Subdivision thresholds for two classes of graphs

Abstract The subdivision threshold for a graph F is the maximum number of edges, ex(n; FS), a graph of order n can have without containing a subdivision of F as a subgraph. We consider two instances: 1. (i) F is the graph formed by a cycle C one vertex of which is adjacent to k vertices not on C, and 2. (ii) F is the graph formed by a cycle C one vertex of which is adjacent to k vertices on C. In the first problem we determine the threshold and characterize the extremal graphs for all k ⩾ 1. In the second problem we do this for k = 2 only.