Math professor contributes to solving Hadwiger's conjecture

Mathematics professor Michelle Delcourt first read about a famous mathematical problem, called Hadwiger’s conjecture, in high school. Although problems like these are what initially drew her to mathematics, she never thought she would make a serious contribution towards solving such a notorious and difficult problem, until she did.

Last year, Delcourt and her co-author Professor Luke Postle from the University of Waterloo developed an original proof, making substantial progress towards Hadwiger’s conjecture. This significantly advanced the understanding of a central question in her field of graph theory and was subsequently published in the Journal of the American Mathematical Society, one of the top peer-reviewed journals in all of mathematics. This accomplishment has resulted in several high-profile conference invitations and shaped the current direction of her research area of graph colouring. In part for this result, she was recently awarded a prestigious Sloan Research Fellowship (external link)  to support her research, a two-year fellowship that “recognizes and rewards outstanding early-career fellows who have the potential to revolutionize their fields of study”.

Delcourt is elated about the opportunity to contribute. “Hadwiger’s is the problem in graph colouring, it is the one everyone wants to solve and the one that all the best minds in graph theory have made partial progress towards over the years. It is one that I never thought I would make substantial progress on, so I am very proud of that,” she says.

Delcourt’s work is now included in a footnote in the textbook where she first learned about the decades-old mathematical conjecture.

Sweeter still is the fact that Delcourt completed her graduate work at the University of Illinois at Urbana-Champaign, home to the researchers who solved the related four-colour theorem (of which Hadwiger’s is a generalization). The four-colour theorem was raised in the 1850s, when cartographers printing maps and atlases were trying to figure out how many colours were needed to colour countries so that no two that share a land border have the same colour. This was an essential question when it originated, as nineteenth-century printers needed a different metal plate to print each new colour—four versus six made a lot of difference. Problems like these also spurred the wider field of graph theory, which mathematicians, such as Delcourt, have been working on ever since. 

These problems from graph theory have many practical applications. In scheduling, for example, it applies when creating exam timetables that will fit a massive undergraduate population into a number of courses while avoiding conflicts. In telecommunications, the approach helps where you are trying to allocate radio stations on a band but do not want to assign them too close to each other. 

Delcourt says that she was inspired to work on Hadwiger’s conjecture after reading another paper on arXiv, an open-source research-sharing platform, during the pandemic that also advanced the work. She says that the mathematics community has benefited significantly from such information sharing. “Now, especially with the internet and the availability of research, you do not have to go and find some obscure text and translate it from German or Russian, it is all available online. I think that is a trend in math, that it's becoming more social. It is definitely a good thing for math,” she says, adding that she has found a strong research community at TMU too, with many who share her interests. “Everybody is very collegial. We love to chat, and we are not competitors, but colleagues.”

Another small but meaningful bonus to Delcourt’s discovery? The latest revised edition of the textbook, where she originally learned graph theory, now has a footnote in the section on Hadwiger’s conjecture with her name on it.