Symmetric cubic graphs of small girth

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=<5. We show that a symmetric cubic graph of girth at most 9 is either 1-regular or 2'-regular (following the notation of Djokovic), or belongs to a small family of exceptional graphs. On the other hand, we show that there are infinitely many 3-regular cubic graphs of girth 10, so that the statement for girth at most 9 cannot be improved to cubic graphs of larger girth. Also we give a characterisation of the 1- or 2'-regular cubic graphs of girth g=<9, proving that with five exceptions these are closely related with quotients of the triangle group @D(2,3,g) in each case, or of the group 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.

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