On the order of semiregular automorphisms of cubic vertex-transitive graphs (Q6655711)

A much-studied question about vertex-transitive graphs concerns the existence of semiregular automorphisms of the graph. For cubic vertex-transitive graphs, the existence of such automorphisms was determined by \textit{D. Marušič} and \textit{R. Scapellato} [Eur. J. Comb. 19, No. 6, 707--712 (1998; Zbl 0912.05038)], but their proof does not consider the order of the semiregular automorphisms.\N\NCameron, Sheehan, and Spiga subsequently proved that there is always a semiregular automorphism of order at least 3 [\textit{P. Cameron} et al., Eur. J. Comb. 27, No. 6, 924--930 (2006; Zbl 1090.05032)], and conjectured that the maximal order of such an automorphism tends to infinity as the number of vertices tends to infinity. This conjecture was disproven by \textit{P. Spiga} [Math. Proc. Camb. Philos. Soc. 157, No. 1, 45--61 (2014; Zbl 1297.05113)], who found an infinite family of cubic vertex-transitive graphs in which the maximal order of a semiregular automorphism is 6.\N\NIn this paper, the authors characterise all cubic vertex-transitive graphs for which the maximal order of a semiregular automorphism is at most \(5\). In particular, since this is a finite list (with the largest having \(4 \cdot 5^{34}\) vertices), the family previously found by Spiga is best possible.