N. Alon, B. Mohar, D.P. Sanders, and N. Robertson "Acyclic colouring of graphs on surfaces", Preprint 1994, proved with probabilistic arguments
They also proved by discharging arguments that
Hence the asymptotic behaviour of a(Sg) has been determined almost exactly.
The interesting particular cases of a(torus) and a(projective plane) are still unanswered.
December 28, 1994. Tommy Jensen and Bjarne Toft.
N. Alon, B. Mohar and D.P. Sanders, On Acyclic Colorings of Graphs on Surfaces, Israel J. Math. 94 (1996), 273-283.
August 14, 1997. Tommy Jensen and Bjarne Toft.