Spanning closed walks with bounded maximum degrees of graphs on surfaces

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 $\chi \le 0$.