Graphs and digraphs with all 2-factors isomorphic

We show that a digraph which contains a directed 2-factor and has minimum in-degree and out-degree at least four has two non-isomorphic directed 2-factors. As a corollary, we deduce that every graph which contains a 2-factor and has minimum degree at least eight has two non-isomorphic 2-factors. In addition we construct: an infinite family of 3-diregular digraphs with the property that all their directed 2-factors are Hamilton cycles, an infinite family of 2-connected 4-regular graphs with the property that all their 2-factors are isomorphic, and an infinite family of cyclically 6-edge-connected cubic graphs with the property that all their 2-factors are Hamilton cycles.