The combined research of Abby Waldron ‘16 and Elias Svensson ‘19, along with faculty lead Dr. Kara Shavo, will be published this summer in Discrete Mathematics, where the team discusses graph theory.

by Sarah Murphy

Abby Waldron ‘16 and Elias Svensson ‘19 graduated three years apart at Presbyterian College. Though both major in mathematics, the two never actually met.

Yet both Waldron and Svensson made significant contributions at different points in time to a research project that will be published as a paper in the July 2024 issue of Discrete Mathematics, a biweekly, peer-reviewed scientific journal. Their paper, titled, “On the girth of forbidden subgraphs of coloring graphs,” explores the world of graph theory and a particular network called a “coloring graph.”

PC professor of mathematics Dr. Kara Shavo is the common thread that helped sew together the work between Waldron and Svensson.

Shavo first had the idea for the graph theory research following a conference that she attended in the mid-2010s. Knowing the work would require small-group collaboration, she began to keep an eye out for a student who would be a good fit to join the project.

“When Abby took my linear algebra class, I was very impressed by her intelligence and work ethic,” Shavo said. “She was a chemistry and math double major, so she hesitated a bit when I asked her to do honors research with me. But it wasn’t all that hard to convince her.”

Instead of opting to explore expository work, Waldron agreed to try tackling the new, original work Shavo had been turning around in her brain. The two began work in Waldron’s final year as an undergraduate, and things started out exactly as Shavo had hoped, at first.

“Once we got to the main point of the project, I asked Abby to prove a conjecture,” Shavo said. “I was pretty sure it was correct, but I held back on trying to prove it so that Abby could do it. She wrote a lovely proof. As I was reading it, though, I realized that there was one tiny little case that was missing in Abby’s proof, and my stomach dropped. The missing case led me to a counterexample. I felt terrible. I had led Abby down a dead-end street.”

Fortunately, with only one month until Waldron’s graduation, Shavo thought of another proposition, and Waldron wrote another proof. This time, the proof was solid — and it gave Shavo even more ideas for future research.

A few years went by. Shavo met Elias Svensson in one of her classes and immediately recognized his talent and curiosity, asking him to join her for honors research starting in the spring of 2019. Since Elias was graduating the following December, the two started working together right away, meeting by Zoom over the summer while Svensson was back home in Sweden.

“The research went amazingly well,” says Shavo. “Working with Elias was like working with a seasoned mathematician. We were able to prove the conjecture that summer, and during the fall, Elias handed me a proof of another conjecture using a technique I don’t think I ever would have tried. I had to read his proof over and over again to be sure it was correct – and it was!”

It was at this point that Shavo realized that the combination of both Waldron’s and Svensson’s work would make an interesting paper that could hopefully be published. In 2021, she combined all results into an organized paper and submitted it to Discrete Mathematics that fall. The peer-review process took two years — not uncommon in mathematics — and in early 2024 Shavo learned the paper would be published.

The research paper shows that there are some networks that cannot exist inside any coloring graph — called forbidden subgraphs — no matter which network you start with and no matter how many colors you use. The team also conjectured that there are forbidden subgraphs of arbitrarily large girth, where the girth of a network is the smallest cycle it contains.

Shavo says that though their work never overlapped, both Waldron and Svensson were essential to the project.

“I could not have done it on my own,” she said. “Collaboration between faculty and students is incredibly beneficial to both students and faculty — and that’s an understatement! The exploration of a question in which the truth is uncertain and the way to find the truth is uncertain — those are immensely powerful experiences. They result in large intellectual gains and increased confidence in problem-solving.”

Today, Waldon is a scientist at the Savannah River National Laboratory while Svensson is working in the energy sector in Sweden.

To learn more about mathematics at Presbyterian College, visit the department website.