On the Turán number of odd-ballooning of 3-chromatic graphs

Given a graph F, the Turán number 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 Turán number of the odd-ballooning of F has been established for several important cases. For a star, it was determined by Erdős, Füredi, 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 Kₒ, ₓ (t s 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 Turán 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 ex (n, F^o) when each substituted odd cycle in F^o has length at least five. As corollaries, we obtain the Turán 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.