On transitive group actions on finite graphs
by
Dragan Marušič
University of Ljubljana

A subgroup of automorphisms of a graph X is said to act half-arc-transitively on X if it acts vertex- and edge- but not arc-transitvely on X, and is said to act semisymmetrically on a regular graph X if it acts edge- but not vertex-transitively on X.

In this lecture I will discuss some recent results as well as some (not so recent) open problems touching the following five graph- and group-theoretic topics: half-arc-transitive actions, semisymmetric actions, 2-arc-transitive actions, semiregular elements in transitive permutation groups, and non-Cayley vertex-transitive graphs. First, following a presentation of some new results in half-arc-transitive and semisymmetric group actions on graphs, I will give an interesting, albeit somewhat special, connection between these two kinds of group actions arising from a generalization of the first known constructions of semisymmetric graphs (due to Folkman in the late sixties). Second, I will discuss the problem of classifying 2-arc-transitive Cayley graphs of abelian and dihedral groups, viewing it in a somewhat larger setting of 2-arc-transitive covers of certain special graphs. To close the lecture I will touch upon two open problems that are particularly dear to my heart. A brief account on the developments in the problem of existence of semiregular elements in 2-closed transitive groups will be given and a slight modification to the problem of non-Cayley numbers suggested.

Date received: November 29, 2000