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 × R + → N such that for each ε > 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 , ε ) , 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 ε | V ( G ) | . 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 Böhme, Mohar and Thomassen [T. Böhme, B. Mohar, C. Thomassen, Long cycles in graphs on a fixed surface, J. Combin. Theory Ser. B 85 (2002) 338–347].