My primary research interest is in Combinatorics and Graph Theory. In particular, I am interested in the study of oriented complete graphs, also known as tournaments.

Suppose three teams participate in a round-robin tournament (in which no ties are allowed). The total number of games played in the tournament is three (each team plays two games, and each game involves two teams so the number of games is (2+2+2)/2 = 3). Since each game has two possible outcomes the number of distinct outcomes for the tournament is 2^3 = 8. We can create a mathematical model of this round-robin tournament by representing each team as a vertex, and for each game played we draw an arrow from the winner to the loser. But if we don't know which team is represented by which vertex, then there are really only two possibilities; either some team wins both games it plays or every team wins one game and loses one game.
The two possibilities described above can be depicted visually:

Tournaments involving three teams aren't very complicated, and as a result the mathematics isn't very interesting either. But the number of possible outcomes for a tournament involving twelve teams is quite surprising. If the teams are labeled in advance, then we can count the possible outcomes just as we did above; each team plays 11 games, and if we add these numbers up for each of the twelve teams, we have a total of 12*11=132 games played. But this counts each game twice (once when we counted the "home team" and once when we counted the "visiting team") so the total number of games should be (12 * 11)/2 = 132/2 = 66. Each of the 66 games has two possible outcomes, and so the number of distinct tournaments is 2^66. This number is approximately 73,000,000,000,000,000,000; a number so large it is hard to even think about. If you could somehow run one million of these round robin tournaments every second, and each one had a different outcome, it would still take well over 2 billion years to run through every possibility.

For more information about the specific research problems I study, you can read my Ph.D. Thesis (if you really want (93 pages & 406 KB)), or some of my publications.

Publications Brown, D., Busch, A., and Lundgren, J.R., "Interval Tournaments"(177 KB) Journal of Graph Theory 56 (2007), 72-81. Note: A much shorter proof of one of the main results in this paper is available here. Busch, A., Ferrara, M., and Kahl, N., "Generalizing D-graphs" Discrete Applied Mathematics 155 (2007), 2487-2495. Busch, A., and Isaak, G., "Recognizing Bipartite Tolerance Graphs in Linear Time" Graph Theoretic Concepts in Computer Science, Lecture Notes in Computer Science 4769 (2007), 12-20. Busch, A., and Jacobson, M., "Extremal results on Arc-traceable Tournaments"(160 KB) Journal of Combinatorial Mathematics and Combinatorial Computing 63 (2007), 3-15. Busch, A., Jacobson, M., and Reid, K.B., "On Arc-traceable Tournaments" Journal of Graph Theory 53 (2006), 157-166. Busch, A., "A characterization of triangle-free tolerance graphs" Discrete Applied Mathematics 154 (2006), 471-477. Busch, A., "A note on the number of hamiltonian paths in strong tournaments" Electronic Journal of Combinatorics 13 (2006), Note 3. Busch, A., Jacobson, M., and Reid, K.B., "On a conjecture of Quintas, and arc-traceability in upset tournaments" Discussiones Mathematicae Graph Theory 25 (2005), 345-354. Busch, A., Chen, G. and Jacobson, M., "All tournament score sequences are pan-partition transitive," submitted. Abueida, A., Busch, A. and Sritharan, R., "A min-max theorem for chordal bipartite graphs with applications," submitted.

The following are mathematicians whom I have had an opportunity to work with. Some of the work resulted in papers that are either submitted for publication or being prepared for submission, and some of it led nowhere (so far?).

Collaborators Atif Abueida Dave Brown Guantao Chen Raph Faudree Mike Ferrara Ron Gould Brian Heinold Garth Isaak Mike Jacobson (my Ph.D. advisor) Nate Kahl Rich Lundgren Brooks Reid R. Sritharan Dustin Stewart Charlie Suffel