New definitions are in bold and key topics covered are in a bulleted list.
This schedule is approximate and subject to change!
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, HavelHakimi 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, selfcomplementary 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
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:
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,
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:
3/14:
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, selfdual 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 nonplanar.
Algorithms (5 classes)
4/2: algorithm, correctness, matching, perfect matching, Hungarian algorithm, Malternating path, Maugmenting path
 The Peterson graph is nonplanar.
 Notes from Section 7.2 and more (Notes pages 79–86)
 Algorithms.
 Maximal, maximum, perfect matchings.
 Hungarian algorithm.
4/4:
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:
4/25: FordFulkerson algorithm, companion graph, transshipment, dynamic network
 FordFulkerson 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:
5/7:
5/9:
5/14:
5/23, 46 pm (Final Exam Day)
