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SOAR Spotlight: Alexis Thiel '16

SOAR Spotlight: Alexis Thiel '16

Quasi-Crowns

Major: Mathematics
Hometown: Allentown, PA
Advisor: Dr. Shannon Talbott

Briefly describe your project.

This project considers a problem in Graph Theory, where a graph is a set of points called vertices with a set of edges that connect those vertices. One question that has motivated research for well over a century is the challenge of coloring a graph; that is, how many colors must one use to color vertices so that any two vertices that are connected by an edge are not the same color? This summer we are looking at a specific type of graph called a crown, which is a visual representation of a partially ordered set or poset. Our main focus is to look at the critical pairs of these crowns. In the past, the operation of layering a crown has been defined. The critical pairs of these layered crowns give rise to an infinite family of graphs for which we know an upper bound on the chromatic number. This summer I began looking at the properties of layering different crowns, with my main focus on analyzing where critical pairs show up. Layering different crowns gives us a new object which does not fit our previous definition, so I created a definition for a layered quasi-crown and proved that this new object still held the three properties of a poset; reflexivity, transitivity, and antisymmetry. I am currently working on proving my conjectures and writing a paper on my work.

Why did you decide to turn your idea into a SOAR project?

Dr. Talbott approached me with the idea this past Spring semester.

How did your faculty advisor guide you through your research?

Dr. Talbott helped me look at problems from a different angle. When I was stuck with something, hearing her ideas and being able to talk through the problem with her was very beneficial. I also struggle with the writing process, so Dr. Talbott was very helpful when it came time to write a paper on my research.

What was your biggest obstacle?

The biggest obstacle was the writing process. Being able to collect all of my research into a paper was something that I struggled with.

What was your biggest takeaway from this experience?

This project has given me an understanding of mathematical research that can only be learned through participation. My biggest takeaway is that I have learned, and experienced, that every part of the research process is important to mathematical research as a whole.

What was the result of your project? Was it congruent with your hypothesis?

We created a definition for a layered quasi-crown and proved that this new object still held the three properties of a poset. After analyzing various other properties of the layered quasi-crown, we were able to state and prove multiple lemmas regarding the location of critical pairs in layered quasi-crowns. Various cases arise depending on the size of the two crowns and the order in which they’re layered.

Will you expand on your research after this summer is over? If so, where would you like to see it go?

There are some conjectures which I have not yet proved. I hope to prove those conjectures and add them to my paper in the near future. Looking at the big picture of this project, we ultimately want to color the graphs of critical pairs of these layered quasi-crowns.