Symmetric cubic graphs of small girth

A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ. Tutte (1947, 1959) 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 1or 2-regular cubic graphs of girth g ≤ 9, proving that with five exceptions these are closely related with quotients of the triangle group ∆(2, 3, g) in each case, or of the group 〈x, y |x2 = y3 = [x, y]4 = 1 〉 in the case g = 8. All the 3-transitive cubic graphs and exceptional 1and 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 Dobcsányi (2002); the largest is the 3-regular graph F570 of order 570 (and girth 9). The proofs of the main results are computer-assisted.