On cubic symmetric non-Cayley graphs with solvable automorphism groups

It was proved in Feng et al. (2015) that a cubic symmetric graph with a solvable automorphism group is either a Cayley graph or a 2-regular graph of type 22, that is, a graph with no automorphism of order 2 interchanging two adjacent vertices. In this paper an infinite family of non-Cayley cubic 2-regular graphs of type 22 with a solvable automorphism group is constructed, and the smallest graph has order 6174. This answers a question posed by Estélyi and Pisanski in 2016. Moreover, it includes a subfamily of graphs which are connected 2-regular covers of the Pappus graph with covering transformation group Zp3, and these graphs were missed in Oh (2009).