Math 355: Graph Theory
(Credit Hours:Lecture Hours:Lab Hours)
Graphs and subgraphs, coloring problems, applications.
Desired Learning Outcomes
Minimal learning outcomes
Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.
Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles.
Possible textbooks for this course include (but are not limited to):
- Graph Theory by Rusell Merris (Chap. 1-5).
A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.
Possible other topics include spectral graph theory (networkings, expanders, Ramanujan graphs), characterization of Ramanujan graphs by the Riemann hypothesis for its zeta function. A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.