Extremal graphs for odd-ballooning of bipartite graphs

Abstract: Given a graph $H$ and an odd integer $t$ ($t\geq 3$), the odd-ballooning of $H$, denoted by $H(t)$, is the graph obtained from replacing each edge of $H$ by an odd cycle of length at least $t$ where the new vertices of the cycles are all distinct. In this paper, we determine the range of Tur\'{a}n numbers for odd-ballooning of bipartite graphs when $t\geq 5$. As applications, we may deduce the Tur\'{a}n numbers for odd-ballooning of stars, paths and even cycles.