Extremal graphs for odd-ballooning of bipartite graphs

Abstract

Given a graph $H$ and an odd integer $t$ ($t\ge 3$), 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 Turán numbers for odd-ballooning of bipartite graphs when $t\ge 5$. As applications, we may deduce the Turán numbers for odd-ballooning of stars, paths and even cycles.