Symmetric cubic graphs of small girth

  • Authors:

  • Marston Conder;Roman Nedela

  • Affiliations:

  • Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand;Mathematical Institute, Slovak Academy of Sciences, 974 00 Banská Bystrica, Slovakia and Department of Mathematics, Matej Bel University, Banská Bystrica, Slovakia

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

Quantified Score

Hi-index 0.00

Visualization

Abstract

A graph @C is symmetric if its automorphism group acts transitively on the arcs of @C, and s-regular if its automorphism group acts regularly on the set of s-arcs of @C. Tutte [W.T. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. 43 (1947) 459-474; W.T. Tutte, On the symmetry of cubic graphs, Canad. J. Math. 11 (1959) 621-624] showed that every cubic finite symmetric cubic graph is s-regular for some s= in the case g=8. All the 3-transitive cubic graphs and exceptional 1- and 2-regular cubic graphs of girth at most 9 appear in the list of cubic symmetric graphs up to 768 vertices produced by Conder and Dobcsanyi [M. Conder, P. Dobcsanyi, Trivalent symmetric graphs up to 768 vertices, J. Combin. Math. Combin. Comput. 40 (2002) 41-63]; the largest is the 3-regular graph F570 of order 570 (and girth 9). The proofs of the main results are computer-assisted.