Math 645: A Survey of Mathematical Problems I
Frank Sottile

• This week, I will have you read Whittaker's biography of William Rowan Hamilton. He did create the topic of Hamiltonian cycles on graphs. This is a minor part of his legacy. If you used i, j, and k to denote vectors, or took a course in advanced mechanics (junior or senior level undergraduate class in physics), you have run into his legacy. There is a branch of geometry, called "symplectic geometry" that arose out of his work in physics.

  Frank Harary's selection describes Hamilton's contribution to graph theory. I met Harary; one of my best friends in mathematics from High school, Martin Erickson, took classes from him and worked with him, and when I visited Martin, I also met Harary. Harary was quite a character, a font of stories about mathematicians and rather charismatic. He had an outsized influence on graph theory in the 1960's—1980's.

• Read number 8 in the course packet by Whittaker
• Read number 10 in the course packet by Harary
• Read number 11 in the course packet by Trudeau
• Chapter 3 of A Survey of Mathematical Problems.
• Dr. Geller's Lecture 5. Note that it is not Alexander Hamilton (US Founding Father), but rather William Rowan Hamilton (Greatest Irish Mathematician) who started the topic of Hamiltonian cycles, and not by the travelling salesman problem, but by the problem of finding a Hamiltonian path in the edge graph of the dodecahedron. This illustrates how it is important to fact check everything, or at least revise your writing critically.

1. Exercises 3.4-3.6, and Problems 3.5-3.8.
   Problem 3.7 has a slightly ambiguous statement. Here it is, disambiguated:
   At a dinner party, there are 2n guests to be seated at a round table. Assume that each guest knows at least n of the others. Show that it is possible for each guest to be seated next to two of her acquaintances. (Between two of her acquaintances.)