Graph Theory Spring 2012
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Graph Theory – Spring 2012
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, 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 Pn, cycle Cn, complete graph Kn, bipartite graph, complete bipartite graph Km,n, wheel graph Wn, star graph Stn, cube graph 

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

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.


  • 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,

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


  • Question and Answer Day


  • 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.


  • 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


  • 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



  • Question and Answer Day


  • Exam 2


  • Presentations

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

  • Presentations
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Christopher HanusaQueens CollegeMathematics Department.