Tomaž Pisanski: Polycirculant graphs and polycyclic configurations

Povzetek/Abstract In their 1990 paper, Branko Grünbaum and John Rigby described in detail a geometric point-line ( (21_4) )-configuration that was first studied in the complex plane by Felix Klein as early as 1879. Although the configuration attracted the attention of several eminent mathematicians of nineteen and twenty century, it took over 110 years to discover a realization of the configuration in the Euclidean plane (see figure attached). Its exceptional geometric and combinatorial symmetries is a prime example of a polycyclic configuration. Its Levi graph is an example of a polycirculant graph, meaning a graph that has semi-regular automorphism.

Independently, the study of polycirculant graphs grew in popularity due to the long standing, still unsolved Marušič conjecture (1981), claiming that every vertex-transitive graph is polycirculant. Several special classes of polycirculants, such as circulants and bicirculants, are being studied at several world centers of algebraic graph theory.

In this talk, we give a unified overview of polycirculant graphs, polycyclic configurations, and their generalizations. We also present some well-known classes of polycirculants, such as generalized Petersen graphs, rose window graphs, and others. Several classification problems that have been solved for some special classes of polycirculants inspire open problems for more general types of polycirculants.