2-connected 7-coverings of 3-connected graphs on surfaces

An m-covering of a graph G is a spanning subgraph of G with maximum degree at most m. In this paper, we shall show that every 3-connected graph on a surface with Euler genus k ≥ 2 with sufficiently large representativity has a 2-connected 7-covering with at most 6k - 12 vertices of degree 7. We also construct, for every surface F² with Euler genus k ≥ 2, a 3-connected graph G on F² with arbitrarily large representativity each of whose 2-connected 7-coverings contains at least 6k - 12 vertices of degree 7.