Extremal Graphs for Odd-Ballooning of Paths and Cycles

The odd-ballooning of a graph F is the graph obtained from F by replacing each edge in F by an odd cycle of length between 3 and $$q\ (q\ge 3)$$ where the new vertices of the odd cycles are all different. Given a forbidden graph H and a positive integer n, the extremal number, ex(n, H), is the maximum number of edges in a graph on n vertices that does not contain H as a subgraph. Erdos et al. and Hou et al. determined the extremal number of odd-ballooning of stars. Liu and Glebov determined the extremal number of odd-ballooning of paths and cycles respectively when replacing each edge of the paths or the cycles by a triangle. In this paper we determine the extremal number and find the extremal graphs for odd-ballooning of paths and cycles, when replacing each edge of the paths or the cycles by an odd cycle of length between 3 and $$q \ (q \ge 3)$$ and n is sufficiently large.