ASMS students Will Howton (senior) and Gregory Li (junior) are working with a faculty member, Dr. Andrei Pavelescu, at the University of South Alabama (USA) to conduct research on Jørgensen’s Conjecture. This problem has further implications on the Robertson-Seymour theorem, a principal result for graph minor theory. Together, they are investigating simple graphs under certain cases to determine whether the conjecture holds.
Li described, “Our research is interesting because we are using a combinatorial approach to address this case of a topological problem.” Specifically, they are working on graph minor theory and testing the conjecture for graphs possible with a certain number of vertices. Howton is working (on graphs) with 13 vertices, and Li is working (on graphs) with 14 vertices. They are utilizing theorems and previous work on graph minors to simplify the span of possibilities and establish the conjecture for all graphs of these two orders.
Pavelescu, a mathematics faculty member at USA and Research Adviser for Howton and Li, described the conjecture and the students’ work in this way, “Jørgensen’s Conjecture is a 25 years old claim with deep implications in the realm of graph theory. While the conjecture is known to hold for large order graphs, little is known about small order. Will and Gregory have been working on validating the conjecture for order 13 and order 14, respectively.”
“Graph theory is an important part of modern mathematics with numerous important applications to computer science, the social sciences, biology, physics and chemistry. Graphs are used to model relations between objects,” said Dr. Cornelius Pillen who is the Graduate Advisor at the Department of Mathematics and Statistics at USA.
Howton and Li were chosen to present their work at the 70th Annual Meeting of the AACTM (Alabama Association of College Teachers of Mathematics), which was held Saturday, February 15, 2020, at the University of South Alabama, Mobile, AL. Their talk was entitled: “Investigation of Jørgensen’s Conjecture on Graphs of Order 13 and 14.” They were the only students to give a presentation.
“Their presentation was enthusiastic, informative, and well-received by an audience of more than 30 teachers and graduate students,” said Pavelescu.
Although their research is far from complete, they are enjoying the opportunity. Li expressed, “It’s exciting to get to work on something you do not know the answer to. It’s refreshingly different from the more usual, standard mathematics. We have to creatively apply what we know and utilize critical thinking, analysis, and problem solving.”