Bridging semisymmetric and half-arc-transitive actions on graphs

  • Authors:

  • Dragan Marusic;Primoz Potocnik

  • Affiliations:

  • IMFM, Oddelek za matematiko, Univerza v Ljubljani, Jadranska 19, SI-1O00 Ljubljana, Slovenija;IMFM, Oddelek za matematiko, Univerza v Ljubljani, Jadranska 19, SI-1O00 Ljubljana, Slovenija

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2002

Quantified Score

Hi-index 0.00

Visualization

Abstract

A generalization of some of Folkman's constructions (see (1967) J. Comb. Theory, 3, 215-232) of the so-called semisymmetric graphs, that is regular graphs which are edge- but not vertex-transitive, was given by Marusic and Potocnik (2001, Europ. J. Combinatorics, 22, 333-349) together with a natural connection between graphs admitting ½-arc-transitive group actions and certain graphs admitting semisymmetric group actions. This connection is studied in more detail in this paper. Among others, a sufficient condition for the semisymmetry of the so-called generalized Folkman graphs arising from certain graphs admitting a ½-arc-transitive group action is given. Furthermore, the concepts of alter-sequence and alter-exponent is introduced and studied in great detail and then used to study the interplay of three classes of graphs: cubic graphs admitting a one-regular group action, the corresponding line graphs which admit a ½-arc-transitive action of the same group and the associated generalized Folkman graphs. At the end an open problem is posed, suggesting an in-depth analysis of the structure of tetravalent ½-arc-transitive graphs with alter-exponent 2.