Syllabus, Mathematics 243: Combinatorial and Discrete Geometry (Fall, 2005)

Prepared by:

Joseph Malkevitch

Mathematics and Computer Science

York College (CUNY)

Jamaica, New York 11451

Combinatorial and Discrete Geometry

3 hours, 3 credits

Prerequisite: Mathematics 122 or permission of instructor

Course description:

A survey of Euclidean and non-Euclidean geometrical ideas primarily from a non-metrical perspective. Basic topics in the in the theory of graphs such as trees, planarity, network flows, coloring problems, and Euler's polyhedral formula. Basic topics in convexity, discrete, and computational geometry such as Helly's theorem, curves of constant breadth, the Sylvester-Gallai theorem, and Voronoi diagrams.

Text:

Gary Chartrand and Ping Zhang, Introduction to Graph Theory, McGraw Hill, 2005.

Tentative syllabus:

Week 1. Visualization and solving geometrical problems

a. Listing the properties of geometrical objects

b. The distinction between metrical and combinatorial geometry

c. Sample problems in combinatorial and discrete geometry

d. Methods of geometrical proof (including induction)

e. Situations where geometry can be applied

Week 2. Basic graph theory

a. Paths, cycles, and trails

b. Vertex degree and degree sequences

c. Graph and digraph models

Week 3. Basic graph theory continued

a. Induced subgraphs

b. Isomorphism of geometrical structures

c. Abstract distance and graph distance

d. Phylogenetic structures and edit distance

Week 4. Traversability problems in graphs

a. Eulerian circuits and paths. Chinese postman problem.

b. Hamiltonian circuits and paths. Traveling salesman problem.

c. Trees, spanning trees. Minimum cost spanning trees.

d. Prim's and Kruskal's algorithms and their complexity.

Week 5. Coloring problems

a. Coloring problems

b. 2, 3, and 4 color theorems

c. Edge and vertex coloring problems

d. Art gallery theorems via coloring problems

e. Coloring applied to scheduling problems.

Week 6. Planar graphs and graphs of polyhedra

a. Planar, plane, and non-planar graphs

b. Coloring the faces of graphs

c. Embeddings of graphs of surfaces

Week 7. Euler's polyhedral formula

a. Euler's polyhedra formula, and proofs of it.

b. Genus of surfaces and graphs. Orientability

c. Proof that there are 5 regular solids

Week 8. Polyhedra, Tilings, and Steinitz's Theorem

a. Convexity

b. Steinitz's Theorem for convex polyhedra

c.. Tilings of the sphere and plane. Hyperbolic tilings

d. Pseudolines and tilings

e. Geometric transformations

Week 9. Polyhedra and tilings continued.

a. Transformations of graphs of polyhedra

b. Medial graphs, line graphs, and truncation of graphs

c. Fullerenes and their applications to chemistry

Week 10. Network flows

a. Flows in networks

b. The max flow min cut theorem of Ford and Fulkerson.

c. Applications of flow problems to water, oil, and electric distribution systems.

Week 11. Matchings and Philip Hall's Theorem

a. Matchings in graphs.

b. Systems of distinct representatives and applications to job assignments problems.

c. Philip Hall's Theorem

Week 12. The Sylvester-Gallai problem.

a. Configurations of points and lines. Ordinary points and lines.

b. Sylvester's problem and Gallai's proof

c. Euler relation proof of Sylvester-Gallai, and Kelly's metrical proof.

d. Finite geometries and axioms for them. (The Fano plane.)

e. Bolyai-Lobachevsky planes, projective planes, and affine planes.

f. Error correction codes and the Hamming distance.

Week 13. Convexity and curves of constant breadth

a. Helly's theorem and its relatives.

b. Curves of constant breadth and the Wankel engine

c. Reauleux triangle, rotors, and Barbier's Theorem

Week 14. Computational geometry

a. Computer representation of geometrical objects.

b. Geometrical skeletons.

c. Computer vision applications.

d. Independent discovery of Voronoi diagrams in geography, physics, and crystallography.

e. Euclidean construction and Voronoi diagrams.

**Assessment**

Hour examinations (2 exams) (40 %)

(No make-up examinations.)

Final examination (2 hours) (40 %)

(This examination will cover the work from the whole semester.)

Homework (5%)

Writing project (at most 10%, if it helps your grade)

Class work (5%)

Phone: 718-262-2551 (Leave a message if I am not there.)

email: joeyc@cunyvm.cuny.edu

http://www.york.cuny.edu/~malk

Date: Fall, 2005