What was the question?
Associated to any graph one can construct its chromatic polynomial. This is a polynomial over the integers and thus generates a field extension of the rational numbers. In this extension we have certain prime decomposition behavior. The question I asked the students was: can we understand the prime decomposition behavior from the graph alone?
What motivated me?
I really like colors. The topic of graph coloring is a topic that drives me mad with emotion because of that. I have always returned to study it. At the time, I had gotten into the book Graph Colouring and the Probabilistic Method by Michael Molloy and Bruce Reed, and thought a work on that direction would be good.
However, I wanted to do something related to number theory which is my main area of study. Thus, I asked myself the above question and decided to see what the students came up with. This was a bit of a risk because the topics are not easy, so a lot of work had to be done by the mentees!
What did the students do?
This was a group of three students. They had a period of learning: what does it mean to have prime splitting in quadratic extensions. I helped them with this part. They learned how to work computationally with the extensions to be able to determine the splitting behaviour. They also learned what is a chromatic polynomial and how does one compute it.
Then came a part of research on their own, guided with ideas from myself, but also with thoughts they had. They were two main directions: on the one hand, they decided to focus on quadratic extensions because they were the most approachable to them. This required to find a family of graphs that admitted complexity but that had quadratic chromatic polynomial. This led them to look for papers in the area and eventually found such a family.
On the other hand, with that family at hand they required to find parameters to produce the family and then check the splitting behavior, came up with patterns and try to prove them. This led them to try to solve certain Diophantine equations involving ellipses, which was very interesting. Finally, they were able to prove a pattern on the parameters that implies certain primes split.
Special moments I remember 🙂
I enjoyed significantly the computational behavior of searching for integer solutions to the ellipse to construct graphs. They did it with computing and they found very interesting solutions.
Outcomes of the work
The students presented the work on the final presentation. We merged the presentation, which was a priori individual, into a single longer one. This allowed them to give a very nice overview of the whole work and the prerequisites. This was a project where I felt the students had indeed touched in a more significant work what it meant to prove something by themselves.
I continue to have contact with one of the students of this project, which has decided to pursue the path of mathematics and go into number theory!
Malors 