Portland State University Discrete Mathematics Seminar

Fall 2013: Neuberger Hall 373, Tuesdays, 11am-noon (all quarters)

October 7, 2013:

Austin Williams, Portland State University
Hunting for a Moore graph of valency 57

October 15, 2013:

Nichole Schimanski, Portland State University
Counting orthomorphisms of (Z/2Z)5

October 22, 2013:

James Mahoney, Portland State University
Universal cycles for spanning trees of Kn

October 29, 2013:

John Caughman, Portland State University
The fun of rainbow spanning trees: two ways to decompose a complete graph

An edge coloring is "proper" if any two edges that meet at a vertex are assigned different colors. So a proper edge coloring of the complete graph K2n on 2n vertices requires at least 2n - 1 colors. Seemingly unrelated, a "spanning tree" for K2n is just a connected, acyclic subgraph with 2n - 1 edges. What could these two notions have in common? Relative to a given proper edge coloring, a "rainbow spanning tree" is a spanning tree whose 2n - 1 edges have 2n - 1 distinct colors. The BH-conjecture, open since 1996, states that for any proper coloring of K2n with 2n - 1 colors, there exist n disjoint rainbow spanning trees whose union is all of K2n . We explore this conjecture and prove a few special cases.

November 5, 2013:

Derek Garton, Portland State University
Factoring, random maps, and polynomial maps

   slides
Pollard's "rho" algorithm is a method of factoring integers that relies on the conjectural "randomness" of certain polynomial maps over finite fields. I will show that polynomial maps over finite fields do indeed act like random maps, for certain classes of maps, a certain definition of "random", and a certain definition of "act like".

November 12, 2013:

Clayton Petsche, Oregon State University (postponed until Winter 2014)

John Caughman, Portland State University
The probability of winning Fibonacci Solitaire

November 19, 2013:

Derek Garton, Portland State University
The Ternary Goldbach Conjecture

On May 13, 2013, Harald Helfgott announced a proof of the Ternary Goldbach Conjecture. I will introduce this conjecture and talk about Helfgott's proof.

November 26, 2013:

Colin Weir, Simon Fraser University
Factoring primes in cubic fields

We learned in elementary school that a prime number only factors as one times itself. However, this definition assumes that we are only factoring over the natural numbers. If we adjoin the square root of 2 for example, then the prime 2 now factors as ( 21/2 )2. We will explore what happens to primality and unique factorization in these types of extensions. We will eventually turn our focus to cubic extensions and present new ways to construct and count cubic extensions that force certain prime factorizations to occur.

December 3, 2013:

Matthew Ridge, Portland State University
Some results on (Kq , k)-stable graphs