  Gao and Richter (1994) showed that every $3$-connected graph which embeds on
the plane or the projective plane has a spanning closed walk meeting each
vertex at most $2$ times. Brunet, Ellingham, Gao, Metzlar, and Richter (1995)
extended this result to the torus and Klein bottle. Sanders and Zhao (2001)
obtained a sharp result for higher surfaces by proving that every $3$-connected
graph embeddable on a surface with Euler characteristic $\chi \le -46$ admits a
spanning closed walk meeting each vertex at most $\lceil
\frac{6-2\chi}{3}\rceil$ times. In this paper, we develop these results to the
remaining surfaces with Euler characteristic $\chi \le 0$.
