Bridging Semisymmetry and Half-Arc-Transitivity
by
D. Marušič
Ljubljana

Given a graph X and a subgroup G of the automorphism group Aut X of X we say that G acts semisymmetrically on X and that X is G-semisymmetric if G acts edge- but not vertex-transitively on X. Similarly, we say that G acts half-arc-transitively on X and that X is G-half-arc-transitive if G acts vertex- and edge- but not arc-transitively on X. In particular, if G=Aut X we say that X is semisymmetric and half-arc-transitive, respectively.

The first written accounts on semisymmetry and half-arc-transitivity in graphs go back to the sixties and are due to, respectively, Folkman (see [``Regular line-symmetric graphs'', J. Combin. Theory 3 (1967), 215-232]) and Tutte (see [``Connectivity in graphs'', University of Toronto Press, Toronto, 1966]). A number of mathematicians have been attracted to different problems touching upon group- and graph-theoretic aspects of these two concepts. As a result, we have a vast literature on the subject in question. However, I am aware of no paper dealing with possible connections between these two concepts. The object of this lecture is twofold. Following a a brief presentation of some of the recent results in semisymmetric and half-arc-transitive group actions, I will give an interesting, albeit somewhat special, connection between these two kinds of group actions. More precisely, the so called generalized Folkman graphs and their semisymmetry will be linked to graphs admitting half-arc-transitive group actions via the concept of alternets. The latter is a joint work with Primoz Potocnik.

Date received: May 26, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafd-17.