Cubic graphical regular representations of Ree groups

A graphical regular representation of a group G is a Cayley graph of G whose full automorphism group is equal to the right regular permutation representation of G. In this paper, we prove that for Ree groups Ree(q) with q > 3, with probability tending to 1 as q ? 8, a random involutionytogether with a fixed elementx with order q-1 gives rise to a cubic graphical regular representation of Ree(q). A similar result involving a fixed element with order q + v3q + 1 is also proved with the help of certain properties of Ree(q) given in [Leemans, D. Liebeck, M. W. (2017). Chiral polyhedra and finite simple groups. Bull. London Math. Soc. 49: 581-592].