2-Connected spanning subgraphs with low maximum degree in locally planar graphs

In this paper, we prove that there exists a function a: N-0 x R+ -> N such that for each epsilon > 0, if G is a 4-connected graph embedded on a surface of Euler genus k such that the face-width of G is at least a (k, epsilon), then G has a 2-connected spanning subgraph with maximum degree at most 3 in which the number of vertices of degree 3 is at most epsilon vertical bar V(G)vertical bar. This improves results due to Kawarabayashi, Nakamoto, and Ota [K. Kawarabayashi, A. Nakamoto, K. Ota, Subgraphs of graphs on surfaces with high representativity, J. Combin. Theory Ser. B 89 (2003) 207-229], and Bohme, Mohar and Thomassen [T. Bohme, B. Mohar, C. Thomassen, Long cycles in graphs on a fixed surface, J. Combin. Theory Ser. B 85 (2002) 338-347]. (c) 2006 Elsevier Inc. All rights reserved.