A classification of tightly attached half-arc-transitive graphs of valency 4

  • Authors:

  • Primož Šparl

  • Affiliations:

  • IMFM, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2008

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Abstract

A graph is said to be half-arc-transitive if its automorphism group acts transitively on the set of its vertices and edges but not on the set of its arcs. With each half-arc-transitive graph of valency 4 a collection of the so-called alternating cycles is associated, all of which have the same even length. Half of this length is called the radius of the graph in question. Moreover, any two adjacent alternating cycles have the same number of common vertices. If this number, the so-called attachment number, coincides with the radius, we say that the graph is tightly attached. In [D. Marusic, Half-transitive group actions on finite graphs of valency 4, J. Combin. Theory Ser. B 73 (1998) 41-76], Marusic gave a classification of tightly attached half-arc-transitive graphs of valency 4 with odd radius. In this paper the even radius tightly attached graphs of valency 4 are classified, thus completing the classification of all tightly attached half-arc-transitive graphs of valency 4.