Tentative lecture schedule
Here is a tentative list of lectures. I probably won't be able to keep to this schedule exactly, because of holidays, but I thought that this would be a good guide for the material I want to cover in the course.

• Week 1: intro, basic definitions, kinds of graphs, degrees, sum of degrees Eulerian circuit theorem
  
  
• Week 2 (trees)
  Lecture 1: Trees, Krushkal's Algorithm, Prufer Codes
  Lecture 2: Matrix Tree Theorem, estimates on trees
  
  
• Week 3: (planar graphs)
  Lecture 1: polygonal Jordan curve and arc Theorems
  Lecture 2: Euler characteristic, non-planarity proofs
  
  
• Week 4: (graph coloring)
  Lecture 1: Basic Examples, 5-color theorem
  Lecture 2 Brooks' Theorem, Chromatic polynomial
  
  
• Week 5: (graph matching)
  Lecture 1: Hall's Matching Theorem, augmenting paths
  Lecture 2: Konig-Egevary Theorem, Gale-Shapely Proposal Algorithm
  
  
• Week 6: (connectivity in graphs)
  Lecture 1: Max Flow Min Cut
  Lecture 2: Menger's Theorem
  
  
• Week 7: (Electric Networks)
  Lecture 1: Short cut theorem
  Lecture 2: Polya's recurrance theorem
  
  
• Week 8: (Ramsey theory)
  Lecture 1: The basic Ramsey Theorem
  Lecture 2: Link Universality
  
  
• Week 9: (Random Graphs)
  Lecture 1: Rado's Theorem
  Lecture 2: graphs of large girth/chromatic number
  
  
• Week 10: (planar graphs revisited)
  Lecture 1: Graph Minors
  Lecture 2: Kuratowski's Theorem
  
  
• Week 11-12 (additional topics:)
  Sperner's Lemma
  Cayley Graphs
  Matroids