Let G be a 3-connected graph with n vertices on a non-spherical closed surface Fk2 of Euler genus k with sufficiently large representativity. In this paper, we first study a new cutting method which produces a spanning planar subgraph of G with a certain good property. This is used to show that such a graph G has a spanning 4-tree with at most max{2k−5,0} vertices of degree 4. Using this result, we prove that for any integer t, if n is sufficiently large, then G has a connected subgraph with t vertices whose degree sum is at most 8t−1. We also give a nearly sharp bound for the projective plane, torus and Klein bottle. Furthermore, using our cutting method, we prove that a 3-connected graph G on Fk2 with high representativity has a 3-walk in which at most max{2k−4,0} vertices are visited three times, and an 8-covering with at most max{4k−8,0} vertices of degree 7 or 8. Moreover, a 4-connected G has a 4-covering with at most max{4k−6,0} vertices of degree 4.
