Long cycles and 3-connected spanning subgraphs of bounded degree in 3-connected K_{1, d}-free graphs

Let G be a 5-connected K_{1, 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 1 2n^c where c = (log₂ 6 + 2 log₂(2d - 1))^-1.