Extremal graphs for odd-ballooning of paths and cycles (Q2175804)

The odd-ballooning of a graph \(G\) is the graph obtained from \(G\) by replacing each edge in \(G\) by an odd cycle of length between \(3\) and \(q\) (\(q\geq 3\)), where the new vertices of the odd cycles are all different. Given a 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. In this paper, the authors 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 \(q\) (\(q\geq 3\)), and \(n\) is sufficiently large.