Tetravalent edge-transitive graphs of girth at most 4

  • Authors:

  • Primož Potočnik;Steve Wilson

  • Affiliations:

  • University of Auckland, Department of Mathematics, PB 92019 Auckland, New Zealand;Department of Mathematics and Statistics, Northern Arizona University, Box 5717, Flagstaff, AZ 86011, USA

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2007

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Abstract

This is the first in the series of articles stemming from a systematical investigation of finite edge-transitive tetravalent graphs, undertaken recently by the authors. In this article, we study a special but important case in which the girth of such graphs is at most 4. In particular, we show that, except for a single arc-transitive graph on 14 vertices, every edge-transitive tetravalent graph of girth at most 4 is the skeleton of an arc-transitive map on the torus or has one of these two properties:(1)there exist two vertices sharing the same neighbourhood, (2)every edge belongs to exactly one girth cycle. Graphs with property (1) or (2) are then studied further. It is shown that they all arise either as subdivided doubles of smaller arc-transitive tetravalent graphs, or as line graphs of triangle-free (G,1)-regular and (G,2)-arc-transitive cubic graphs, or as partial line graphs of certain cycle decompositions of smaller tetravalent graphs.