity

2/21: bridge, disconnecting set, connectivity (κ(G)), edge connectivity (κ'(G)), minimum, minimal, tree, forest

• Vertex and edge connectivity (No new notes today)
• Um versus Al
• 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.
• Theorem 1.3.1.

2/22:

• Discussion of Homework 2

2/27: tree, forest, girth g(G), distance between vertices, diameter diam(G), clique, clique number ω(G), independent set, independence number α(G)

• Trees and forests.
• Theorems 1.3.2, 1.3.3, and 1.3.5.
• Theorems 2.4.1 and 3.2.1
• Graph statistics (Notes pages 35–36) 

Coloring (2 classes)

2/29: (vertex) coloring, proper coloring,

• Notes from Sections 2.1 and 2.2 (Notes pages 37–49) 
• (Vertex) coloring, proper coloring
• Chromatic number
• Lemma C. If H is a subgraph of G, then χ(H)≤χ(G).
• Groupwork on graph statistics

3/5: critical graph, bipartite graph, edge coloring, edge chromatic number

• Critical graphs
• Bipartite graphs
• Edge coloring
• Vizing's Theorem

3/7: snark, turning trick

• Snarks
• Edge chromatic number of complete graphs

3/12:

• Question and Answer Day

3/14:

• Exam 2

Planarity (4 classes)

3/19: 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 50–61) 
• Planar graphs.
• Euler's Formula.
• Maximal planar graphs.

3/21: dual graph, map, normal map, kempe chain, deletion, contraction, minor, subdivision

• dual graph, self-dual graph
• Maps, normal maps.
• Four Color Theorem (not proved).
• History of the four color theorem.
• Notes from Sections 8.2, 8.3, and 9.1 (Notes pages 62–71) 
• Six Color Theorem (proved).
• Five Color Theorem (proved).

3/26: kempe chain, deletion, contraction, minor, subdivision

• Five Color Theorem (proved).
• Kempe Chains argument
• Modifications of graphs.
• Kuratowski's Theorem.

3/28: crossing number, thickness, and genus of a graph, torus,

• Notes from Sections 9.1, 9.2, and 10.3 (Notes pages 72–78) 
• Statistics of nonplanarity.
• Crossing number of a graph
• Thickness of a graph
• Genus of a graph
• The Peterson graph is non-planar.

Algorithms (5 classes)

4/2: algorithm, correctness, matching, perfect matching, Hungarian algorithm, M-alternating path, M-augmenting path

• The Peterson graph is non-planar.
• Notes from Section 7.2 and more (Notes pages 79–86) 
• Algorithms.
• Maximal, maximum, perfect matchings.
• Hungarian algorithm.

4/4:

• Discussion of Homework 4

4/16: stable matching

• Correctness of the Hungarian algorithm.
• Notes about stable matchings (Notes pages 87–95) 
• Stable matchings
• The play
• Proof of correctness
• Proof of male optimality

4/18: directed edges, network, flow, cut, max flow, min cut, augment a flow

• Notes about network flow (Notes pages 96–108) 
• Network
• Flow in a network, MAX FLOW
• Edge cuts, MIN CUT

4/23:

• Peer Review Day

4/25: Ford-Fulkerson algorithm, companion graph, transshipment, dynamic network

• Ford-Fulkerson algorithm and examples
• Notes about transshipment (Notes pages 109–115) 
• Transshipment
• Dynamic Network

4/30: weighted graph, spanning tree, Kruskal's Algorithm, Hamiltonian Cycle, Traveling Salesman Tour

• Notes from Section 7.1 and TSP (Notes pages 116–123) 
• Minimum Weight Spanning Trees
• Traveling Salesman Problem

5/2:

• Proof of Kruskal's algorithm

5/7:

• Question and Answer Day

5/9:

• Exam 2

5/14:

• Presentations

5/23, 4-6 pm (Final Exam Day)

• Presentations