Math 381, Graph Theory
Spring 2021
Syllabus

Textbook

Nora Hartsfield and Gerhard Ringel, Pearls in Graph Theory, third edition (Dover, 2003).
(Essentially the same as the second edition, Academic Press, 1994.)

Topics

• Chapter 1. Basic Graph Theory
  • 1.1. Graphs and Degrees of Vertices
  • 1.2. Subgraphs, Isomorphic Graphs, Automorphisms
    Extra: Automorphisms (handout)
  • 1.3. Trees
• Chapter 2. Colorings of Graphs
  • 2.1. Vertex Colorings
    Extra: Chromatic Polynomial (handout)
  • 2.2. Edge Colorings
    Extra: Line Graphs (handout)
  • 2.3. Decompositions and Hamilton Cycles
  • 2.4. More Decomposition
• Chapter 3. Circuits and Cycles
  • 3.1. Eulerian Circuits
  • 3.2. The Oberwohlfach Problem
• Chapter 4. Extremal Problems
  • 4.1. A Theorem of Turan
  • 4.2. Cages
  • Extra: Line Graphs (again) (handout)
• Chapter 5. Counting
  • 5.1. Counting 1-Factors
  • 5.2. Cayley's Spanning Tree Formula
  • Extra: The Matrix-Tree Theorem (handout)
  • 5.3. More Spanning Trees
• Linear algebra bonus: Graphs as Arrangements of Lines, Planes, or Hyperplanes (will not be tested)
• Chapter 6. Labeling Graphs
  • 6.1. Magic Graphs and Graceful Trees
  • 6.2. Conservative Graphs
• Chapter 7. Applications and Algorithms
  • 7.1. Spanning Tree Algorithms
• Chapter 8. Drawings of Graphs
  • 8.1. Planar Graphs
  • 8.2. The Four Color Theorem
  • 8.3. The Five Color Theorem
• Chapter 9. Measurements of Closeness to Planarity
  • 9.1. Crossing Number
  • 9.2. Thickness and Splitting Number
  • 9.3. Heawood's Empire Problem
• Chapter 10 and Extra. Graphs on Other Surfaces
  • Diagrams for the Torus, Double Torus, etc. (in class)
  • 10.3. The Genus of a Graph
    (Omit anything about "schemes", "maximal rotations", and "current graphs", and pp. 235-237.)