The Traveling Philosopher and Avoiding Szemerédi's Regularity Lemma

Document Type

Presentation Abstract

Presentation Date

10-15-2012

Abstract

Suppose a wandering philosopher regularly visits every major city in an area, but she gets bored if she travels on the same path, and therefore never wants to use a road twice. How many times can she travel to every city before she is forced to reuse a road? In graph theory, this problem is finding as many edge-disjoint Hamiltonian cycles as possible.

Recently, in the case where each vertex has large degree (each city has many roads leading out of it), Christofides, Kühn, and Osthus proved you could find a surprisingly large number of Hamiltonian cycles. They used one of the most powerful tools in graph theory, originated by Szemerédi, known as the Regularity Lemma. However, there are some drawbacks to using the Regularity Lemma, and recently there has been a push to develop tools and proof methods that replace it.

We developed a partition theorem similar in flavor to the Regularity Lemma that is easier to use. Using our partition theorem, we were able to give a shorter proof of the Christofides, Kühn, Osthus result that applies in more cases. Proving our partition theorem will allow us to talk about another of the most important ideas in graph theory: the probabilistic method.

Additional Details

Monday, 15 October 2012
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

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