On the Tur\'{a}n number of odd-ballooning of $3$-chromatic graphs

Given a graph $F$, the Tur\'{a}n number ${\rm ex}(n,F)$ is the maximum number of edges in any $n$-vertex $F$-free graph. The odd-ballooning of $F$, denoted by $F^{o}$, is a graph obtained by replacing each edge of $F$ with an odd cycle, where all new vertices of the odd cycles are distinct. The Tur\'{a}n number of the odd-ballooning of $F$ has been established for several important cases. For a star, it was determined by Erd\H{o}s, F\"{u}redi, Gould, and Gunderson (1995), Hou, Qiu, and Liu (2018), and Yuan (2018); for trees under certain conditions, by Zhu and Chen (2023); and for complete bipartite graphs $K_{s,t}$ ($t\geq s \geq 2$) where each substituted odd cycle has length at least five, by Peng and Xia (2024). In this paper, we apply Simonovits'celebrated method of progressive induction to determine the Tur\'{a}n number for the odd-ballooning of a class of $3$-chromatic graphs. Specifically, let $F$ be a graph formed by connecting a single vertex to all vertices of another graph whose components are either non-trivial trees or even cycles. We determine ${\rm ex}(n,F^{o})$ when each substituted odd cycle in $F^{o}$ has length at least five. As corollaries, we obtain the Tur\'{a}n number for the odd-ballooning of several well-known graph classes, including odd wheels, fan graphs, book graphs, and friendship graphs, where each substituted odd cycle in the ballooning has length at least five.