Long Cycles and 3-Connected Spanning Subgraphs of Bounded Degree in 3-Connected K1, d-Free Graphs

Let G be a 5-connected K1, d-free graph on n vertices. We show that G contains a 3-connected spanning subgraph of maximum degree at most 2d − 1. Using an earlier result of ours, we deduce that G contains a cycle of length at least 12nc where c = (log2 6 + 2 log2(2d − 1))−1.