In 1979, then 26-year-old doctoral student Dragan Marušič posed a question during a lecture in Cambridge that went on to become one of the major open problems in discrete mathematics: Does there exist a finite vertex-transitive graph without a semiregular automorphism?

The question remained unanswered for almost half a century. Over the years, it attracted many prominent researchers and became an important open problem at the intersection of algebraic graph theory and the theory of finite permutation groups, while also connecting with other areas of mathematics.

Now, young mathematicians Saul D. Freedman and Melissa Lee have provided an answer. In a preprint posted on arXiv on July 26, 2026, they showed that such a graph does exist, providing a counterexample to the policirculant conjecture.

Their work is particularly striking given the long history of the problem. Many earlier results had pointed in the opposite direction: the policirculant conjecture had been proved for various families of graphs, including graphs of valency 3 and 4. The final breakthrough therefore came unexpectedly. Instead of proving that the conjecture holds in general, the researchers found a counterexample.

The smallest example they present has 16,464 vertices and valency 63. Their research also made use of computational tools, but the core of the result rests on a mathematical construction and deep theoretical work.

When a Good Mathematical Question Survives for Decades

For the University of Primorska, the achievement is particularly significant because Prof. Dragan Marušič has been a long-standing researcher at the university and is one of Slovenia’s most prominent mathematicians in the field of graph theory. His question, first posed in 1979, has thus received an answer 47 years later.

Prof. Dr. Tomaž Pisanski, a mathematician who also works at the University of Primorska, is a leading expert at the intersection of theoretical computer science and discrete mathematics. In an article for the newspaper Delo about the question that remained unanswered for 47 years, he highlights the importance of asking good mathematical questions. Good questions often drive the development of new methods and theories, while their answers may emerge only after decades of research.

Prof. Dr. Dragan Marušič’s achievement and the latest development in the problem therefore offer a remarkable example of the long-term impact of research. A question posed by a young Slovenian mathematician almost half a century ago remains part of active mathematical research today.

Prof. Dr. Tomaž Pisanski’s own reflections also offer an interesting perspective on the passage of time. Last year, he received the Zois Award for Lifetime Achievement in discrete mathematics. As early as 1970, in an interview with Mladina, he predicted: “Whoever controls computers will rule the world.” More than five decades later, in his article for Delo, he reflects on the growing influence of artificial intelligence on mathematical research and on how it could eventually change the very nature of mathematical work. His earlier prediction now sounds remarkably relevant—not simply as a statement about the future of computers, but as a question about how humans will work alongside increasingly powerful computer systems.

The newspaper Delo has also published an extensive report on the achievement, in which Prof. Dr. Tomaž Pisanski presents the history of the problem, its significance for the development of mathematics, and the unexpected breakthrough brought about by Saul D. Freedman and Melissa Lee.

Meet the Mathematicians Who Answered Marušič’s Question

The story will continue in Slovenia next year. An international graph theory conference will be held in Bled in June 2027, with Saul D. Freedman and Melissa Lee among the invited participants.

More information about the conference is available on the SICGT website.

This article was prepared based on recent developments in graph theory research and an article by Prof. Dr. Tomaž Pisanski published on 13 Auggust 2026 in the newspaper Delo.